Research ArticleShip-Borne Phased Array Radar Using GA Based Adaptive 120572-120573-120574Filter for Beamforming Compensation and Air Target Tracking

J Mar12 Chen-Chih Liu1 and M B Basnet1

1Department of Communications Engineering Yuan-Ze University 135 Yuan-Tung Road Jungli Taoyuan 320 Taiwan2Communication Research Center Yuan-Ze University 135 Yuan-Tung Road Jungli Taoyuan 320 Taiwan

Correspondence should be addressed to J Mar eejmarsaturnyzuedutw

Received 7 March 2014 Revised 27 August 2014 Accepted 11 September 2014

Academic Editor Hang Hu

Copyright copy 2015 J Mar et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Beam pointing error caused by ship motion over the ocean affects the tracking performance of the ship-borne phased array radarDue to the dynamic nature of the sea environments the ship-borne phased array radar must be able to compensate for the shiprsquosmotion adaptively In this paper the adaptive 120572-120573-120574 filter is proposed for the ship-borne phased array radar to compensate for thebeam pointing error and to track the air target The genetic algorithm (GA) and the particle swarm optimization (PSO) methodsare applied to estimate the gain parameters of adaptive 120572-120573-120574 filters while achieving the optimum objective of minimum rootmean square error (RMSE) The roll and pitch data measured from a gyroscope of the sea vehicle and generated from ship motionmathematical model are used in the experiments The tracking accuracy of adaptive 120572-120573-120574 filter using the GAmethod is comparedwith PSO method under different ship motion conditions The convergent time and tracking accuracy of ship-borne phased arrayradar using the proposed GA based adaptive 120572-120573-120574 filter are also compared with the adaptive extended Kalman filter (AEKF)Finally it is proved that the proposed GA based adaptive 120572-120573-120574 filter is a real time applicable algorithm for ship-borne phased arrayradar

1 Introduction

Sea wave causes the effect of roll and pitch motions onships These ship rotational motions result in measurementerror in phased array radar aboard the ship The antennastabilization to achieve the beam pointing accuracy over thelong dwell time is an important issue for ship-borne phasedarray radar [1] There are two ship motion compensationscompensation for rotational motion (ie pitch roll andheading angle) and compensation for translational motion(ie radial speed relative to the earth) Gyroscope providespitch roll heading angles of the ship speed course andvertical velocity of antenna installed on the ship at a datarate of 10Hz To compensate for the translational motionthe speed course and vertical velocity acquired from thegyroscope are averaged for the duration of radar-dwelltime Then the Doppler shift introduced by the translationalmotion is estimated in the digital signal processor (DSP)of the radar to compensate the radial velocity The radar

control computer (RCC) provides the target locations relativeto earth schedules beam directions and predicts beampointing error to the beam steering controller (BSC) whichcompensates for the beam pointing error and controls thephased array antenna to point the beam at the target directionrelative to the ship coordinates

The motion compensation method based on coordinateconversion has been described in [2 3] which requires themeasurement of a devicersquos coordinates the shiprsquos coordinatesand the earthrsquos coordinates systems to compensate the deviceerrors due to motion disturbances In [1] Kalman filteringalong with coordinates conversion is applied to reducethe beam pointing error and to stabilize a tracking beamThe 120572-120573-120574 filter which is more easily implemented than ageneralized Kalman filter is used for motion compensationunder different sea states in [1]The sea environments are verydynamic hence there is need of an adaptive system for con-trolling and compensating devices regardless of ship motionMost recent works have used Kalman filtering (KF) [4]

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 563726 16 pageshttpdxdoiorg1011552015563726

2 International Journal of Antennas and Propagation

Adaptive

filter for rollpitchprediction

Shipearth coordinates conversion

Radar detection target data in

earth coordination

Adaptive

filter for target

tracking

Performanceverifications for

1 ship motion compensation2 target tracking accuracy

Earthshipcoordinates conversion

120601R(t) 120579P(t)

Δ120579E Δ120601E

120579E 120601E

120579EC 120601EC 120579A 120601A+

minus

1205721-1205731-1205741

1205722-1205732-1205742

Figure 1 High level structure of ship rotational motion compensation for phased array radar

and extended Kalman filtering (EKF) [5] to estimate theshiprsquos attitude Estimation accuracy of KF and EKF dependson the values of different parameters such as error covari-ance matrices thus the KF and EKF require knowledgeof covariance In [6] the automatic beam pointing errorcompensation mechanism employs the parallel fuzzy basisfunction network (FBFN) architecture to estimate the beampointing error caused by roll and pitch of the ship The effectof automatic beam pointing error compensation mechanismon the tracking performance of adaptive extended Kalmanfilter (AEKF) implemented in ship-borne phased array radaris investigated It shows that the tracking error of AEKF con-verges to less than about 20m at 550 iterations (seciteration)and the estimation errorwill remainwithin the range of about20m when beam pointing error is compensated by FBFNcontroller

The 120572-120573-120574 filter has a similar structure with the KF butdepends on the gain value of 120572 120573 and 120574 which are limitedand interdependent [7ndash9] The uses of various methods toadjust parameter values for Kalman filter EKF and 120572-120573-120574 filter have been proposed and implemented over the yearsFuzzy membership function is used in [10] to estimate andcorrect the target position for the signal being trackedGenetic algorithm (GA) is used in [9] to search the suitableparameter values for the 120572-120573-120574 filter which can providethe approximate position velocity and acceleration signaland simultaneously decrease themeasurement noise Particleswarm optimization (PSO) is used in [11] to tune the noisecovariance of a Kalman filter In [12] the PSO is used to findthe optimal gain parameter values for an 120572-120573-120574 filter PSOand GA methods are suitable for the real time applicationbecause they do not require differentiation and are lesscomplicated than other methods

In this paper GA based adaptive 120572-120573-120574 filter is proposedfor automatic beam pointing error correction and air targettracking in three-dimensional space Continuous monitor-ing of the environment and adapting filter gain parameterwith less computational burden is needed for real timeapplication The adaptive 120572-120573-120574 filter requires knowledgeof the coefficients for which GA algorithm is used only incertain time intervals GA is used to find the gain values byminimizing the objective function that is root mean squareerror (RMSE) The proposed adaptive 120572-120573-120574 filter algorithm

is implemented and compared with using PSO method Theroll and pitch data were recorded by a gyroscope of thesea vehicle to simulate the tracking performance of ship-borne phased array radar using the proposed adaptive 120572-120573-120574 filter In addition the roll and pitch data generated fromship motionmathematical model at sea states 2 and 3 are alsoapplied for the proposed adaptive 120572-120573-120574 filter to verify thecorrectness of the test results

The rest of this paper is organized as follows the modelof ship rotational motion coordinates transform and planararray antenna of ship-borne phased array radar are describedin detail in Section 2The algorithm of the proposed adaptive120572-120573-120574 filter based on the GA is presented in Section 3 wherethe 120572-120573-120574 filter GA gain estimator PSO gain estimatorand optimization problem formulation are described InSection 4 six different experimental cases are performed thevariations of roll and pitch angles in sea states 2 and 3 areanalyzed the GA and PSO methods are used to determinethe optimal filter gain of adaptive 120572-120573-120574 filter that minimizesRMSE the tracking performance of ship-borne phased arrayradar using the proposed adaptive 120572-120573-120574 filter is simulatedFinally conclusions are made in Section 5

2 Ship Rotational Motion Compensation

The ship-borne phased array radar must be able to com-pensate the shiprsquos motion and track the maneuvering tar-gets automatically The adaptive 120572-120573-120574 filtering algorithmis designed to real time compensate the errors caused bythe shiprsquos motion The block diagram of rotational motioncompensation system for ship-borne phased array radar isshown in Figure 1 which consists of adaptive 120572-120573-120574 filterfor beam pointing error prediction ship coordinatesearthcoordinates conversion earth coordinatesship coordinatesconversion and adaptive 120572-120573-120574 filter for target trackingThe roll angle 120601

119877(119905) and the pitch angle 120579

119875(119905) of the ship

angularmotion aremeasured by the gyroscope (120579119860 120601119860) is the

antenna beam pointing angle relative to the ship body where120579119860is the angle off antenna boresight and 120601

119860is the azimuth

angle counterclockwise from the bowof the ship Assume thatthe beam steering angle at a certain time instant is (120579

0 1206010) the

antenna point angle offset caused by the ship motion (pitch

International Journal of Antennas and Propagation 3

Table 1 Sea state parameters

Roll Pitch119860119877(∘) 119879

119877(sec) 119860

119875(∘) 119879

119875(sec)

Sea state 2 1 25 02 12Sea state 3 1 20 02 06

roll) is (120579119890 120601119890) and then the current antenna pointing angle

is (120579119864 120601119864) with respect to the earth coordinates

120579119864= 1205790+ 120579119890 120601

119864= 1206010+ 120601119890 (1)

If the beampointing error is predicted as (Δ120579119864 Δ120601119864) then the

beam pointing angle is corrected as

120579119864119862

= 1205790+ 120579119890minus Δ120579119864 120601

119864119862= 1206010+ 120601119890minus Δ120601119864 (2)

The value of (120579119864119862 120601119864119862) is approximate to (120579

0 1206010)

21 Ship Rotational Motion Model Ship is affected by thewaves in the ocean resulting in six degrees of freedomof movement In this paper assuming zero yaw angle thesimplified roll and pitch model [1 13] is adopted to describethe ship rotational motion in the earth coordinates Shiprsquosrotational motion is modeled with sinusoidal signal The rollangle is

120601119877 (119905) = 119860

119877sin (120596

119877119905) + 119899

119877 (119905) (3)

The pitch angle is

120579119875 (119905) = 119860

119875sin (120596

119875119905) + 119899

119875 (119905) (4)

where 119860119877and 119860

119875are the amplitude of shiprsquos roll and pitch

angles 119899119877(119905) and 119899

119875(119905) are assumed to be zeromeanGaussian

noise 119879119877and 119879

119875are the roll and pitch periods and 120596

119877=

2120587119879119877and120596

119875= 2120587119879

119875are the roll and pitch angular frequen-

ciesThe random roll and pitch angle errors caused by other

unknown factors including the change of shiprsquos travelingdirection and weather in the actual ship navigation envi-ronment will be considered into the standard deviation ofthe noise terms The amplitude and period parameters of seastates 2 and 3 are listed in Table 1 [1]

22 Coordinates Transform Since the phased array antennais installed on the ship the beam steering control employsthe ship body coordinates But the target tracking of adaptive120572-120573-120574 filter employs the Earth coordinates The Euler coor-dinates transform formula is used to convert antenna beampointing angle relative to the earth (120579

119864 120601119864) where 120579

119864is the

angle from vertical and 120601119864is the angle clockwise from true

north into antenna beam pointing angle relative to the shipbody (120579

119860 120601119860) [13]

[

[

sin 120579119860cos120601119860

sin 120579119860sin120601119860

cos 120579119860

]

]

= 119879 (120601) 119879 (120579) 119879 (120595)[

[

sin 120579119864cos120601119864

minus sin 120579119864sin120601119864

cos 120579119864

]

]

(5)

where the coordinates transform matrices are defined as

119879 (120601) =[

[

1 0 0

0 cos120601 sin1206010 minus sin120601 cos120601

]

]

119879 (120579) =[

[

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579]

]

119879 (120595) =[

[

cos120595 minus sin120595 0

sin120595 cos120595 0

0 0 1

]

]

(6)

where the ship roll angle 120601 is positive with downward roll ofstarboard the ship pitch angle 120579 is positive with bow pitchupward and the ship yaw angle 120595 is positive clockwise fromthe north

23 Planar Array Antenna An 119872 times 119873 element planararray [6] is designed for the ship-borne phased array radarsystem which includes the beamforming (BF) mode anddirection of arrival (DOA) mode The planar array canproduce multibeams in the azimuth and elevation by usingthe beamformer network (BFN) which consists of a set ofpower dividers and phase shifters The planar array usingamplitude comparison method [14] generates the differencesignal patterns for theDOAestimationThedifference signalsobtained from two neighboring beams canmeasure the DOAof the target signal of the ship-borne phased array radarsystem The beam pattern is expressed as [14 15]

119860119865 (120579 120601) =

119873

sum

119899 = 1

1198681119899[

119872

sum

119898=1

1198681198981119890119895(119898minus 1)(119896119889

119909sin 120579 cos120601+120573

119909)]

times 119890119895(119899 minus 1)(119896119889

119910sin 120579 sin120601+120573

119910)= 119878119909119898119878119910119899

(7)

where

119878119909119898

=

119872

sum

119898=1

1198681198981119890119895(119898minus 1)(119896119889

119909sin 120579 cos120601+120573

119909)

119878119910119898

=

119873

sum

119899 = 1

1198681119899119890119895(119899 minus 1)(119896119889

119910sin 120579 sin120601+120573

119910)

120573119909= minus 119896119889

119909sin 120579119905cos120601119905

120573119910= minus 119896119889

119910sin 120579119905sin120601119905

(8)

where 1198681198981

= 1198681119899= Chebyshev weighting [13]119872 = the number

of array elements in the 119909-axis 119873 = the number of arrayelements in 119910-axis 120601

119905= the azimuth steering angle of the

main beam 120579119905= the elevation steering angle of the main

beam 119889119909= the interelement spacing in 119909-axis 119889

119910= the

interelement spacing in 119910-axis 119896 = 2120587120582 = the phasepropagation constant and 120582 is the wavelength of the carrier

The phase of the RF signal at each array element isadjusted to steer the beam to the coordinates (120579

119905 120601119905)

4 International Journal of Antennas and Propagation

Estimation Prediction

Delay

Pitch gain estimator

Roll gain estimator

Estimation Prediction

Delay

Roll pitchs anglemeasurement

1205721-1205731-1205741 filter(g1r)

1205721-1205731-1205741 filter(g1p)

120579mr(k)

120579mp(k)

r(k)

r(k)

ar(k)

p(k)

p(k)

ap(k)

r(k + 1)r(k + 1)ar(k + 1)

p(k + 1)

p(k + 1)

ap(k + 1)

120579er(k)

120596er(k)

aer(k)

120579ep(k)

120596ep(k)

aep(k)

Figure 2 Parallel adaptive 120572-120573-120574 filter for beam pointing error prediction

3 Adaptive 120572-120573-120574 Filter

As shown in Figure 1 the adaptive 120572-120573-120574 filters are applied tobeam pointing error prediction and target tracking respec-tively

31 Parallel Adaptive 120572-120573-120574 Filter for Beam Pointing ErrorPrediction Figure 2 shows the beam pointing error predic-tion system which consists of separate adaptive 120572

1-1205731-1205741

filters to predict the roll and pitch angles of ship motionand their outputs are fed into the beam steering controller(BSC)The BSC compensates the beam pointing errors of thephased array antenna onboard the ship to reduce the impactof the roll and pitch motion on the phased array radar Rolland pitch data are random but they are similar in natureand they can be characterized with different amplitudes andfrequencies [1]The 120572

1-1205731-1205741filter predicts the next positions

using current error also known as innovation [8] Thisinnovation process has two steps prediction and estimation

The pitch prediction equations are expressed as

120579119901 (119896 + 1) = 120579

119890119901 (119896) + 1198791

120596119890119901 (

119896) +

1198792

1

2

119886119890119901 (

119896)

119901 (119896 + 1) = 120596

119890119901 (119896) + 1198791

119886119890119901 (

119896)

119886119901 (119896 + 1) = 119886

119890119901 (119896)

(9)

The pitch estimation equations are expressed as

120579119890119901 (

119896) =120579119901 (119896) + 1205721119901

[120579119898119901 (

119896) minus120579119901 (119896)] + 1198991119901 (

119896)

120596119890119901 (

119896) = 119901 (119896) +

1205731119901

1198791

[120579119898119901 (

119896) minus120579119901 (119896)]

119886119890119901 (

119896) = 119886119901 (119896) +

1205741119901

21198792

1

[120579119898119901 (

119896) minus120579119901 (119896)]

(10)

where 120579119901(119896 + 1) 119908

119901(119896 + 1) and 119886

119901(119896 + 1) are pitch angu-

lar position pitch angular velocity and pitch angular accel-eration values predicted at (119896 + 1)th interval respectivelyThe sampling time 119879

1= 01 sec 120579

119890119901(119896) 119908

119890119901(119896) and 119886

119890119901(119896)

are pitch angular position pitch angular velocity and pitchangular acceleration estimated at 119896th interval 120572

1119901 1205731119901 and

1205741119901

are the smoothing parameters whose region of stabilityand parameter constraint have been studied in [8] Therelationship among 120572

1119901 1205731119901 and 120574

1119901parameters can be

related to pitch gain 1198921119901 where 0 lt 119892

1119901lt 1 The formula

is described as follows

1205721119901

= 1 minus 1198923

1119901

1205731119901

= 15 sdot (1 + 1198921119901) (1 minus 119892

1119901)

2

1205741119901

= (1 minus 1198921119901)

3

(11)

The roll prediction equations roll estimation equations androll smoothing parameters have similar forms as the pitchprediction equations pitch estimation equations and pitchsmoothing parameters respectively

32 Adaptive 120572-120573-120574 Filter for Target Tracking in Three-Dimensional Space The GA based 120572

2-1205732-1205742filter algorithm

for target tracking is used by ship-borne phased array radarto predict the next target positions using current error andfurther improves its tracking accuracy The tracking processof 1205722-1205732-1205742filter is shown in Figure 3 where the prediction

equations are

X (119896 + 1) = X119890 (119896) + 1198792

V119890 (119896) +

1198792

2

2

A119890 (119896)

V (119896 + 1) = V119890 (119896) + 1198792

A119890 (119896)

A (119896 + 1) = A119890 (119896)

(12)

International Journal of Antennas and Propagation 5

Tracking gain estimator

Estimation Prediction

Positionmeasurement

Delay

1205722-1205732-1205742 filter(g2)

Xm(k)

X(k)

V(k)

A(k)

Xe(k)

Ve(k)

Ae(k)

X(k + 1)

V(k + 1)

A(k + 1)

Figure 3 Adaptive 120572-120573-120574 filter for target tracking in three-dimensional space

The estimation equations are

X119890 (119896) =

X (119896) + 1205722[X119898 (

119896) minusX (119896)] + n

2 (119896)

V119890 (119896) = V (119896) +

1205732

1198792

[X119898 (

119896) minus X (119896)]

A119890 (119896) = A (119896) +

1205742

21198792

2

[X119898 (

119896) minus X (119896)]

(13)

where X(119896) = [119909(119896) 119910(119896) 119911(119896)]119879 V(119896) = [V

119909(119896) V119910(119896)

V119911(119896)]119879 and A(119896) = [119886

119909(119896) 119886119910(119896) 119886119911(119896)]119879 are the position

vector velocity vector and acceleration vector respectivelyat time instant 119896 n

2(119896) is the zero mean white Gaussian noise

and the sampling time 1198792= 1 sec The relationship among

1205722 1205732 and 120574

2parameters can be related to gain factor vector

g2= [1198922 1198922 1198922]119879 The formula is described as follows

1205722=[

[

1205722119909

0 0

0 1205722119910

0

0 0 1205722119911

]

]

=[

[

1 minus 1198923

20 0

0 1 minus 1198923

20

0 0 1 minus 1198923

2

]

]

1205732=[

[

1205732119909

0 0

0 1205732119910

0

0 0 1205732119911

]

]

=[

[

[

15 (1 + 1198922) (1 minus 119892

2)2

0 0

0 15 (1 + 1198922) (1 minus 119892

2)2

0

0 0 15 (1 + 1198922) (1 minus 119892

2)2

]

]

]

1205742=[

[

1205742119909

0 0

0 1205742119910

0

0 0 1205742119911

]

]

=[

[

[

(1 minus 1198922)3

0 0

0 (1 minus 1198922)3

0

0 0 (1 minus 1198922)3

]

]

]

(14)

33 GA Gain Estimator The flow chart of GA rollpitch gainestimators of adaptive 120572-120573-120574 filter is shown in Figure 4 TheGA rollpitch gain estimators are used to estimate the roll andpitch angles gain (119892

11199031198921119901) of adaptive 120572

1-1205731-1205741filters The

GA tracking gain estimator is used to estimate the optimalgain g

2of adaptive 120572

2-1205732-1205742filter The flow chart of GA

tracking gain estimator is similar to Figure 4TheGAmethoduses selection crossover andmutation [16] techniques to findthe solutions At first the initial population which containsrandomly generated chromosomes is made Chromosomesare chains of 0 and 1 bits whose length is defined as stringlength Chromosomes number is defined as the populationsizeThese chromosomes from themating pool are shuffled to

create more randomness and then applied to the problem tofind their outputs and fitness scores accordingly Fitness valuedefines how well the chromosome solves the problem Twochromosomes are selected from the mating pool dependingon selection method Depending on the crossover rate allbits between two randomly chosen points selected from twodifferent chromosomes are interchanged which is known astwo-point crossover Chromosomes may undergo mutationwhere random bits of chromosomes are flipped dependingon the mutation rate Termination criteria are defined by thenumber of generations

Selection is done usually by using two methods asfollows roulette wheel selection and tournament selection

6 International Journal of Antennas and Propagation

Get optimal filter gain

Evaluate

YesNo

End

Roulette wheelselection

Crossover

Mutation

Generate the next generation

Is termination criteria met

Generate the initial population randomly

between range of 0 and 1

gr1gp1

Input the rollpitch sea state data

Figure 4 Flow chart of GA rollpitch gain estimator

In roulette wheel selection the chromosomes selected frompopulation are put into a mating pool The chromosomeselection probability is proportion to the fitness value Intournament selection chromosomes have to go throughseveral tournaments The chromosome which has the fittestvalue is selected and put into the mating pool Here theroulette wheel selection method is chosen to determine thesolution

Two-point crossover calls for two random points selectedfrom two different parent chromosome strings all bitsbetween two points are swapped for two parent chromo-somes Crossover rate is commonly chosen in a range of060 to 080 [16] Mutation rate describes the probability thata bit in a chromosome will be flipped (0 flips to 1 and 1flips to 0) Mutation brings diversity to the population asmutation of a chromosome is a random process which willbring randomness to present chromosome group Generationdefines iteration the GA method runs for a defined numberof generations and the final best solution is considered as theresult

34 PSO Gain Estimator [9 12 17] PSO method was firstproposed in [18] PSO is based on the number of particlesflying around the solution space to find the best solutionthat minimizes the cost function Particles move around thesolution space as soon as a particle detects a better solutionthe information is passed to other particles and then particleschange their position with respect to their best positionobserved among all the particles Figure 5 describes the PSOmethod in detail The PSO changes the velocity and positionof the particle according to (15) and (17) respectively toachieve the fitness [16]

V119894119889(119899 + 1)

= 119870 lowast [V119894119889(119899)

+ 1198881sdot rand ( ) sdot (119901119894119889(119899) minus 119909119894119889(119899))

+ 1198882sdot Rand ( ) sdot (119901119892119889(119899) minus 119909119894119889(119899))]

(15)

119870 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic1205932minus 4120593

1003816100381610038161003816100381610038161003816

where 120593 = 1198881+ 1198882 120593 gt 4 (16)

119909119894119889(119899 + 1)

= 119909119894119889(119899)

+ V119894119889(119899)

(17)

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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DistributedSensor Networks

International Journal of

2 International Journal of Antennas and Propagation

Adaptive

filter for rollpitchprediction

Shipearth coordinates conversion

Radar detection target data in

earth coordination

Adaptive

filter for target

tracking

Performanceverifications for

1 ship motion compensation2 target tracking accuracy

Earthshipcoordinates conversion

120601R(t) 120579P(t)

Δ120579E Δ120601E

120579E 120601E

120579EC 120601EC 120579A 120601A+

minus

1205721-1205731-1205741

1205722-1205732-1205742

Figure 1 High level structure of ship rotational motion compensation for phased array radar

and extended Kalman filtering (EKF) [5] to estimate theshiprsquos attitude Estimation accuracy of KF and EKF dependson the values of different parameters such as error covari-ance matrices thus the KF and EKF require knowledgeof covariance In [6] the automatic beam pointing errorcompensation mechanism employs the parallel fuzzy basisfunction network (FBFN) architecture to estimate the beampointing error caused by roll and pitch of the ship The effectof automatic beam pointing error compensation mechanismon the tracking performance of adaptive extended Kalmanfilter (AEKF) implemented in ship-borne phased array radaris investigated It shows that the tracking error of AEKF con-verges to less than about 20m at 550 iterations (seciteration)and the estimation errorwill remainwithin the range of about20m when beam pointing error is compensated by FBFNcontroller

The 120572-120573-120574 filter has a similar structure with the KF butdepends on the gain value of 120572 120573 and 120574 which are limitedand interdependent [7ndash9] The uses of various methods toadjust parameter values for Kalman filter EKF and 120572-120573-120574 filter have been proposed and implemented over the yearsFuzzy membership function is used in [10] to estimate andcorrect the target position for the signal being trackedGenetic algorithm (GA) is used in [9] to search the suitableparameter values for the 120572-120573-120574 filter which can providethe approximate position velocity and acceleration signaland simultaneously decrease themeasurement noise Particleswarm optimization (PSO) is used in [11] to tune the noisecovariance of a Kalman filter In [12] the PSO is used to findthe optimal gain parameter values for an 120572-120573-120574 filter PSOand GA methods are suitable for the real time applicationbecause they do not require differentiation and are lesscomplicated than other methods

In this paper GA based adaptive 120572-120573-120574 filter is proposedfor automatic beam pointing error correction and air targettracking in three-dimensional space Continuous monitor-ing of the environment and adapting filter gain parameterwith less computational burden is needed for real timeapplication The adaptive 120572-120573-120574 filter requires knowledgeof the coefficients for which GA algorithm is used only incertain time intervals GA is used to find the gain values byminimizing the objective function that is root mean squareerror (RMSE) The proposed adaptive 120572-120573-120574 filter algorithm

is implemented and compared with using PSO method Theroll and pitch data were recorded by a gyroscope of thesea vehicle to simulate the tracking performance of ship-borne phased array radar using the proposed adaptive 120572-120573-120574 filter In addition the roll and pitch data generated fromship motionmathematical model at sea states 2 and 3 are alsoapplied for the proposed adaptive 120572-120573-120574 filter to verify thecorrectness of the test results

The rest of this paper is organized as follows the modelof ship rotational motion coordinates transform and planararray antenna of ship-borne phased array radar are describedin detail in Section 2The algorithm of the proposed adaptive120572-120573-120574 filter based on the GA is presented in Section 3 wherethe 120572-120573-120574 filter GA gain estimator PSO gain estimatorand optimization problem formulation are described InSection 4 six different experimental cases are performed thevariations of roll and pitch angles in sea states 2 and 3 areanalyzed the GA and PSO methods are used to determinethe optimal filter gain of adaptive 120572-120573-120574 filter that minimizesRMSE the tracking performance of ship-borne phased arrayradar using the proposed adaptive 120572-120573-120574 filter is simulatedFinally conclusions are made in Section 5

2 Ship Rotational Motion Compensation

The ship-borne phased array radar must be able to com-pensate the shiprsquos motion and track the maneuvering tar-gets automatically The adaptive 120572-120573-120574 filtering algorithmis designed to real time compensate the errors caused bythe shiprsquos motion The block diagram of rotational motioncompensation system for ship-borne phased array radar isshown in Figure 1 which consists of adaptive 120572-120573-120574 filterfor beam pointing error prediction ship coordinatesearthcoordinates conversion earth coordinatesship coordinatesconversion and adaptive 120572-120573-120574 filter for target trackingThe roll angle 120601

119877(119905) and the pitch angle 120579

119875(119905) of the ship

angularmotion aremeasured by the gyroscope (120579119860 120601119860) is the

antenna beam pointing angle relative to the ship body where120579119860is the angle off antenna boresight and 120601

119860is the azimuth

angle counterclockwise from the bowof the ship Assume thatthe beam steering angle at a certain time instant is (120579

0 1206010) the

antenna point angle offset caused by the ship motion (pitch

International Journal of Antennas and Propagation 3

Table 1 Sea state parameters

Roll Pitch119860119877(∘) 119879

119877(sec) 119860

119875(∘) 119879

119875(sec)

Sea state 2 1 25 02 12Sea state 3 1 20 02 06

roll) is (120579119890 120601119890) and then the current antenna pointing angle

is (120579119864 120601119864) with respect to the earth coordinates

120579119864= 1205790+ 120579119890 120601

119864= 1206010+ 120601119890 (1)

If the beampointing error is predicted as (Δ120579119864 Δ120601119864) then the

beam pointing angle is corrected as

120579119864119862

= 1205790+ 120579119890minus Δ120579119864 120601

119864119862= 1206010+ 120601119890minus Δ120601119864 (2)

The value of (120579119864119862 120601119864119862) is approximate to (120579

0 1206010)

21 Ship Rotational Motion Model Ship is affected by thewaves in the ocean resulting in six degrees of freedomof movement In this paper assuming zero yaw angle thesimplified roll and pitch model [1 13] is adopted to describethe ship rotational motion in the earth coordinates Shiprsquosrotational motion is modeled with sinusoidal signal The rollangle is

120601119877 (119905) = 119860

119877sin (120596

119877119905) + 119899

119877 (119905) (3)

The pitch angle is

120579119875 (119905) = 119860

119875sin (120596

119875119905) + 119899

119875 (119905) (4)

where 119860119877and 119860

119875are the amplitude of shiprsquos roll and pitch

angles 119899119877(119905) and 119899

119875(119905) are assumed to be zeromeanGaussian

noise 119879119877and 119879

119875are the roll and pitch periods and 120596

119877=

2120587119879119877and120596

119875= 2120587119879

119875are the roll and pitch angular frequen-

ciesThe random roll and pitch angle errors caused by other

unknown factors including the change of shiprsquos travelingdirection and weather in the actual ship navigation envi-ronment will be considered into the standard deviation ofthe noise terms The amplitude and period parameters of seastates 2 and 3 are listed in Table 1 [1]

22 Coordinates Transform Since the phased array antennais installed on the ship the beam steering control employsthe ship body coordinates But the target tracking of adaptive120572-120573-120574 filter employs the Earth coordinates The Euler coor-dinates transform formula is used to convert antenna beampointing angle relative to the earth (120579

119864 120601119864) where 120579

119864is the

angle from vertical and 120601119864is the angle clockwise from true

north into antenna beam pointing angle relative to the shipbody (120579

119860 120601119860) [13]

[

[

sin 120579119860cos120601119860

sin 120579119860sin120601119860

cos 120579119860

]

]

= 119879 (120601) 119879 (120579) 119879 (120595)[

[

sin 120579119864cos120601119864

minus sin 120579119864sin120601119864

cos 120579119864

]

]

(5)

where the coordinates transform matrices are defined as

119879 (120601) =[

[

1 0 0

0 cos120601 sin1206010 minus sin120601 cos120601

]

]

119879 (120579) =[

[

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579]

]

119879 (120595) =[

[

cos120595 minus sin120595 0

sin120595 cos120595 0

0 0 1

]

]

(6)

where the ship roll angle 120601 is positive with downward roll ofstarboard the ship pitch angle 120579 is positive with bow pitchupward and the ship yaw angle 120595 is positive clockwise fromthe north

23 Planar Array Antenna An 119872 times 119873 element planararray [6] is designed for the ship-borne phased array radarsystem which includes the beamforming (BF) mode anddirection of arrival (DOA) mode The planar array canproduce multibeams in the azimuth and elevation by usingthe beamformer network (BFN) which consists of a set ofpower dividers and phase shifters The planar array usingamplitude comparison method [14] generates the differencesignal patterns for theDOAestimationThedifference signalsobtained from two neighboring beams canmeasure the DOAof the target signal of the ship-borne phased array radarsystem The beam pattern is expressed as [14 15]

119860119865 (120579 120601) =

119873

sum

119899 = 1

1198681119899[

119872

sum

119898=1

1198681198981119890119895(119898minus 1)(119896119889

119909sin 120579 cos120601+120573

119909)]

times 119890119895(119899 minus 1)(119896119889

119910sin 120579 sin120601+120573

119910)= 119878119909119898119878119910119899

(7)

where

119878119909119898

=

119872

sum

119898=1

1198681198981119890119895(119898minus 1)(119896119889

119909sin 120579 cos120601+120573

119909)

119878119910119898

=

119873

sum

119899 = 1

1198681119899119890119895(119899 minus 1)(119896119889

119910sin 120579 sin120601+120573

119910)

120573119909= minus 119896119889

119909sin 120579119905cos120601119905

120573119910= minus 119896119889

119910sin 120579119905sin120601119905

(8)

where 1198681198981

= 1198681119899= Chebyshev weighting [13]119872 = the number

of array elements in the 119909-axis 119873 = the number of arrayelements in 119910-axis 120601

119905= the azimuth steering angle of the

main beam 120579119905= the elevation steering angle of the main

beam 119889119909= the interelement spacing in 119909-axis 119889

119910= the

interelement spacing in 119910-axis 119896 = 2120587120582 = the phasepropagation constant and 120582 is the wavelength of the carrier

The phase of the RF signal at each array element isadjusted to steer the beam to the coordinates (120579

119905 120601119905)

4 International Journal of Antennas and Propagation

Estimation Prediction

Delay

Pitch gain estimator

Roll gain estimator

Estimation Prediction

Delay

Roll pitchs anglemeasurement

1205721-1205731-1205741 filter(g1r)

1205721-1205731-1205741 filter(g1p)

120579mr(k)

120579mp(k)

r(k)

r(k)

ar(k)

p(k)

p(k)

ap(k)

r(k + 1)r(k + 1)ar(k + 1)

p(k + 1)

p(k + 1)

ap(k + 1)

120579er(k)

120596er(k)

aer(k)

120579ep(k)

120596ep(k)

aep(k)

Figure 2 Parallel adaptive 120572-120573-120574 filter for beam pointing error prediction

3 Adaptive 120572-120573-120574 Filter

As shown in Figure 1 the adaptive 120572-120573-120574 filters are applied tobeam pointing error prediction and target tracking respec-tively

31 Parallel Adaptive 120572-120573-120574 Filter for Beam Pointing ErrorPrediction Figure 2 shows the beam pointing error predic-tion system which consists of separate adaptive 120572

1-1205731-1205741

filters to predict the roll and pitch angles of ship motionand their outputs are fed into the beam steering controller(BSC)The BSC compensates the beam pointing errors of thephased array antenna onboard the ship to reduce the impactof the roll and pitch motion on the phased array radar Rolland pitch data are random but they are similar in natureand they can be characterized with different amplitudes andfrequencies [1]The 120572

1-1205731-1205741filter predicts the next positions

using current error also known as innovation [8] Thisinnovation process has two steps prediction and estimation

The pitch prediction equations are expressed as

120579119901 (119896 + 1) = 120579

119890119901 (119896) + 1198791

120596119890119901 (

119896) +

1198792

1

2

119886119890119901 (

119896)

119901 (119896 + 1) = 120596

119890119901 (119896) + 1198791

119886119890119901 (

119896)

119886119901 (119896 + 1) = 119886

119890119901 (119896)

(9)

The pitch estimation equations are expressed as

120579119890119901 (

119896) =120579119901 (119896) + 1205721119901

[120579119898119901 (

119896) minus120579119901 (119896)] + 1198991119901 (

119896)

120596119890119901 (

119896) = 119901 (119896) +

1205731119901

1198791

[120579119898119901 (

119896) minus120579119901 (119896)]

119886119890119901 (

119896) = 119886119901 (119896) +

1205741119901

21198792

1

[120579119898119901 (

119896) minus120579119901 (119896)]

(10)

where 120579119901(119896 + 1) 119908

119901(119896 + 1) and 119886

119901(119896 + 1) are pitch angu-

lar position pitch angular velocity and pitch angular accel-eration values predicted at (119896 + 1)th interval respectivelyThe sampling time 119879

1= 01 sec 120579

119890119901(119896) 119908

119890119901(119896) and 119886

119890119901(119896)

are pitch angular position pitch angular velocity and pitchangular acceleration estimated at 119896th interval 120572

1119901 1205731119901 and

1205741119901

are the smoothing parameters whose region of stabilityand parameter constraint have been studied in [8] Therelationship among 120572

1119901 1205731119901 and 120574

1119901parameters can be

related to pitch gain 1198921119901 where 0 lt 119892

1119901lt 1 The formula

is described as follows

1205721119901

= 1 minus 1198923

1119901

1205731119901

= 15 sdot (1 + 1198921119901) (1 minus 119892

1119901)

2

1205741119901

= (1 minus 1198921119901)

3

(11)

The roll prediction equations roll estimation equations androll smoothing parameters have similar forms as the pitchprediction equations pitch estimation equations and pitchsmoothing parameters respectively

32 Adaptive 120572-120573-120574 Filter for Target Tracking in Three-Dimensional Space The GA based 120572

2-1205732-1205742filter algorithm

for target tracking is used by ship-borne phased array radarto predict the next target positions using current error andfurther improves its tracking accuracy The tracking processof 1205722-1205732-1205742filter is shown in Figure 3 where the prediction

equations are

X (119896 + 1) = X119890 (119896) + 1198792

V119890 (119896) +

1198792

2

2

A119890 (119896)

V (119896 + 1) = V119890 (119896) + 1198792

A119890 (119896)

A (119896 + 1) = A119890 (119896)

(12)

International Journal of Antennas and Propagation 5

Tracking gain estimator

Estimation Prediction

Positionmeasurement

Delay

1205722-1205732-1205742 filter(g2)

Xm(k)

X(k)

V(k)

A(k)

Xe(k)

Ve(k)

Ae(k)

X(k + 1)

V(k + 1)

A(k + 1)

Figure 3 Adaptive 120572-120573-120574 filter for target tracking in three-dimensional space

The estimation equations are

X119890 (119896) =

X (119896) + 1205722[X119898 (

119896) minusX (119896)] + n

2 (119896)

V119890 (119896) = V (119896) +

1205732

1198792

[X119898 (

119896) minus X (119896)]

A119890 (119896) = A (119896) +

1205742

21198792

2

[X119898 (

119896) minus X (119896)]

(13)

where X(119896) = [119909(119896) 119910(119896) 119911(119896)]119879 V(119896) = [V

119909(119896) V119910(119896)

V119911(119896)]119879 and A(119896) = [119886

119909(119896) 119886119910(119896) 119886119911(119896)]119879 are the position

vector velocity vector and acceleration vector respectivelyat time instant 119896 n

2(119896) is the zero mean white Gaussian noise

and the sampling time 1198792= 1 sec The relationship among

1205722 1205732 and 120574

2parameters can be related to gain factor vector

g2= [1198922 1198922 1198922]119879 The formula is described as follows

1205722=[

[

1205722119909

0 0

0 1205722119910

0

0 0 1205722119911

]

]

=[

[

1 minus 1198923

20 0

0 1 minus 1198923

20

0 0 1 minus 1198923

2

]

]

1205732=[

[

1205732119909

0 0

0 1205732119910

0

0 0 1205732119911

]

]

=[

[

[

15 (1 + 1198922) (1 minus 119892

2)2

0 0

0 15 (1 + 1198922) (1 minus 119892

2)2

0

0 0 15 (1 + 1198922) (1 minus 119892

2)2

]

]

]

1205742=[

[

1205742119909

0 0

0 1205742119910

0

0 0 1205742119911

]

]

=[

[

[

(1 minus 1198922)3

0 0

0 (1 minus 1198922)3

0

0 0 (1 minus 1198922)3

]

]

]

(14)

33 GA Gain Estimator The flow chart of GA rollpitch gainestimators of adaptive 120572-120573-120574 filter is shown in Figure 4 TheGA rollpitch gain estimators are used to estimate the roll andpitch angles gain (119892

11199031198921119901) of adaptive 120572

1-1205731-1205741filters The

GA tracking gain estimator is used to estimate the optimalgain g

2of adaptive 120572

2-1205732-1205742filter The flow chart of GA

tracking gain estimator is similar to Figure 4TheGAmethoduses selection crossover andmutation [16] techniques to findthe solutions At first the initial population which containsrandomly generated chromosomes is made Chromosomesare chains of 0 and 1 bits whose length is defined as stringlength Chromosomes number is defined as the populationsizeThese chromosomes from themating pool are shuffled to

create more randomness and then applied to the problem tofind their outputs and fitness scores accordingly Fitness valuedefines how well the chromosome solves the problem Twochromosomes are selected from the mating pool dependingon selection method Depending on the crossover rate allbits between two randomly chosen points selected from twodifferent chromosomes are interchanged which is known astwo-point crossover Chromosomes may undergo mutationwhere random bits of chromosomes are flipped dependingon the mutation rate Termination criteria are defined by thenumber of generations

Selection is done usually by using two methods asfollows roulette wheel selection and tournament selection

6 International Journal of Antennas and Propagation

Get optimal filter gain

Evaluate

YesNo

End

Roulette wheelselection

Crossover

Mutation

Generate the next generation

Is termination criteria met

Generate the initial population randomly

between range of 0 and 1

gr1gp1

Input the rollpitch sea state data

Figure 4 Flow chart of GA rollpitch gain estimator

In roulette wheel selection the chromosomes selected frompopulation are put into a mating pool The chromosomeselection probability is proportion to the fitness value Intournament selection chromosomes have to go throughseveral tournaments The chromosome which has the fittestvalue is selected and put into the mating pool Here theroulette wheel selection method is chosen to determine thesolution

Two-point crossover calls for two random points selectedfrom two different parent chromosome strings all bitsbetween two points are swapped for two parent chromo-somes Crossover rate is commonly chosen in a range of060 to 080 [16] Mutation rate describes the probability thata bit in a chromosome will be flipped (0 flips to 1 and 1flips to 0) Mutation brings diversity to the population asmutation of a chromosome is a random process which willbring randomness to present chromosome group Generationdefines iteration the GA method runs for a defined numberof generations and the final best solution is considered as theresult

34 PSO Gain Estimator [9 12 17] PSO method was firstproposed in [18] PSO is based on the number of particlesflying around the solution space to find the best solutionthat minimizes the cost function Particles move around thesolution space as soon as a particle detects a better solutionthe information is passed to other particles and then particleschange their position with respect to their best positionobserved among all the particles Figure 5 describes the PSOmethod in detail The PSO changes the velocity and positionof the particle according to (15) and (17) respectively toachieve the fitness [16]

V119894119889(119899 + 1)

= 119870 lowast [V119894119889(119899)

+ 1198881sdot rand ( ) sdot (119901119894119889(119899) minus 119909119894119889(119899))

+ 1198882sdot Rand ( ) sdot (119901119892119889(119899) minus 119909119894119889(119899))]

(15)

119870 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic1205932minus 4120593

1003816100381610038161003816100381610038161003816

where 120593 = 1198881+ 1198882 120593 gt 4 (16)

119909119894119889(119899 + 1)

= 119909119894119889(119899)

+ V119894119889(119899)

(17)

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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International Journal of Antennas and Propagation 3

Table 1 Sea state parameters

Roll Pitch119860119877(∘) 119879

119877(sec) 119860

119875(∘) 119879

119875(sec)

Sea state 2 1 25 02 12Sea state 3 1 20 02 06

roll) is (120579119890 120601119890) and then the current antenna pointing angle

is (120579119864 120601119864) with respect to the earth coordinates

120579119864= 1205790+ 120579119890 120601

119864= 1206010+ 120601119890 (1)

If the beampointing error is predicted as (Δ120579119864 Δ120601119864) then the

beam pointing angle is corrected as

120579119864119862

= 1205790+ 120579119890minus Δ120579119864 120601

119864119862= 1206010+ 120601119890minus Δ120601119864 (2)

The value of (120579119864119862 120601119864119862) is approximate to (120579

0 1206010)

21 Ship Rotational Motion Model Ship is affected by thewaves in the ocean resulting in six degrees of freedomof movement In this paper assuming zero yaw angle thesimplified roll and pitch model [1 13] is adopted to describethe ship rotational motion in the earth coordinates Shiprsquosrotational motion is modeled with sinusoidal signal The rollangle is

120601119877 (119905) = 119860

119877sin (120596

119877119905) + 119899

119877 (119905) (3)

The pitch angle is

120579119875 (119905) = 119860

119875sin (120596

119875119905) + 119899

119875 (119905) (4)

where 119860119877and 119860

119875are the amplitude of shiprsquos roll and pitch

angles 119899119877(119905) and 119899

119875(119905) are assumed to be zeromeanGaussian

noise 119879119877and 119879

119875are the roll and pitch periods and 120596

119877=

2120587119879119877and120596

119875= 2120587119879

119875are the roll and pitch angular frequen-

ciesThe random roll and pitch angle errors caused by other

unknown factors including the change of shiprsquos travelingdirection and weather in the actual ship navigation envi-ronment will be considered into the standard deviation ofthe noise terms The amplitude and period parameters of seastates 2 and 3 are listed in Table 1 [1]

22 Coordinates Transform Since the phased array antennais installed on the ship the beam steering control employsthe ship body coordinates But the target tracking of adaptive120572-120573-120574 filter employs the Earth coordinates The Euler coor-dinates transform formula is used to convert antenna beampointing angle relative to the earth (120579

119864 120601119864) where 120579

119864is the

angle from vertical and 120601119864is the angle clockwise from true

north into antenna beam pointing angle relative to the shipbody (120579

119860 120601119860) [13]

[

[

sin 120579119860cos120601119860

sin 120579119860sin120601119860

cos 120579119860

]

]

= 119879 (120601) 119879 (120579) 119879 (120595)[

[

sin 120579119864cos120601119864

minus sin 120579119864sin120601119864

cos 120579119864

]

]

(5)

where the coordinates transform matrices are defined as

119879 (120601) =[

[

1 0 0

0 cos120601 sin1206010 minus sin120601 cos120601

]

]

119879 (120579) =[

[

cos 120579 0 sin 1205790 1 0

minus sin 120579 0 cos 120579]

]

119879 (120595) =[

[

cos120595 minus sin120595 0

sin120595 cos120595 0

0 0 1

]

]

(6)

where the ship roll angle 120601 is positive with downward roll ofstarboard the ship pitch angle 120579 is positive with bow pitchupward and the ship yaw angle 120595 is positive clockwise fromthe north

23 Planar Array Antenna An 119872 times 119873 element planararray [6] is designed for the ship-borne phased array radarsystem which includes the beamforming (BF) mode anddirection of arrival (DOA) mode The planar array canproduce multibeams in the azimuth and elevation by usingthe beamformer network (BFN) which consists of a set ofpower dividers and phase shifters The planar array usingamplitude comparison method [14] generates the differencesignal patterns for theDOAestimationThedifference signalsobtained from two neighboring beams canmeasure the DOAof the target signal of the ship-borne phased array radarsystem The beam pattern is expressed as [14 15]

119860119865 (120579 120601) =

119873

sum

119899 = 1

1198681119899[

119872

sum

119898=1

1198681198981119890119895(119898minus 1)(119896119889

119909sin 120579 cos120601+120573

119909)]

times 119890119895(119899 minus 1)(119896119889

119910sin 120579 sin120601+120573

119910)= 119878119909119898119878119910119899

(7)

where

119878119909119898

=

119872

sum

119898=1

1198681198981119890119895(119898minus 1)(119896119889

119909sin 120579 cos120601+120573

119909)

119878119910119898

=

119873

sum

119899 = 1

1198681119899119890119895(119899 minus 1)(119896119889

119910sin 120579 sin120601+120573

119910)

120573119909= minus 119896119889

119909sin 120579119905cos120601119905

120573119910= minus 119896119889

119910sin 120579119905sin120601119905

(8)

where 1198681198981

= 1198681119899= Chebyshev weighting [13]119872 = the number

of array elements in the 119909-axis 119873 = the number of arrayelements in 119910-axis 120601

119905= the azimuth steering angle of the

main beam 120579119905= the elevation steering angle of the main

beam 119889119909= the interelement spacing in 119909-axis 119889

119910= the

interelement spacing in 119910-axis 119896 = 2120587120582 = the phasepropagation constant and 120582 is the wavelength of the carrier

The phase of the RF signal at each array element isadjusted to steer the beam to the coordinates (120579

119905 120601119905)

4 International Journal of Antennas and Propagation

Estimation Prediction

Delay

Pitch gain estimator

Roll gain estimator

Estimation Prediction

Delay

Roll pitchs anglemeasurement

1205721-1205731-1205741 filter(g1r)

1205721-1205731-1205741 filter(g1p)

120579mr(k)

120579mp(k)

r(k)

r(k)

ar(k)

p(k)

p(k)

ap(k)

r(k + 1)r(k + 1)ar(k + 1)

p(k + 1)

p(k + 1)

ap(k + 1)

120579er(k)

120596er(k)

aer(k)

120579ep(k)

120596ep(k)

aep(k)

Figure 2 Parallel adaptive 120572-120573-120574 filter for beam pointing error prediction

3 Adaptive 120572-120573-120574 Filter

As shown in Figure 1 the adaptive 120572-120573-120574 filters are applied tobeam pointing error prediction and target tracking respec-tively

31 Parallel Adaptive 120572-120573-120574 Filter for Beam Pointing ErrorPrediction Figure 2 shows the beam pointing error predic-tion system which consists of separate adaptive 120572

1-1205731-1205741

filters to predict the roll and pitch angles of ship motionand their outputs are fed into the beam steering controller(BSC)The BSC compensates the beam pointing errors of thephased array antenna onboard the ship to reduce the impactof the roll and pitch motion on the phased array radar Rolland pitch data are random but they are similar in natureand they can be characterized with different amplitudes andfrequencies [1]The 120572

1-1205731-1205741filter predicts the next positions

using current error also known as innovation [8] Thisinnovation process has two steps prediction and estimation

The pitch prediction equations are expressed as

120579119901 (119896 + 1) = 120579

119890119901 (119896) + 1198791

120596119890119901 (

119896) +

1198792

1

2

119886119890119901 (

119896)

119901 (119896 + 1) = 120596

119890119901 (119896) + 1198791

119886119890119901 (

119896)

119886119901 (119896 + 1) = 119886

119890119901 (119896)

(9)

The pitch estimation equations are expressed as

120579119890119901 (

119896) =120579119901 (119896) + 1205721119901

[120579119898119901 (

119896) minus120579119901 (119896)] + 1198991119901 (

119896)

120596119890119901 (

119896) = 119901 (119896) +

1205731119901

1198791

[120579119898119901 (

119896) minus120579119901 (119896)]

119886119890119901 (

119896) = 119886119901 (119896) +

1205741119901

21198792

1

[120579119898119901 (

119896) minus120579119901 (119896)]

(10)

where 120579119901(119896 + 1) 119908

119901(119896 + 1) and 119886

119901(119896 + 1) are pitch angu-

lar position pitch angular velocity and pitch angular accel-eration values predicted at (119896 + 1)th interval respectivelyThe sampling time 119879

1= 01 sec 120579

119890119901(119896) 119908

119890119901(119896) and 119886

119890119901(119896)

are pitch angular position pitch angular velocity and pitchangular acceleration estimated at 119896th interval 120572

1119901 1205731119901 and

1205741119901

are the smoothing parameters whose region of stabilityand parameter constraint have been studied in [8] Therelationship among 120572

1119901 1205731119901 and 120574

1119901parameters can be

related to pitch gain 1198921119901 where 0 lt 119892

1119901lt 1 The formula

is described as follows

1205721119901

= 1 minus 1198923

1119901

1205731119901

= 15 sdot (1 + 1198921119901) (1 minus 119892

1119901)

2

1205741119901

= (1 minus 1198921119901)

3

(11)

The roll prediction equations roll estimation equations androll smoothing parameters have similar forms as the pitchprediction equations pitch estimation equations and pitchsmoothing parameters respectively

32 Adaptive 120572-120573-120574 Filter for Target Tracking in Three-Dimensional Space The GA based 120572

2-1205732-1205742filter algorithm

for target tracking is used by ship-borne phased array radarto predict the next target positions using current error andfurther improves its tracking accuracy The tracking processof 1205722-1205732-1205742filter is shown in Figure 3 where the prediction

equations are

X (119896 + 1) = X119890 (119896) + 1198792

V119890 (119896) +

1198792

2

2

A119890 (119896)

V (119896 + 1) = V119890 (119896) + 1198792

A119890 (119896)

A (119896 + 1) = A119890 (119896)

(12)

International Journal of Antennas and Propagation 5

Tracking gain estimator

Estimation Prediction

Positionmeasurement

Delay

1205722-1205732-1205742 filter(g2)

Xm(k)

X(k)

V(k)

A(k)

Xe(k)

Ve(k)

Ae(k)

X(k + 1)

V(k + 1)

A(k + 1)

Figure 3 Adaptive 120572-120573-120574 filter for target tracking in three-dimensional space

The estimation equations are

X119890 (119896) =

X (119896) + 1205722[X119898 (

119896) minusX (119896)] + n

2 (119896)

V119890 (119896) = V (119896) +

1205732

1198792

[X119898 (

119896) minus X (119896)]

A119890 (119896) = A (119896) +

1205742

21198792

2

[X119898 (

119896) minus X (119896)]

(13)

where X(119896) = [119909(119896) 119910(119896) 119911(119896)]119879 V(119896) = [V

119909(119896) V119910(119896)

V119911(119896)]119879 and A(119896) = [119886

119909(119896) 119886119910(119896) 119886119911(119896)]119879 are the position

vector velocity vector and acceleration vector respectivelyat time instant 119896 n

2(119896) is the zero mean white Gaussian noise

and the sampling time 1198792= 1 sec The relationship among

1205722 1205732 and 120574

2parameters can be related to gain factor vector

g2= [1198922 1198922 1198922]119879 The formula is described as follows

1205722=[

[

1205722119909

0 0

0 1205722119910

0

0 0 1205722119911

]

]

=[

[

1 minus 1198923

20 0

0 1 minus 1198923

20

0 0 1 minus 1198923

2

]

]

1205732=[

[

1205732119909

0 0

0 1205732119910

0

0 0 1205732119911

]

]

=[

[

[

15 (1 + 1198922) (1 minus 119892

2)2

0 0

0 15 (1 + 1198922) (1 minus 119892

2)2

0

0 0 15 (1 + 1198922) (1 minus 119892

2)2

]

]

]

1205742=[

[

1205742119909

0 0

0 1205742119910

0

0 0 1205742119911

]

]

=[

[

[

(1 minus 1198922)3

0 0

0 (1 minus 1198922)3

0

0 0 (1 minus 1198922)3

]

]

]

(14)

33 GA Gain Estimator The flow chart of GA rollpitch gainestimators of adaptive 120572-120573-120574 filter is shown in Figure 4 TheGA rollpitch gain estimators are used to estimate the roll andpitch angles gain (119892

11199031198921119901) of adaptive 120572

1-1205731-1205741filters The

GA tracking gain estimator is used to estimate the optimalgain g

2of adaptive 120572

2-1205732-1205742filter The flow chart of GA

tracking gain estimator is similar to Figure 4TheGAmethoduses selection crossover andmutation [16] techniques to findthe solutions At first the initial population which containsrandomly generated chromosomes is made Chromosomesare chains of 0 and 1 bits whose length is defined as stringlength Chromosomes number is defined as the populationsizeThese chromosomes from themating pool are shuffled to

create more randomness and then applied to the problem tofind their outputs and fitness scores accordingly Fitness valuedefines how well the chromosome solves the problem Twochromosomes are selected from the mating pool dependingon selection method Depending on the crossover rate allbits between two randomly chosen points selected from twodifferent chromosomes are interchanged which is known astwo-point crossover Chromosomes may undergo mutationwhere random bits of chromosomes are flipped dependingon the mutation rate Termination criteria are defined by thenumber of generations

Selection is done usually by using two methods asfollows roulette wheel selection and tournament selection

6 International Journal of Antennas and Propagation

Get optimal filter gain

Evaluate

YesNo

End

Roulette wheelselection

Crossover

Mutation

Generate the next generation

Is termination criteria met

Generate the initial population randomly

between range of 0 and 1

gr1gp1

Input the rollpitch sea state data

Figure 4 Flow chart of GA rollpitch gain estimator

In roulette wheel selection the chromosomes selected frompopulation are put into a mating pool The chromosomeselection probability is proportion to the fitness value Intournament selection chromosomes have to go throughseveral tournaments The chromosome which has the fittestvalue is selected and put into the mating pool Here theroulette wheel selection method is chosen to determine thesolution

Two-point crossover calls for two random points selectedfrom two different parent chromosome strings all bitsbetween two points are swapped for two parent chromo-somes Crossover rate is commonly chosen in a range of060 to 080 [16] Mutation rate describes the probability thata bit in a chromosome will be flipped (0 flips to 1 and 1flips to 0) Mutation brings diversity to the population asmutation of a chromosome is a random process which willbring randomness to present chromosome group Generationdefines iteration the GA method runs for a defined numberof generations and the final best solution is considered as theresult

34 PSO Gain Estimator [9 12 17] PSO method was firstproposed in [18] PSO is based on the number of particlesflying around the solution space to find the best solutionthat minimizes the cost function Particles move around thesolution space as soon as a particle detects a better solutionthe information is passed to other particles and then particleschange their position with respect to their best positionobserved among all the particles Figure 5 describes the PSOmethod in detail The PSO changes the velocity and positionof the particle according to (15) and (17) respectively toachieve the fitness [16]

V119894119889(119899 + 1)

= 119870 lowast [V119894119889(119899)

+ 1198881sdot rand ( ) sdot (119901119894119889(119899) minus 119909119894119889(119899))

+ 1198882sdot Rand ( ) sdot (119901119892119889(119899) minus 119909119894119889(119899))]

(15)

119870 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic1205932minus 4120593

1003816100381610038161003816100381610038161003816

where 120593 = 1198881+ 1198882 120593 gt 4 (16)

119909119894119889(119899 + 1)

= 119909119894119889(119899)

+ V119894119889(119899)

(17)

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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International Journal of

4 International Journal of Antennas and Propagation

Estimation Prediction

Delay

Pitch gain estimator

Roll gain estimator

Estimation Prediction

Delay

Roll pitchs anglemeasurement

1205721-1205731-1205741 filter(g1r)

1205721-1205731-1205741 filter(g1p)

120579mr(k)

120579mp(k)

r(k)

r(k)

ar(k)

p(k)

p(k)

ap(k)

r(k + 1)r(k + 1)ar(k + 1)

p(k + 1)

p(k + 1)

ap(k + 1)

120579er(k)

120596er(k)

aer(k)

120579ep(k)

120596ep(k)

aep(k)

Figure 2 Parallel adaptive 120572-120573-120574 filter for beam pointing error prediction

3 Adaptive 120572-120573-120574 Filter

As shown in Figure 1 the adaptive 120572-120573-120574 filters are applied tobeam pointing error prediction and target tracking respec-tively

31 Parallel Adaptive 120572-120573-120574 Filter for Beam Pointing ErrorPrediction Figure 2 shows the beam pointing error predic-tion system which consists of separate adaptive 120572

1-1205731-1205741

filters to predict the roll and pitch angles of ship motionand their outputs are fed into the beam steering controller(BSC)The BSC compensates the beam pointing errors of thephased array antenna onboard the ship to reduce the impactof the roll and pitch motion on the phased array radar Rolland pitch data are random but they are similar in natureand they can be characterized with different amplitudes andfrequencies [1]The 120572

1-1205731-1205741filter predicts the next positions

using current error also known as innovation [8] Thisinnovation process has two steps prediction and estimation

The pitch prediction equations are expressed as

120579119901 (119896 + 1) = 120579

119890119901 (119896) + 1198791

120596119890119901 (

119896) +

1198792

1

2

119886119890119901 (

119896)

119901 (119896 + 1) = 120596

119890119901 (119896) + 1198791

119886119890119901 (

119896)

119886119901 (119896 + 1) = 119886

119890119901 (119896)

(9)

The pitch estimation equations are expressed as

120579119890119901 (

119896) =120579119901 (119896) + 1205721119901

[120579119898119901 (

119896) minus120579119901 (119896)] + 1198991119901 (

119896)

120596119890119901 (

119896) = 119901 (119896) +

1205731119901

1198791

[120579119898119901 (

119896) minus120579119901 (119896)]

119886119890119901 (

119896) = 119886119901 (119896) +

1205741119901

21198792

1

[120579119898119901 (

119896) minus120579119901 (119896)]

(10)

where 120579119901(119896 + 1) 119908

119901(119896 + 1) and 119886

119901(119896 + 1) are pitch angu-

lar position pitch angular velocity and pitch angular accel-eration values predicted at (119896 + 1)th interval respectivelyThe sampling time 119879

1= 01 sec 120579

119890119901(119896) 119908

119890119901(119896) and 119886

119890119901(119896)

are pitch angular position pitch angular velocity and pitchangular acceleration estimated at 119896th interval 120572

1119901 1205731119901 and

1205741119901

are the smoothing parameters whose region of stabilityand parameter constraint have been studied in [8] Therelationship among 120572

1119901 1205731119901 and 120574

1119901parameters can be

related to pitch gain 1198921119901 where 0 lt 119892

1119901lt 1 The formula

is described as follows

1205721119901

= 1 minus 1198923

1119901

1205731119901

= 15 sdot (1 + 1198921119901) (1 minus 119892

1119901)

2

1205741119901

= (1 minus 1198921119901)

3

(11)

The roll prediction equations roll estimation equations androll smoothing parameters have similar forms as the pitchprediction equations pitch estimation equations and pitchsmoothing parameters respectively

32 Adaptive 120572-120573-120574 Filter for Target Tracking in Three-Dimensional Space The GA based 120572

2-1205732-1205742filter algorithm

for target tracking is used by ship-borne phased array radarto predict the next target positions using current error andfurther improves its tracking accuracy The tracking processof 1205722-1205732-1205742filter is shown in Figure 3 where the prediction

equations are

X (119896 + 1) = X119890 (119896) + 1198792

V119890 (119896) +

1198792

2

2

A119890 (119896)

V (119896 + 1) = V119890 (119896) + 1198792

A119890 (119896)

A (119896 + 1) = A119890 (119896)

(12)

International Journal of Antennas and Propagation 5

Tracking gain estimator

Estimation Prediction

Positionmeasurement

Delay

1205722-1205732-1205742 filter(g2)

Xm(k)

X(k)

V(k)

A(k)

Xe(k)

Ve(k)

Ae(k)

X(k + 1)

V(k + 1)

A(k + 1)

Figure 3 Adaptive 120572-120573-120574 filter for target tracking in three-dimensional space

The estimation equations are

X119890 (119896) =

X (119896) + 1205722[X119898 (

119896) minusX (119896)] + n

2 (119896)

V119890 (119896) = V (119896) +

1205732

1198792

[X119898 (

119896) minus X (119896)]

A119890 (119896) = A (119896) +

1205742

21198792

2

[X119898 (

119896) minus X (119896)]

(13)

where X(119896) = [119909(119896) 119910(119896) 119911(119896)]119879 V(119896) = [V

119909(119896) V119910(119896)

V119911(119896)]119879 and A(119896) = [119886

119909(119896) 119886119910(119896) 119886119911(119896)]119879 are the position

vector velocity vector and acceleration vector respectivelyat time instant 119896 n

2(119896) is the zero mean white Gaussian noise

and the sampling time 1198792= 1 sec The relationship among

1205722 1205732 and 120574

2parameters can be related to gain factor vector

g2= [1198922 1198922 1198922]119879 The formula is described as follows

1205722=[

[

1205722119909

0 0

0 1205722119910

0

0 0 1205722119911

]

]

=[

[

1 minus 1198923

20 0

0 1 minus 1198923

20

0 0 1 minus 1198923

2

]

]

1205732=[

[

1205732119909

0 0

0 1205732119910

0

0 0 1205732119911

]

]

=[

[

[

15 (1 + 1198922) (1 minus 119892

2)2

0 0

0 15 (1 + 1198922) (1 minus 119892

2)2

0

0 0 15 (1 + 1198922) (1 minus 119892

2)2

]

]

]

1205742=[

[

1205742119909

0 0

0 1205742119910

0

0 0 1205742119911

]

]

=[

[

[

(1 minus 1198922)3

0 0

0 (1 minus 1198922)3

0

0 0 (1 minus 1198922)3

]

]

]

(14)

33 GA Gain Estimator The flow chart of GA rollpitch gainestimators of adaptive 120572-120573-120574 filter is shown in Figure 4 TheGA rollpitch gain estimators are used to estimate the roll andpitch angles gain (119892

11199031198921119901) of adaptive 120572

1-1205731-1205741filters The

GA tracking gain estimator is used to estimate the optimalgain g

2of adaptive 120572

2-1205732-1205742filter The flow chart of GA

tracking gain estimator is similar to Figure 4TheGAmethoduses selection crossover andmutation [16] techniques to findthe solutions At first the initial population which containsrandomly generated chromosomes is made Chromosomesare chains of 0 and 1 bits whose length is defined as stringlength Chromosomes number is defined as the populationsizeThese chromosomes from themating pool are shuffled to

create more randomness and then applied to the problem tofind their outputs and fitness scores accordingly Fitness valuedefines how well the chromosome solves the problem Twochromosomes are selected from the mating pool dependingon selection method Depending on the crossover rate allbits between two randomly chosen points selected from twodifferent chromosomes are interchanged which is known astwo-point crossover Chromosomes may undergo mutationwhere random bits of chromosomes are flipped dependingon the mutation rate Termination criteria are defined by thenumber of generations

Selection is done usually by using two methods asfollows roulette wheel selection and tournament selection

6 International Journal of Antennas and Propagation

Get optimal filter gain

Evaluate

YesNo

End

Roulette wheelselection

Crossover

Mutation

Generate the next generation

Is termination criteria met

Generate the initial population randomly

between range of 0 and 1

gr1gp1

Input the rollpitch sea state data

Figure 4 Flow chart of GA rollpitch gain estimator

In roulette wheel selection the chromosomes selected frompopulation are put into a mating pool The chromosomeselection probability is proportion to the fitness value Intournament selection chromosomes have to go throughseveral tournaments The chromosome which has the fittestvalue is selected and put into the mating pool Here theroulette wheel selection method is chosen to determine thesolution

Two-point crossover calls for two random points selectedfrom two different parent chromosome strings all bitsbetween two points are swapped for two parent chromo-somes Crossover rate is commonly chosen in a range of060 to 080 [16] Mutation rate describes the probability thata bit in a chromosome will be flipped (0 flips to 1 and 1flips to 0) Mutation brings diversity to the population asmutation of a chromosome is a random process which willbring randomness to present chromosome group Generationdefines iteration the GA method runs for a defined numberof generations and the final best solution is considered as theresult

34 PSO Gain Estimator [9 12 17] PSO method was firstproposed in [18] PSO is based on the number of particlesflying around the solution space to find the best solutionthat minimizes the cost function Particles move around thesolution space as soon as a particle detects a better solutionthe information is passed to other particles and then particleschange their position with respect to their best positionobserved among all the particles Figure 5 describes the PSOmethod in detail The PSO changes the velocity and positionof the particle according to (15) and (17) respectively toachieve the fitness [16]

V119894119889(119899 + 1)

= 119870 lowast [V119894119889(119899)

+ 1198881sdot rand ( ) sdot (119901119894119889(119899) minus 119909119894119889(119899))

+ 1198882sdot Rand ( ) sdot (119901119892119889(119899) minus 119909119894119889(119899))]

(15)

119870 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic1205932minus 4120593

1003816100381610038161003816100381610038161003816

where 120593 = 1198881+ 1198882 120593 gt 4 (16)

119909119894119889(119899 + 1)

= 119909119894119889(119899)

+ V119894119889(119899)

(17)

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 5

Tracking gain estimator

Estimation Prediction

Positionmeasurement

Delay

1205722-1205732-1205742 filter(g2)

Xm(k)

X(k)

V(k)

A(k)

Xe(k)

Ve(k)

Ae(k)

X(k + 1)

V(k + 1)

A(k + 1)

Figure 3 Adaptive 120572-120573-120574 filter for target tracking in three-dimensional space

The estimation equations are

X119890 (119896) =

X (119896) + 1205722[X119898 (

119896) minusX (119896)] + n

2 (119896)

V119890 (119896) = V (119896) +

1205732

1198792

[X119898 (

119896) minus X (119896)]

A119890 (119896) = A (119896) +

1205742

21198792

2

[X119898 (

119896) minus X (119896)]

(13)

where X(119896) = [119909(119896) 119910(119896) 119911(119896)]119879 V(119896) = [V

119909(119896) V119910(119896)

V119911(119896)]119879 and A(119896) = [119886

119909(119896) 119886119910(119896) 119886119911(119896)]119879 are the position

vector velocity vector and acceleration vector respectivelyat time instant 119896 n

2(119896) is the zero mean white Gaussian noise

and the sampling time 1198792= 1 sec The relationship among

1205722 1205732 and 120574

2parameters can be related to gain factor vector

g2= [1198922 1198922 1198922]119879 The formula is described as follows

1205722=[

[

1205722119909

0 0

0 1205722119910

0

0 0 1205722119911

]

]

=[

[

1 minus 1198923

20 0

0 1 minus 1198923

20

0 0 1 minus 1198923

2

]

]

1205732=[

[

1205732119909

0 0

0 1205732119910

0

0 0 1205732119911

]

]

=[

[

[

15 (1 + 1198922) (1 minus 119892

2)2

0 0

0 15 (1 + 1198922) (1 minus 119892

2)2

0

0 0 15 (1 + 1198922) (1 minus 119892

2)2

]

]

]

1205742=[

[

1205742119909

0 0

0 1205742119910

0

0 0 1205742119911

]

]

=[

[

[

(1 minus 1198922)3

0 0

0 (1 minus 1198922)3

0

0 0 (1 minus 1198922)3

]

]

]

(14)

33 GA Gain Estimator The flow chart of GA rollpitch gainestimators of adaptive 120572-120573-120574 filter is shown in Figure 4 TheGA rollpitch gain estimators are used to estimate the roll andpitch angles gain (119892

11199031198921119901) of adaptive 120572

1-1205731-1205741filters The

GA tracking gain estimator is used to estimate the optimalgain g

2of adaptive 120572

2-1205732-1205742filter The flow chart of GA

tracking gain estimator is similar to Figure 4TheGAmethoduses selection crossover andmutation [16] techniques to findthe solutions At first the initial population which containsrandomly generated chromosomes is made Chromosomesare chains of 0 and 1 bits whose length is defined as stringlength Chromosomes number is defined as the populationsizeThese chromosomes from themating pool are shuffled to

create more randomness and then applied to the problem tofind their outputs and fitness scores accordingly Fitness valuedefines how well the chromosome solves the problem Twochromosomes are selected from the mating pool dependingon selection method Depending on the crossover rate allbits between two randomly chosen points selected from twodifferent chromosomes are interchanged which is known astwo-point crossover Chromosomes may undergo mutationwhere random bits of chromosomes are flipped dependingon the mutation rate Termination criteria are defined by thenumber of generations

Selection is done usually by using two methods asfollows roulette wheel selection and tournament selection

6 International Journal of Antennas and Propagation

Get optimal filter gain

Evaluate

YesNo

End

Roulette wheelselection

Crossover

Mutation

Generate the next generation

Is termination criteria met

Generate the initial population randomly

between range of 0 and 1

gr1gp1

Input the rollpitch sea state data

Figure 4 Flow chart of GA rollpitch gain estimator

In roulette wheel selection the chromosomes selected frompopulation are put into a mating pool The chromosomeselection probability is proportion to the fitness value Intournament selection chromosomes have to go throughseveral tournaments The chromosome which has the fittestvalue is selected and put into the mating pool Here theroulette wheel selection method is chosen to determine thesolution

Two-point crossover calls for two random points selectedfrom two different parent chromosome strings all bitsbetween two points are swapped for two parent chromo-somes Crossover rate is commonly chosen in a range of060 to 080 [16] Mutation rate describes the probability thata bit in a chromosome will be flipped (0 flips to 1 and 1flips to 0) Mutation brings diversity to the population asmutation of a chromosome is a random process which willbring randomness to present chromosome group Generationdefines iteration the GA method runs for a defined numberof generations and the final best solution is considered as theresult

34 PSO Gain Estimator [9 12 17] PSO method was firstproposed in [18] PSO is based on the number of particlesflying around the solution space to find the best solutionthat minimizes the cost function Particles move around thesolution space as soon as a particle detects a better solutionthe information is passed to other particles and then particleschange their position with respect to their best positionobserved among all the particles Figure 5 describes the PSOmethod in detail The PSO changes the velocity and positionof the particle according to (15) and (17) respectively toachieve the fitness [16]

V119894119889(119899 + 1)

= 119870 lowast [V119894119889(119899)

+ 1198881sdot rand ( ) sdot (119901119894119889(119899) minus 119909119894119889(119899))

+ 1198882sdot Rand ( ) sdot (119901119892119889(119899) minus 119909119894119889(119899))]

(15)

119870 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic1205932minus 4120593

1003816100381610038161003816100381610038161003816

where 120593 = 1198881+ 1198882 120593 gt 4 (16)

119909119894119889(119899 + 1)

= 119909119894119889(119899)

+ V119894119889(119899)

(17)

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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International Journal of

6 International Journal of Antennas and Propagation

Get optimal filter gain

Evaluate

YesNo

End

Roulette wheelselection

Crossover

Mutation

Generate the next generation

Is termination criteria met

Generate the initial population randomly

between range of 0 and 1

gr1gp1

Input the rollpitch sea state data

Figure 4 Flow chart of GA rollpitch gain estimator

In roulette wheel selection the chromosomes selected frompopulation are put into a mating pool The chromosomeselection probability is proportion to the fitness value Intournament selection chromosomes have to go throughseveral tournaments The chromosome which has the fittestvalue is selected and put into the mating pool Here theroulette wheel selection method is chosen to determine thesolution

Two-point crossover calls for two random points selectedfrom two different parent chromosome strings all bitsbetween two points are swapped for two parent chromo-somes Crossover rate is commonly chosen in a range of060 to 080 [16] Mutation rate describes the probability thata bit in a chromosome will be flipped (0 flips to 1 and 1flips to 0) Mutation brings diversity to the population asmutation of a chromosome is a random process which willbring randomness to present chromosome group Generationdefines iteration the GA method runs for a defined numberof generations and the final best solution is considered as theresult

34 PSO Gain Estimator [9 12 17] PSO method was firstproposed in [18] PSO is based on the number of particlesflying around the solution space to find the best solutionthat minimizes the cost function Particles move around thesolution space as soon as a particle detects a better solutionthe information is passed to other particles and then particleschange their position with respect to their best positionobserved among all the particles Figure 5 describes the PSOmethod in detail The PSO changes the velocity and positionof the particle according to (15) and (17) respectively toachieve the fitness [16]

V119894119889(119899 + 1)

= 119870 lowast [V119894119889(119899)

+ 1198881sdot rand ( ) sdot (119901119894119889(119899) minus 119909119894119889(119899))

+ 1198882sdot Rand ( ) sdot (119901119892119889(119899) minus 119909119894119889(119899))]

(15)

119870 =

2

1003816100381610038161003816100381610038161003816

2 minus 120593 minus radic1205932minus 4120593

1003816100381610038161003816100381610038161003816

where 120593 = 1198881+ 1198882 120593 gt 4 (16)

119909119894119889(119899 + 1)

= 119909119894119889(119899)

+ V119894119889(119899)

(17)

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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International Journal of

International Journal of Antennas and Propagation 7

Initialize particles with random positions and random velocities

For each particle calculate the fitness values

Is fitness value ltbest fitness value

Particle exhausted

Set global best fitness value as best fitness value

Update particle positions and velocities using (18) and (20)

Maximum iterationreached

Give global best fitnessvalue and position

Set global best as local best fitness value

Yes

No

No

No

Yes

Yes

Figure 5 Flow chart of PSO method

where 119909119894119889(119899)

and V119894119889(119899)

are position and velocity of (119894119889)thparticle at (119899)th iteration respectively 119901

119894119889(119899)and 119901

119892119889(119899)are

the best position and global best position of (119894119889)th particle at(119899)th iteration respectively Rand( ) and rand( ) are randomnumber between 0 and 1 respectively Equation (15) updatesthe velocity of the particles and (17) updates the position ofparticles 119870 means inertia 119888

1and 1198882are correction factors

and swarm size means number of particles for an iterationas described in [16]

Particles in the PSO method sometimes go out of con-straints while changing their position Reinitialization of theparticlersquos position randomly to fall within the constraints isdone as soon as a particle moves out of constraints Thisensures full use of particles and also increases the chance ofdiscovering a better solution To reduce the chance of gettingtrapped in a local minimum the advantages of using PSOmethod are that it is derivative-free easy to implement andeasy to comprehend and the solution is independent of theinitial point [17] One of the drawbacks of the PSOmethod isthat it does not guarantee success that is the solution foundmay not be the best solution

35 GAPSO Based Adaptive 120572-120573-120574 Filter The flow chart ofthe adaptive 120572-120573-120574 filter algorithm is shown in Figure 6where 119873 is the number of sampling data to determine thefilter gain In this algorithm the parameters120572120573 and 120574 whichare related to the filter gain 119892 are optimized by minimizingthe RMSE using GA or PSO method The proposed adaptive120572-120573-120574 filter processes the current sampled data 119909

119894which

predicts the next sample data 119909119894+1

as iteration continuesThe adaptive GA or PSO process only occurs at certain timeintervalThe previous119873 data that is 119909

119901(119894minus119873) sdot sdot sdot 119909

119901(119894minus1) is

sent to the adaptiveGAor PSO algorithmwhere the optimumvalue of filter gain (119892) is determined for the 120572-120573-120574 filterThen 119909

119894+1is predicted and the process continues The value

of119873 determines running time and accuracy of the algorithmso 119873 must have small enough value to support real timeimplementation and large enough value to provide acceptableaccuracy

Since the roll and pitch angles of ship motion changeover time randomly in order to speed up the processingtime and to reach the objectives of minimum estimationerror the filter gain values are real time updated by using thepipelined architectureThe recorded roll and pitch signals are

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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DistributedSensor Networks

International Journal of

8 International Journal of Antennas and Propagation

Collect previous N data using PSO or GA algorithm

Predicted value for120572-120573-120574 filterxi

xi+1

ximinus1 ximinusFind g for Nximinus1 ximinusN

that minimizes RMSE

Figure 6 Block diagram of adaptive 120572-120573-120574 filter

segmented into block of 1000 sampled data (100 sec) for lineartrajectory target and 420 sampled data (42 sec) for circulartrajectory target respectivelyThe gain value of 120572

2-1205732-1205742filter

in 101ndash200 sec is estimated by GA and PSO methods usingthe collected data during 1ndash100 sec the gain value of 120572

2-

1205732-1205742filter in 201ndash300 sec is estimated by the GA and PSO

methods using the collected data in 101ndash200 sec and so onSimilarly the measured flight target signals are segmentedinto block of 100 sampled data (100 sec) for linear trajectorytarget and 42 sampled data (42 sec) for circular trajectorytarget respectively The gain value of 120572

2-1205732-1205742filter in 101ndash

200 sec is estimated by the GA and PSO methods using thecollected data during 1ndash100 sec the gain value of 120572

2-1205732-1205742

filter in the interval of 201ndash300 sec is estimated by the GA andPSO methods using the collected data in 101ndash200 sec and soon

For the PSO process it was found that gain parameterconverged within 100 iterations so the number of iterationsfor an adaptive 120572-120573-120574 filter was kept at constant 100 iterationsper processing cycle swarm size = 30 inertia = 07298 andcorrection factors 119888

1= 21 119888

2= 2 For the GA process we

have considered population size = 8 string length = 12crossover rate = 08 and mutation rate = 005

If 120579119894119901 119894 = 1 2 119873 and

120579119894119901 119894 = 1 2 119873 are real

measurement pitchrsquos values and estimated pitchrsquos values ofadaptive 120572

1-1205731-1205741filter respectively then 120579

119894119903 119894 = 1 2 119873

and 120579119894119903 119894 = 1 2 119873 are realmeasurement rollrsquos values and

estimated rollrsquos values of adaptive 1205721-1205731-1205741filter respectively

where 119894 is the current iteration and 119873 is the total number ofsampling data The RMSEs of pitch and roll estimations aredefined as follows

RMSE119901= radic

1

119873

119873

sum

119894 = 1

(120579119894119901minus 120579119894119901)

2

(18)

RMSE119903= radic

1

119873

119873

sum

119894 = 1

(120579119894119903minus 120579119894119903)

2

(19)

where the main purpose of GA and PSO methods is to findthe optimum values of 119892

1119901and 119892

1119903which minimize the

RMSE119901and RMSE

119903 respectively

If X119894and X

119894 119894 = 1 2 119873 are real measurement target

position vector values and estimated target position vector

values of adaptive 1205722-1205732-1205742filter the RMSE of target tracking

is defined as

RMSE = radic1

119873

119873

sum

119894 = 1

((119909119894minus 119909119894)2+ (119910119894minus 119910119894)2+ (119911119894minus 119894)2) (20)

where 119909119894 119910119894 and 119911

119894are the components of target position

vector X119894in 119909 119910 and 119911 axes The main purpose of GA and

PSO algorithms is to find the optimum gain value of g2which

minimizes the RMSE

4 Experimental Results

Six scenarios are used to simulate the tracking performanceof ship-borne phased array radar using the proposed adaptive120572-120573-120574 filter for beam pointing error compensation and targettracking in three-dimensional space

Case 1 (roll and pitch estimations using GA based adaptive120572-120573-120574 filter and measured data) The roll and pitch datarecorded by a gyroscope of the ship were used in the exper-iments The total number of data is 1000 and the samplingtime is 01 sec The angle variation of roll and pitch signalsmeasured by ship gyroscope are shown in Figure 7 whichwillresult in the beam pointing errors of phased array antennaTheGA based adaptive 120572-120573-120574 filter is used to estimate the rolland pitch angles Figures 8(a) and 8(b) show that the conver-gent time of roll and pitch angles are 3 sec and 5 sec respec-tively under simulated sea states 2 and 3 and measured dataThe shiprsquos roll and pitch signals are simulated according to (3)and (4) with roll and pitch angle parameters of sea states 2and 3 listed in Table 1 and standard deviation of 001 Itconcludes that the proposed GA based adaptive 120572-120573-120574 filter issuitable to compensate the ship motion As shown in Table 1the period of sea state 2 is longer than sea state 3 Thereforethe convergent RMSE (about 019∘ for roll about 01∘ forpitch) of sea state 2 is less than sea state 3 (about 029∘ for rollabout 022∘ for pitch) because the roll and pitch signals withshorter period in sea state 3 have rapid oscillations which aremore difficult to be predicted precisely

Case 2 (tracking linear moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Assuming the ship is not affected by thesea waves (antenna beam pointing error is zero) the trackingperformance of GA based adaptive 120572-120573-120574 filter for a straightflight target trajectory is simulated The initial position ofradar is (0 0 0) The ship moves with a speed of 10msec in

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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Chemical EngineeringInternational Journal of Antennas and

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DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 9

0 100 200 300 400 500 600 700 800 900 1000

0

02

04

06

08

Pitc

h an

gle (

deg)

Time (10minus1 s)

minus12

minus08

minus06

minus04

minus02

minus1

(a)

0

1

2

3

Roll

angl

e (de

g)

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

minus1

minus2

minus3

minus4

(b)

Figure 7 Demonstrating angle variation of (a) pitch and (b) roll signals measured by gyroscope

0

005

01

015

02

025

03

035

04

RMSE

(deg

)

Sea 3Sea 2Measurement

0 100 200 300 400 500 600 700 800 900 1000Time (10minus1 s)

(a)

0

005

01

015

02

025

03

035

04RM

SE (d

eg)

Sea 3Sea 2Measurement

Time (10minus1 s)0 100 200 300 400 500 600 700 800 900 1000

(b)

Figure 8 RMSE of (a) roll and (b) pitch under simulated data of sea states 2 and 3 and measured data

the 119884 axial direction The linear path equation of the ship isgiven by

X119904(119909119905 119910119905 119911119905) = X0119904+ V119904119905 (21)

where 119905 = 1 2 3 1000 sec V119904is ship velocity vector

and X0119904is initial ship position vector in three-dimensional

spaceThe radar position is updated every secondThe initialposition of flying target is (minus74840 minus129620 9100)m whichis about 150 km from the radar The flight speed of target is300msec (119883 axial velocity of 150msec and 119884 axial velocityof 260msec) The target location is updated every secondThe linear target trajectory equation is

X119898(119909119905 119910119905 119911119905) = X0119898

+ V119898119905 (22)

where V119898

is the target velocity vector and X0119898

is theinitial target position vector in three-dimensional space Thetracking accuracy of GA based adaptive 120572-120573-120574 filter for astraight flight target is shown in Figure 9 where the trajectoryestimation error is calculated by

119864119905= radic(119909

119905minus 119909119905)2+ (119910119905minus 119910119905)2+ (119911119905minus 119905)2 (23)

where (119909119905 119910119905 119911119905) and (119909

119905 119910119905 119905) denote the real target location

and estimated target location respectively It shows that theGA based adaptive 120572-120573-120574 filter converges to less than about2m after 5 sec

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Submit your manuscripts athttpwwwhindawicom

VLSI Design

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Shock and Vibration

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Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

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Electrical and Computer Engineering

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Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 International Journal of Antennas and Propagation

Table 2 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0574 06399 0652 06904 06537 0674 06286 06313 06044 053921198921199011

0629 06664 0622 07756 0664 07399 0663 06652 06816 064981198922

0285 05827 03282 03851 00054 03480 02916 03736 0547 04864

Table 3 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0557 05986 0357 05345 06142 05774 05544 0529 04592 056261198921199011

0612 05898 04129 05675 05286 03953 05565 06199 05214 062851198922

0326 007 01136 0146 0098 02365 021 02547 02474 01245

0 20 40 60 80 100 1200

20

40

60

80

100

120

140

160

180

Time (s)

Et

(m)

Average estimation error = 15176m

Figure 9 Estimation error for linear target trajectory without beampointing error

Case 3 (tracking circular moving target in three-dimensionalspace using GA based adaptive 120572-120573-120574 filter (without beampointing error)) Figures 10(a) and 10(b) show the simulationscenario of beam pointing error compensation for circularmaneuvering air target and linear moving ship The shiphas the same linear moving speed and path equation asCase 2The flying target is parallel to the119883119884 plane the initialposition of flying target is (8885 4590 9100)m the velocity is300msec and the angle (120579

0) between the initial position and

the119883-axis is 30∘The trajectory of the flight vehicle is a circlewhich has a radius of 10 km The center of circular trajectoryis 119884-axis The rotational cycle time of the flight target is 209seconds and the total observation time is 419 sec

The circular trajectory equation is

X119898(119909119905 119910119905 119911119905) = (119903 cos (120579

0minus 120596119905) 119903 sin (120579

0minus 120596119905) 9100) (24)

where 119905 = 1 2 3 418 419 sec 119903 is the circular flight radiusin meter 120579

0is the initial angle between the flight vehicle

and the 119883-axis 120596 is the angular speed of flight vehicle Theaccuracy of GA based adaptive 120572-120573-120574 filter for tracking acircular flight target is shown in Figure 11 where theGAbasedadaptive 120572-120573-120574 filter converges to less than 3m after 8 secTherefore using the GA based adaptive 120572-120573-120574 filter to trackthe circular trajectory target has the larger average estimationerror and longer convergent time than linear trajectory target

Case 4 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand measured data (with beam pointing error)) The mea-sured roll and pitch signals shown in Figure 7 are used toevaluate the performance of GAPSO based adaptive 120572-120573-120574filter for estimating the roll and pitch angles and tracking thelinear trajectory targetThe beamof 6times6 planar array antennais steered to track the linear trajectory target The samplingfrequency is set as 10Hz Therefore the shiprsquos roll and pitchsignals are sampled 10000 points within 1000 seconds Asthe scenario described in Case 2 the ship is along the 119884

axis of 119883119884 plane in the earth coordinates the ship speedis 10msec and the flying target speed is 300msec Theflying target is parallel to the 119883119884 plane 91 km above theground and the angle between the flight direction and the119884-axis is 120∘ The largest reconnaissance distance of ship-borne radar is 150 km When the air target is flying withinthe radar detection range the beampointing error estimationand linear trajectory target tracking of ship-borne phasedarray radar are simulated

Figure 12 is a simulation flow chart used to verifythe effect of beam pointing error compensation on thetracking accuracy of adaptive 120572-120573-120574 filter Tables 2 and 3demonstrate the optimal gain parameters of adaptive 120572-120573-120574 filter for roll 119892

1199031 pitch 119892

1199011 and target tracking 119892

2

obtained by the GA and PSO methods respectively thatminimizes the RMSE Figures 13ndash17 show the estimationerrors for five different gain values which are summarizedin Table 4 It demonstrates that the greater gain value(119892 = 01 075 and 09) will generate the greater error andneed the longer convergence time The estimation errors

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 11

Table 4 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

05392sim06904 0357sim06142 01 075 091198921199011

0622sim07756 03953sim06285 01 075 091198922

0054sim05827 007sim03259 01 075 09Average 119864

119905(m) 14781 163224 169663 534804 3991188

Table 5 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0543 05653 05424 05455 05665 056 058 0532 05414 053851198921199011

0464 0484 04459 0422 04667 04654 04509 04459 04308 046511198922

0161 00929 01937 00865 00241 00609 01174 01897 01713 01727

Y

X

Z

(a)

Y

X10 km

2730∘

(b)

Figure 10 Simulation scenarios of Case 3 for (a) 3D and (b) top view diagrams

0 10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

Time (s)

Et

(m)

Average estimation error = 24905m

Figure 11 Estimation error for circular target trajectory without beam pointing error

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

12 International Journal of Antennas and Propagation

Generate the target detection data

Using GA or PSO

to track the target

Generate beam steering angle

Measure or simulate roll and pitch angles

Using GA or PSO

roll and pitch angles

Simulate beam pointing error compensation

Verify tracking accuracy ofGAPSO adaptive 120572-120573-120574 filter

120572-120573-120574 filter 120572-120573-120574 filter to estimate

Figure 12 Flow chart of tracking performance simulation forCase 4experiment

0 100 200 300 400 500 600 700 800 900 10000

50

100

150

200

250

300

Time (s)

Et

(m)

Average estimation error = 14781m

Figure 13 Estimation error of GA based adaptive 120572-120573-120574 filter

presented in Figures 16 and 17 are too large to be appli-cable when the gain values are equal to 075 and 09The GA and PSO have smaller errors compared with thefixed gain values and the GA obtains the best estimationaccuracy The estimation errors of GA and PSO meth-ods were found to be close From Figures 13 14 and 15we can observe that there is a peak value that occurredat the time duration about 510 to 515 seconds becausethe horizontal beam direction of ship-borne phased array

0

50

100

150

200

250

300

Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 163224 m

Figure 14 Estimation error of PSO based adaptive 120572-120573-120574 filter

0

20

40

60

80

100

120

140

160

180

200Et

(m)

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 169663 m

Figure 15 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

radar is steered according to the linear trajectory targetwhich is flying through sky just above the ship at thetime instant of 510 second As shown in Figure 18 thehorizontal beam steering angle of ship-borne phased arrayradar at about the time of 510 second is changing fromdescending into ascending When the GA based adaptive120572-120573-120574 filter tracks the linear trajectory target the esti-mation error values are less than 30 meters or less ifthe peak estimation errors during these 5 seconds areignored

Case 5 (estimating roll and pitch angles and tracking lineartrajectory target using GAPSO based adaptive 120572-120573-120574 filterand simulated data (with beam pointing error)) The shiprsquosroll and pitch signals are simulated according to (3) and (4)with parameters of sea state 2 listed in Table 1 and standard

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 13

0 200 400 600 800 10000

500

1000

1500

Time (s)

Et

(m)

Average estimation error = 534804 m

Figure 16 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0 200 400 600 800 10000

1000

2000

3000

4000

5000

6000

7000

8000

9000

Time (s)

Et

(m)

Average estimation error = 3991188 m

Figure 17 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

deviation of 001 The simulated shiprsquos roll and pitch signalsare applied for five different methods to estimate the gain ofGAPSO based adaptive 120572-120573-120574 filter The total observationtime is 1000 sec and sampling time is 01 sec The simulationreplicates 10000 times The input samples are updated byone new sample for the next iteration The other simulationscenario and parameters are the same asCase 4 Tables 5 and 6demonstrate the optimal gain parameters of adaptive 120572-120573-120574filter for roll gain 119892

1199031 pitch gain 119892

1199011 and target tracking gain

vector g2obtained by the GA and PSOmethods respectively

that minimizes RMSE Figures 19 20 21 22 and 23 showthe estimation errors for five different gain values which aresummarized in Table 7 The estimation error of GA basedadaptive 120572-120573-120574 filter for tracking a linear trajectory targetis shown in Figure 19 where the convergence time of GA

0 100 200 300 400 500 600 700 800 900 1000Time (s)

0

05

1

15

2

25

Azi

mut

h ste

erin

g an

gle (

rad)

With beam pointing errorWithout beam pointing error

Figure 18 Azimuth beam steering angle of ship-borne phased arrayradar

0

10

20

30

40

50

60

70

80

90

100

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 160756 m

Et

(m)

Figure 19 Estimation error of GA based adaptive 120572-120573-120574 filter

based adaptive 120572-120573-120574 filter is about 3 sec and the averageestimation error is about 160756m It concludes that theexperimental results in Case 5 have the same trend as inCase 4 therefore the simulated data can be used to observethe effect of shipmotion on the tracking performance of ship-borne phased array radar under different sea state conditionsThe convergence time and average estimation error of GAbased adaptive 120572-120573-120574 filter are better than themethod used in[6] where the tracking error of AEKF converges to less thanabout 20m at 550 iterations (seciteration) and the estimationerror will remain within the range of about 20m when thebeam pointing error is compensated by FBFN controller

Case 6 (estimating roll and pitch angles and tracking circularmoving target using GA based adaptive 120572-120573-120574 filter and

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

14 International Journal of Antennas and Propagation

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 187684 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 20 Estimation error of PSO based adaptive 120572-120573-120574 filter

0 100 200 300 400 500 600 700 800 900 1000Time (s)

Average estimation error = 199594 m

0

10

20

30

40

50

60

70

80

90

100

Et

(m)

Figure 21 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

measured data (with beam pointing error)) As the scenariodescribed in Case 3 the measured roll and pitch signalsshown in Figure 7 are used to evaluate the performance ofGA based adaptive 120572-120573-120574 filter for estimating the roll andpitch angles and tracking the circular trajectory target Theestimation errors of 120572-120573-120574 filter with fixed gains of 119892

1199031=

1198921199011

= 1198922= 01 and 119892

1199031= 1198921199011

= 1198922= 075 are shown in

Figures 24 and 25 respectivelyThe average estimation errorsfor fixed filter gains of 01 and 075 are about 41638m and572386m respectively The optimal roll gain 119892

1199031 pitch gain

1198921199011 and target tracking gain 119892

2of adaptive 120572-120573-120574 filter

estimated by the GA method during the observation timeof 419 sec are listed in Table 8 The estimation error of GAbased adaptive 120572-120573-120574 filter for tracking a circular flight targetis shown in Figure 26 where the convergence time of GAbased adaptive 120572-120573-120574 filter is about 5 sec and the average

0

500

1000

1500

0 200 400 600 800 1000Time (s)

Average estimation error = 1080515 m

Et

(m)

Figure 22 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 200 400 600 800 1000Time (s)

Average estimation error = 4279449 m

Et

(m)

Figure 23 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

09

estimation error is about 36667m It concludes that the GAbased adaptive 120572-120573-120574 filter can greatly improve the trackingaccuracy and convergence time of the conventional 120572-120573-120574filter for tracking the circular trajectory target

5 Conclusions

This paper proposed an intelligent beam pointing error com-pensationmechanism for ship-borne phased array radarTheGA based adaptive 120572-120573-120574 filter estimates the roll and pitchangles of the ship moving in the sea and thus compensatesfor the antenna beam pointing error in order to enhance thetracking accuracy of phased array radar system

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of Antennas and Propagation 15

Table 6 Estimated gain parameters for PSO based adaptive 120572-120573-120574 filter

Time (sec) 1sim100 101sim200 201sim300 301sim400 401sim500 501sim600 601sim700 701sim800 801sim900 901sim10001198921199031

0366 03645 0532 04519 05151 05002 04797 05254 04146 053511198921199011

0366 03647 03634 03644 03643 0363 03644 03684 03643 036421198922

0125 01271 02286 02166 01169 01799 01735 01564 00767 01346

Table 7 Estimation errors for different gains

Calculated by GA Calculated by PSO Fixed gain1198921199031

0459sim0583 03645sim05352 01 075 091198921199011

0498sim0667 0363sim03684 01 075 091198922

00586sim01697 00367sim02291 01 075 09Average 119864

119905(m) 160756 187684 199594 1080515 4279449

Table 8 Estimated gain parameters for GA based adaptive 120572-120573-120574 filter

Time (sec) 1sim42 43sim84 85sim126 127sim168 169sim210 211sim252 253sim294 295sim336 337sim378 379sim4191198921199031

0619 04996 05316 0644 05052 05568 05316 04459 05092 059241198921199011

0582 06727 07201 04913 04168 07169 05983 02036 05858 06211198922

0214 02215 01612 02474 01822 0157 02073 01211 02278 01074

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 41638 m

Figure 24 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

01

Six experimental cases are conducted to verify the perfor-mance of ship-borne phased array radar using the proposedGA based adaptive 120572-120573-120574 filter In Case 1 the GA basedadaptive 120572-120573-120574 filter is used to estimate the roll and pitchangles with the simulated and measured roll and pitchsignals respectively The simulation results show that themeasurement data has the minimum estimation error theestimation error in the sea state 2 is the second and theestimation error in the sea state 3 is the worst Cases 2 and 3show that when the GA based adaptive 120572-120573-120574 filter is appliedto estimate the beam pointing error and to track the targetin a ship-borne phased array radar the circular trajectory

0 50 100 150 200 250 300 350 400 450Time (s)

0

500

1000

1500

Average estimation error = 572386 m

Et

(m)

Figure 25 Estimation error of 120572-120573-120574 filter with 1198921199031

= 1198921199011

= 1198922=

075

target will be tracked with the larger average estimation errorand longer convergence time than linear trajectory targetTheexperimental results ofCases 4 and 5demonstrate thatwhen alinear trajectory target is tracked by ship-borne phased arrayradar the average estimation error and convergence time ofship-borne phased array radar using the GA based adaptive120572-120573-120574 filter are better than PSO based adaptive 120572-120573-120574 filterand fixed filter gain 120572-120573-120574 filter The convergence time andtracking accuracy of ship-borne phased array radar usingthe proposed GA based adaptive 120572-120573-120574 filter are superiorto the AEKF significantly In conclusions the proposed GAbased adaptive 120572-120573-120574 filter is a real time applicable algorithm

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

16 International Journal of Antennas and Propagation

0

20

40

60

80

100

120

140

160

180

200

Et

(m)

0 50 100 150 200 250 300 350 400 450Time (s)

Average estimation error = 36667 m

Figure 26 Estimation error of GA based adaptive 120572-120573-120574 filter

whose accuracy is good with relatively less complexity Inaddition the roll and pitch signalsmeasured in real operationenvironments can be replaced with the simulated roll andpitch signals to test all the relevant properties of practical shipmotion compensation and air target tracking for ship-bornephased array radar

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by research grants fromNational Science Council China (NSC 102-2218-E-155-001)

References

[1] Z Fang and Q Rundong Ship-borne Phased Array RadarMotion Compensation System Engineering and ElectronicTechnologies 1998

[2] L J Love J F Jansen and F G Pin ldquoCompensation of waveinduced motion and force phenomenon for ship based highperformance robotic and human amplifying systemsrdquo OakRidge National Laboratory ORNLTM-2003233 2003

[3] R J Hill Motion Compensation for Ship Borne Radars andLidars US Department of Commerce National Oceanic andAtmospheric Administration Office of Oceanic and Atmo-spheric Research Earth System Research Laboratory PhysicalSciences Division 2005

[4] T I Fossen and T Perez ldquoKalman filtering for positioning andheading control of ships and offshore rigsrdquo IEEEControl SystemsMagazine vol 29 no 6 pp 32ndash46 2009

[5] S Kuchler C Pregizer J K Eberharter K Schneider and OSawodny ldquoReal-time estimation of a shiprsquos attituderdquo in Proceed-ings of theAmericanControl Conference (ACC rsquo11) pp 2411ndash2416San Francisco Calif USA July 2011

[6] J Mar K C Tsai Y T Wang and M B Basnet ldquoIntelligentmotion compensation for improving the tracking performanceof shipborne phased array radarrdquo International Journal ofAntennas and Propagation vol 2013 Article ID 384756 14pages 2013

[7] T Dirk and S Tarunraj ldquoOptimal design of 120572-120573-(120574) filtersrdquo inAmerican Control Conference vol 6 pp 4348ndash4352 ChicagoIll USA June 2000

[8] D Tenne and T Singh ldquoCharacterizing performance of 120572-120573-120574filtersrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 3 pp 1072ndash1087 2002

[9] T E Lee J P Su and K W Yu ldquoParameter optimization fora third-order sampled-data trackerrdquo in Proceedings of the 2ndInternational Conference on Innovative Computing Informationand Control p 336 Kumamoto Japan September 2007

[10] Y H Lho and J H Painter ldquoA fuzzy tuned adaptive Kalmanfilterrdquo in Industrial FuzzyControl and Intelligent System pp 144ndash148 December 1993

[11] N Ramakoti A Vinay and R K Jatoth ldquoParticle swarm opti-mization aided Kalman filter for object trackingrdquo in Proceedingsof the International Conference on Advances in ComputingControl and Telecommunication Technologies (ACT rsquo09) pp 531ndash533 Kerela India December 2009

[12] T-E Lee J-P Su andK-WYu ldquoAparticle swarmoptimization-based state estimation scheme formoving objectsrdquoTransactionsof the Institute of Measurement and Control vol 34 no 2-3 pp236ndash254 2012

[13] D C Law S A McLaughlin M J Post et al ldquoAn electronicallystabilized phased array system for shipborne atmospheric windprofilingrdquo Journal of Atmospheric and Oceanic Technology vol19 no 6 pp 924ndash933 2002

[14] C A Balanis AntennaTheory Analysis and Design JohnWileyamp Sons New York NY USA 3rd edition 2005

[15] J Mar and Y R Lin ldquoImplementation of SDR digital beam-former for microsatellite SARrdquo IEEE Geoscience and RemoteSensing Letters vol 6 no 1 pp 92ndash96 2009

[16] R C Eberhart and Y Shi Computational Intelligence Conceptsto Implementations ElsevierMorgan Kaufmann 2007

[17] K Y Lee and J-B Park ldquoApplication of particle swarm optimi-zation to economic dispatch problem advantages anddisadvan-tagesrdquo in Proceedings of the IEEE PES Power Systems Conferenceand Exposition (PSCE rsquo06) pp 188ndash192 Atlanta Ga USANovember 2006

[18] R C Eberhart and J Kennedy ldquoNew optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of