Trajectory estimation based on extended state observer ... · ESO is used to linearly approximate and ascertain a non-linear and uncertain plane by two channel compensator. 4.1 Derivation

ABSTRACTThis paper intends to develop a target trajectory estimation algorithm with application to the ballistic target estimation in the terminal phase. The proposed design is based on the application of a second-order extended state observer (ESO) technique using target information acquired from the seeker to estimate the trajectory of manoeuverable ballistic targets. Satisfactory results have been received while applying the design in estimation of either two-dimenional or three-dimentional target evasive acceleration via computer simulation.

1.0 INTRODUCTIONTrajectory estimations of ballistic missiles and the control system design for anti-missiles have been receiving a great deal of attention as part of the defense industry(1-4) in the past two decades.

Recently, advanced tactical ballistic missiles (TBMs) already possess the capability of evasion during the reentry phase which makes the trajectory estimation either from on-board radars or from ground-based radars more difficult than before.

In the aspect of trajectory estimations, α-β filter(5) is useful for radar measurement. Kalman filter (KF) and extended Kalman filter (EKF)(6,7) have the abilities of trajectory estimation of aircraft with noise by using recursive algorithms which have been widely applied in navigation, fixed sonar, and industrial automations. In Ref. 8, the authors have presented the results of comparison for estimation performance of re-entry targets using the EKF, statistical linearization, particle filtering, and unscented Kalman filter (UKF). The results favored the EKF.

The AeronAuTicAl JournAl AugusT 2015 Volume 119 no 1218 1017

Paper No. 4004. Manuscript received 31 May 2013, 1st revised version received 10 October 2013, 2nd revised version received 17 June 2014, 3rd revised version received 9 December 2014, accepted 16 January 2015.

Trajectory estimation based on extended state observer with Fal-filterC-L [email protected] Hsieh and Y-P LinDepartment of Electrical Engineering National Chung Hsing University Taichung Taiwan

1018 The AeronAuTicAl JournAl AugusT 2015

However, in real-world systems, there are modelling errors which may lead to estimation error in Kalman filtering design. In addition, complicated matrix-based computation may require much computation time. Besides, the information received from the seeker is limited, one thus needs a more simple and efficient way to estimate the evasive target trajectory.

Owing to the less dependence on model information, strong capabilities for disturbance rejection and simple computation, the use of extended state observer (ESO) for on-line estimating the total uncertainties, which lumps the internal nonlinear and uncertain dynamics and the external disturbance, has been received much attention. In the literature, the authors in Ref. 9 designed a three-dimentional guidance utilized in hit-to-kill interceptor based on ESO. In Refs 10, 11, the ESO was used to estimate the evasive acceleration of TBM and design the guidance law of interceptor.

Accuracy of ESO is affected by the observer gain, because of the disturbance, uncertainty, and measurement noise(11). Recently, the method of expanding first-order state to ESO was considered(12)

while the authors of Ref. 13 used the method of expanding the filter state to ESO. In Refs 14, 15, the problem was solved based on Fal filter and feedback control. The approaches presented in Refs 12,13 need to increase the order of ESO, which result in the increase of setting param-eters. The method in Ref. 14 doesn’t need to increase the order of ESO that simplifies parameter adjustment while maintaining estimation performance. However, the case study given in Ref. 14 for ESO design was actually conducted in the planar environment, which means that pitch and yaw dynamics are assumed to be fully decoupled. However, ignoring the coupling effect in the three-dimensional space may induce significant performance degradation of state estimation when the relative distance between target and interceptor decreases gradually.

This paper intends to develop an improved design of trajectory estimation based on the second-order ESO with Fal filter in the three-dimensional space. In addition to consider the coupling effect of the pitch and yaw dynamics, this research also investigates the effect of parameter changes in ESO design to estimation performance. The results provide useful guidelines while designing ESOs.

Figure 1. Geometric relationship between TBM and seeker.

hsieh et al TrAJecTory esTimATion bAsed on exTended sTATe obserVer wiTh fAl-filTer 1019

2.0 SYSTEM DESCRIPTION

2.1 TBM model in re-entry phase

Assuming a TBM with the co-ordinates (XR, YR, ZR), and the missile seeker is located at the origin Or of the inertial co-ordinate system as illustrated in Fig. 1.

Let the TBM be regarded as a point mass which has a constant weight Wt influenced by drag and gravity in reentry phase so that the dynamic equation can be expressed as

. . . (1)

where Vtx, Vty and Vtz are, respectively, the components of target’s velocity along X, Y and Z axes; atx, aty and atz are the respective components of evasive acceleration; pitch angle γt, yaw angle ψt, and ballistic coefficient βt are defined, respectively, as

. . . (2)

where Sref, Wt and CtD0 denote the reference area, weight, and zero-lift drag coefficient of the ballistic target, respectively.

2.2 Derivation of the equations of motion

Derivation of equations of motion for the pitch and yaw dynamics in LOS co-ordinates is standard. One is referred to Fig. 2 for illustration of the inertial co-ordinate system and LOS co-ordinate system, where the direction cosine matrix (DCM) relating the inertial co-ordinates (Ol Xl Yl Zl) to the LOS co-ordinates (OL XL YL ZL) is given by

. . . (3)

The distance vector between missile and target is denoted as rL = [r 0 0]T where r is the relative distance in the LOS co-ordinates. The acceleration vectors of the missile and target are expressed as and , respectively. The angular velocity vector along X, Y, and Z axes is denoted as ωL = [ωXL

ωXL ωXL

] T and the matrix of rotation (8) is given by

v v g a

v v g a

v

txt

tt t tx

tyt

tt t ty

2

2

2

2

Cos Cos

Cos Sin

ttzt

tt tz

v g g a

2

2Sin

ttz

tx ty

ttx

ty

t vv v

t vv

Tan

Tan

1

2 2

1ψ

ttt

tref tD

WS C

0

XYZ

L

L

L

L L L L L

L L

Cos Cos Cos Sin SinSin Cos

Sin

0

L L L L L

I

I

I

XYZCos Sin Sin Cos

CIL ( , ) L L

I

I

I

XYZ

aLm

xm

ym

zm Ta a a

L L L= [ ] , , aL

txt

yt

zt Ta a a

L L L= [ , , ]


. . . (4)

Considering the target kinematics one can obtain

. . . (5)

i.e.

After conducting some calculations, one can get

. . . (6)

Clearly the two terms are coupled. From Equation (3), the angular velocity vector ωL can be described in terms of LOS angles as

. . . (7)Cos 0 Sin 0 Cos Cos Cos Sin Sin 0

0 1 0 Sin Cos 0 0Sin 0 Cos 0 Sin Cos Sin Sin Cos

Sin

Cos

L

L

L

x

L y

z

L L L L L L L

L L L

L L L L L L L L

L L

L

L L

ω

Figure 2. Illustration of inertial co-ordinate system and LOS co-ordinate system.

L

z y

z x

y x

L L

L L

L L

0

0

0

aLt

Lm

L L L L L L L L

Lm

L L L L L

a r r r ra r r

( ) 2

rr rL L L 2

a

a

a

a r r r

a r rxt

yt

zt

xm

z y

ym

z

L

L

L

L L L

L L

2 2

xx y z

zm

y x z y

L L L

L L L L L

r

a r r r

2

2

ω

z zyt

ym

x y

y yzt

zm

L L

L L

L L

L L

L L

rr

a ar

rr

a ar

2

2

,

xx zL L


Substituting Equation (7) into Equation (6) yields the equations of motion in the LOS co-ordinates as follows

. . . (8)

. . . (9)

It is seen that the pitch and yaw dynamics are closely coupled due to the existence of two coupling terms 2ψ. L θ

.L Tan θ

.L and ψ. 2

L Sin θ.L Cos θ

.L .

3.0 NOISE ANALYSISConsider the missiles detecting target position via an active seeker. During engagement, an active seeker might be disturbed by ambient noises including glint noise, fading noise, and receiver noise during flight.

Fading noise is independent on the relative distance between missile and target which is usually at high frequencies. It can be assumed to be a Gaussian white noise, with its standard deviation denoted by σfade (rad), passing through a low-pass filter.

Effect of receiver noise is proportional to the relative distance between missile and target. The receiver noise of an active seeker is proportional to the square of the relative distance between missile (interceptor) and target. It appears at a higher frequency than the operational frequency of the guiding system. Suppose that the standard deviation is denoted as σreceiver (rad).

Effect of glint noise increases with decrement of the relative distance between missile and target. Its frequency is usually higher than the operational frequency of the guiding system. Usually, it is generated by a Gaussian white noise source passing through a low-pass filter with its standard deviation denoted σglint (rad).

Summation of the standard deviations of noises detected by the active seeker can be expressed as Ref. 3

. . . (10)

where r0 is a reference distance related to the signal intensity of the receiver. For the ballistic target, assume that its RCS is comparatively smaller while comparing to regular jet fighters, the effect due to glint noise is thus ignorable. Therefore, Equation (10) is simplified to

. . . (11)

4.0 ESO AND ESO FILTERESO is used to linearly approximate and ascertain a non-linear and uncertain plane by two channel compensator.

4.1 Derivation of ESO equations

ESO is a new type of observer(11) which is designed for simultaneously observing the states, uncertainty of the system, and the external disturbance.

L L

yt

ym

LL L L

rr

a arL L

2 2Cos

Tan

L Lzt

zm

L L Lrr

a ar

L L

2 2Sin Cos

angle

glintfade

receiver

rr

r2

2

22

2

04

4

angle fade

receiverrr

2 22

04

4


Consider an nth-order nonlinear SISO system with external disturbance and uncertainty, as shown below

. . . (12)

where f (x, x., ..., x(n–1), t) is an unknown function of the plant and w(t) is unknown disturbance input.Let the states of the system be

. . . (13)

where x(n+1), the n+1th state, is the extended state of the system. Equation (12) can be rewritten as

. . . (14)

where h(t), the derivative of f (x, x., ..., x(n–1), t) + w(t), is an unknown function. Based on Equation (14), the corresponding state-space model can be expressed as

. . . (15)

where

,

and E = [0 0 ... 0 1]T. From which, we can design a state feedback estimator for the closed-loop system as follows

. . . (16)

where the observer gain L = [β1 ... βn+1]T. That is

. . . (17)

x f x x x t w tn n( ) ( )( , , , , ) ( ) & L 1

x t x tx t x t

x t x tx t f x x x

nn

nn

1

2

1

1

( ) ( )( ) ( )

( ) ( )

( ) ( , , ,

( )

(

&

M

& L 11) , ) ( )t w t

&

&

M

&

&

&

x t x tx t x t

x t x tx t x tx

n n

n n

n

1 2

2 3

1

1

1

( ) ( )( ) ( )

( ) ( )( ) ( )

(( ) ( )t h t

x t Ax t Eh ty Cx( ) ( ) ( )

A

0 1 0 00 0 1 0

0 0 0 10 0 0 0

L

L

M M M O M

L

L

ˆ ˆ( ) ( ) ( ( ))( )

x t Ax t L y Cx ty Cx t

1 2 1 1 1

2 3 2 1 1

1 1 1

1 1 1 1

ˆ ˆ ˆ( )

ˆ ˆ ˆ( )

ˆ ˆ ˆ( )

ˆ ˆ( )n n n

n n

x x x x

x x x x

x x x x

x x x

C

10

00

T


Let the state estimate zi(t) = x̂i(t) and estimation error ei(t) = zi(t) – xi(t), i = 1, ..., n+1 so that Equation (17) can be converted into the following error dynamic equations:

. . . (18)

It’s easy to see that if the terms β1(ei) ... βn+1(ei) satisfy

. . . (19)

then the system described in Equation (18) would be stable at the origin with respect to the bounded h(t).

It meant that as long as one chooses β1(ei) ... βn+1(ei) appropriately, the states in Equation (14) can be well tracked by the states in Equation (17), i.e.

. . . (20)

Referring to Equation (18) one can obtain the general form of ESO as below

. . . (21)

4.2 ESO with Fal filter

The accuracy of ESO is affected by the observer gain, because of the disturbance, uncertainty, and measurement noise(8). Incorporation of a low-pass filter can help to pre-filter measurement noise. However, it also brings extra phase lag which may deteriorate estimation performance. The problem can be resolved by eliminating measurement noise using the ESO with Fal filter(14).Consider a system with measurement noise given as below

. . . (22)

where the plant model function f (x, x., ..., x(n–1), t) is unknown, wt is an unknown disturbance input, and vy is measurement noise.

For the system of Equation (22), the ESO is designed as

x f x x x t w ty x v

n n

y

( ) ( )( , , , , ) ( )

& L 1

& & &

& & &

e z x z x z xe e t

e z x z x

1 1 1 2 2 1 1 1

2 1 1

2 2 2 3 3

( )( ( ))

2 1 1

3 2 1

1 1 0 1

( )( ( ))

( ) (

z xe e t

e z x z x b u zn n n n n n

M

& & & xxe e t

e z x x z xh

n n

n n n n n

1

1 1

1 1 1 1 1 1 1

)( ( ))

( )

& & & &

(( ) ( ( ))t e tn

1 1

e e e i ni i1 1 10 0 0 0 1 1 ( ) , , ( ) , , ,

z t x t z t x t z t x tn n n n1 1 1 1( ) ( ), , ( ) ( ), ( ) ( )

e z x z yz z e tz z e t

z zn n n

1 1 1 1

1 2 1 1

2 3 2 1

1

&

&

M

&

( ( ))( ( ))

(ee tz h t e tn n

1

1 1 1

( ))( ) ( ( ))&


. . . (23)

and the filter fal_filter (y, kf, af, δf) is defined as

. . . (24)

where

with 0 ≤ a ≤ 1 and 0 < δ. The function fal exhibits vital filtering characteristics on noise input. When a = 1 it acts as a

linear gain of 1 and it becomes a saturation function when a = 0. For the latter, when │e│ > δ the nonlinear function fal fixes at +1 or –1, whereas for │e│ ≤ δ, it acts like a linear gain of 1/δ. Similarly, for a ∈ (0,1) it acts as a linear gain 1/δ1–a for │e│ ≤ δ which decreases with the decrement of a implying a better noise filtering effect. On the other hand, the equivalent gain within │e│ ≤ δ increases with the decreasing value of δ. This gives rise to better estimation performance.

4.3 Estimation of evasive acceleration

The equation of inertial motion has been presented in Equation (8) for the LOS yaw plane and in Equation (9) for the pitch plane. And the estimated LOS rate θ

.^

L = θ.L = σ.angle and ψ

.^

L = ψ. L = σ.angle where θ

.L and ψ. L are the nominal angle rates. Let θ

.^

L and ψ.^

pass through a respective Fal filter. That is

. . . (25)

and

. . . (26)

The evasive accelerations atyl and at

xl are regarded as uncertain terms and expanded as two new states.Let the states x1 = –r θ

.^

L0 and x3 = r ψ.^

L0Cos θ̂L, then Equation (8) and Equation (9) can be written as

. . . (27)

y x vy fal filter y k ae z t y

z t z t

y

f f f

g

0

1 1 0

1 2 1

_ ( , , , )

( )( ) ( )

& eez t z t fal e a

z t z t fal e a

g

n n gn

1

2 3 2 1 2

1 1

&

M

&

( ) ( ) ( , , )

( ) ( ) ( ,

nn

n g n nz t fal e a, )

( ) ( , , )( )

&

1 1 1 1

x k fal e ay xe y x

f f f

( , , )

0

fal e ae e ee e

a

a

( , , )( ),

,

sign if

if 1

0ˆ ˆ( , , , )L L f f fFalFilter k a

0ˆ ˆ( , , , )L L f f fFalFilter k a

21 3

1

ˆTanL L

t mLz z

rx xx a ar


and

. . . (28)

Since r, r., θ̂L, θ̂L0 and ψ.^

L0 are detectable by the seeker, we let the known components

and where and

are the coupling terms and unknown components f2 = atzl and f4 = at

yl .Equations (27) and (28) can be further expressed as

. . . (29)

and

. . . (30)

Expand the unknown term f2 to a first-order state, and assume x2 = wp where wp is unknown. Then Equation (29) becomes . . . (31)

On the basis of the above development, we are able to synthesise a second-order ESO as follows . . . (32)

In Equation (30), let the extended state x.4 = wy where x 4 = f4. We obtain

. . . (33)

and ESO is designed as

. . . (34)

where (ap, δp, β1, β2) and (ay, δy, β3, β4) are parameters of ESO, yp = x1 –r θ̂L0 and yy = x3 = rψ.^

L0Cos θ̂L are detectable by the seeker, the missile lateral acceleration command am

yl and amzl are determined

by the terminal guidance of the missile. In Equation (32) and Equation (34), z1, z2, z3 and z4 are the estimated values of x1, x2, x3 and x4, respectively. Once ep and ey converge, the target evasive accelerations at

zl and atyl are obtained.

Due to the presence of disturbance, uncertainty, and measurement noise, the accuracy of ESO is affected by the parameters sets (ap, δp, β1, β2) and (ay, δy, β3, β4). The function and effect of α

3 1 33

ˆTanL L

t mLy y

rx x xx a ar

21 3

1

ˆTan Lrx xfr

3 1 3

3

ˆTan Lrx x xfr

23

ˆTan L xr 1 3

ˆTan Lx xr

x f f azmL1 1 2

x f f aymL3 3 4

x f f a

x wy x

zm

p

p

L1 1 2

2

1

1

21 3

1 2 1

2 2

ˆTan

( , , )L

p p

mLp z

p p p

e z y

rz zz z e ar

z fal e a

x f f a

x wy x

ym

y

y

L3 3 4

4

3

3

3 1 33 4 3

4 4

ˆTanβ

( , , )L

y y

mLy y

y y y

e z y

rz z zz z e ar

z fal e a


and δ have been depicted previously. In practice, the noisy target information measured by the seeker is post processed via a low-pass filter. However, the guidance filter cannot filter the noise thoroughly. It will cause an extra dynamic lag and may deteriorate estimation performance of ESO.

For ESO design, the following rules can be refereed: the value of βi determines the response speed of the filter. The larger it is, the better tracking performance will be; however, this will also make the system to be easily corrupted by the external noises, and vice versa. Therefore, one can set the values of βi inversely proportional to the interceptor-target relative range. From Equation (23), Equation (32) and Equation (34), it is seen that within the linear operational region, the undamped natural frequency of the linearised second dynamics of the ESO is approximately determined by with the damping ratio . Therefore, one may select β2 > β1 and β4 > β3 and with β2 = β4 for the consistent pitch and yaw dynamics.

Design of δ depends on the estimation error in actual situation. Referring to Equation (32) and Equation (34), setting δ relates to the relative range r for an appropriate setting of the gain. A parameter optimisation algorithm can be used to search for the optimal ESO parameters by minimising the overall estimation error such as genetic algorithm(16).

For an advanced design of ESO, the variable observation gains can be considered. For example, when the interceptor is still far away from the target, a smaller βi will yield better filtering effect to noise corruption. On the other hand, increasing its values when the interceptor gradually approaches the evasive target will bring better estimation to the target evasive acceleration. The design guideline would be beneficial to guidance design for the appropriate strategy of target engagement.

One is also referred to Ref. 16 for the stability analysis of the estimation error dynamics under variable estimation gains.

5.0 CASE STUDYThe simulation study consists of three parts. For the first case, the estimation accuracy with coupling terms and without coupling terms are compared. In this case, the noise effect is ignored, thus it is needless to use a Fal filter. The second case discusses the effect of parameters of Fal filter and ESO. Finally, on the basis of Cases 1 and 2, the simulation study with the coupled pitch and yaw dynamics is considered.

Case 1: Comparison of the estimation results with and without coupling termsThe simulated conditions are set as follows: the missile is at the origin (0,0,0)(m), the missile velocity (Vmx,Vmy,Vmz) = (1,000, 1,000, 2,000) (m/sec), the acceleration of the missile (Amx,Amy ,Amz) = (2,3,–1)(G), the target initial position (X,Y,Z) = = (10,000, 10,000, 20,000) (m), the target velocity (Vtx,Vty,Vtz ) = (–100, –100, –150) (m/sec), and the target evasive acceleration (Atx,Aty) = (–5,–2)(G) from one to five seconds; ESO parameters are set as follows: a = 0.25, δ = 0.01, β01 = 50, β02 = 500, β03 = 50 , and β04 = 500. By the approach given in Ref. 14, if one ignores the coupled dynamics, the estimated and real value of the evasive acceleration in LOS co-ordinate will behave as the green lines appeared in Figs. 3 and 4. However, involving the coupling terms as that considered in this research and with the proposed approach, the results are different as shown with the red lines in Figs. 3 and 4. It’s easily seen that, at the beginning of estimation, the effect of coupling terms is ignorable, however, as with the increasing angular changing rates, the coupling effect become significant and should not be neglected.

Although the increase of LOS angle rates in pitch and yaw planes are not large, as shown in Figs 5 and 6, the estimated states x1 = –rθ

.^

L and x3 = r .ϕ̂ L Cos θ̂L, as shown in Figs 7 and 8, become significant

when multiplying them by the relative distance r, thus the coupling effect should not be treated loosely.

β2 β β1 22


Case 2: Effect of Fal filter and ESO parametersWe estimate the evasive acceleration with measurement noise σangle and ro = 15km. The other simulated conditions are the same as those in Case 1. ESO and Fal filter parameters a = 0.25, and δ = 0.01 are constant. We use different ESO parameters β1, β2 and Fal filter parameters kf in pitch plane, and then observe the influence of measurement noise on these parameters. As the fal function exhibit highly nonlinear characteristics, we examine stability of the resulting estimator design via simulation.

First, we consider the effect of Fal filter. Set β1 = 100, β2 = 500, β3 = 100, β4 =500 and kf = 10. Figs 9 and 10 show, respectively, the ESO design without and with Fal filter when the noise deviation σangle = 0.000707rad with the data measurement rate 0.1s(3) (i.e. the equivalent power spectral density Φσangle

= 5x10–8rad2/Hz ). One is referred to Ref. 3 for characterisation of the noise intensity. It’s evident that the ESO design with Fal filter performs better in the tracking response.

Next, we consider the effect of the gain kf in Fal filter. Remain the same noise intensity. Set β1 =

Figure 3. Estimated and real value of the evasive acceleration in pitch plane.

Figure 4. Estimated and real value of the evasive acceleration in yaw plane.

Figure 5. LOS angle rate in pitch plane. Figure 6. LOS angle rate in yaw plane.


100, β2 = 500 and adjust kf. While choosing kf = 1, 20, 40, 50 the simulation results are illustrated respectively, in Figs 11-14. While kf = 1, the oscillation of estimated values is small, but there is a slight phase lag. When kf is increased to 20, the phase lag is improved and transient behavior of the estimated result is comparatively acceptable. Increasing kf to 40 leads to explicit oscillation in the result. Keeping increment of kf until 50 leads to an unstable response. Which reveals that the admissible kf would be less than 50.

Next, we increase the measurement noise intensity Φσangle = 5x10–6rad2/Hz and remain the

same Fal filter parameter settings. While kf = 20, the measurement noise has a large effect on the estimated result, as shown in Fig. 15. When choosing kf = 1, the noise effect decreases, see Fig. 16; the estimation performance is better than kf = 20. In Equation (24), a larger kf means the faster variation of the output y0. In Equation (32), a larger β2 means the faster variation of z2, and a larger β1 means the slower variation of z1.

Figure 7. Estimated state x1 in pitch plane.

Figure 8. Estimated state x3 in yaw plane. Figure 9. Results of ESO design in pitch plane (a) without Fal filter (b) with Fal filter.


(a) (b)

Figure 13. Estimated and real value of the evasive acceleration while kf = 40.




Figure 10. Results of ESO design in yaw plane (a) without Fal filter (b) with Fal filter.


In brief, the filtering effect will be notable when the Fal filter parameter kf is chosen to be small. However, choosing a smaller kf may lead to a larger phase lag. The function of the ESO parameter β2 is similar to that of kf, whereas, the effect of selection of β1 is contrary to the that of kf.

Case 3: Simulation result based on Cases 1 and 2The simulated conditions of interceptor and target remain invariant as those in Case 1 with exception σfade = 0.0003rad and σreceiver = 0.0001rad. When the interceptor is far away from the target, the measurement noise effect is significant. As the relative distance between both objects decreases, the measurement noise effect is comparatively smaller. Therefore, one has to choose the appropriate parameters of ESO and Fal filter to guarantee estimation performance in the terminal engagement phase. Here, the ESO and Fal filter parameter are chosen as follows: a = 0.25, δ = 0.01, β1 = 100, β2 = 500, β3 = 100, β4 = 500 and kf =10 . Estimation of the target evasive acceleration in pitch and yaw planes are illustrated, respectively, in Figs 17 and 18 which show satisfactory performance of the present approach.



Figure 17. Estimated and real value of the evasive acceleration using the accurate dynamics in pitch

plan.

Figure 18. Estimated and real value of the evasive acceleration using the accurate dynamics in yaw

plan.


6.0 CONCLUSIONSThis research has developed an evasive acceleration estimation scheme for the medium- and higher-tier ballistic targets in the terminal phase. The design is based on a second-order extended states observer for the coupled LOS angle rate dynamics. It is shown that the coupling effect to estimation accuracy would be significant when the interceptor approaches to the target in the terminal phase. Effects of the parameters in ESO and Fal filter have been discussed via a variety of case studies. The case studies have verified applicability of the proposed design.

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Trajectory estimation based on extended state observer ... · ESO is used to linearly approximate and ascertain a non-linear and uncertain plane by two channel compensator. 4.1 Derivation