A Distance Estimation Algorithm for Infrared System Based on Structure Stochastic Jump System Theory

Binghua Zhao System Engineering

Engineering College, Air Force Engineering University

Xi’an, China becomewater2002@yahoo.com.cn

Youli WuDepartment of Aviation Weapon

Engineering Engineering College, Air Force

Engineering University Xi’an, China

wu_youli@126.com

Linhu Zhu System Engineering

Engineering College, Air Force Engineering University

Xi’an, China

Abstract-In order to solve the problem of the distance estimation of the infrared passive system, a distance estimation algorithm is proposed based on the optimal estimation theory of structure stochastic jump system. Considered the rapidity and the limit of the internal storage capacity of the algorithm, a simplified algorithm is derived. And the ability of the algorithm correcting the initial distance deviation is discussed. simulation results show that the algorithm and its simplified algorithm can both accurately estimate the distance between the infrared passive tracker and the infrared target. And the algorithm can quickly correct the initial distance deviation in 1~2 seconds.

Keywords-infrared passive system, distance estimation, structure stochastic jump system, optimal estimation.

I. INTRODUCTION The Infrared passive tracking system can not directly

measure the relative distance between the tracker and the target. But relative distance is necessary for the control and guidance of the infrared space interceptor and the threat estimation of the carrier infrared search system. Specially, most advanced guidance laws are dependent on the relative distance between the missile and the target. A distance estimation algorithm was researched by Xiaoping Shi and Zicai Wang based on a nonlinear observer design method with two canonical forms. But the algorithm is complex[1]. Zhihong Xu, Meng Zheng introduced a algorithm of passive distance measurement technology to measure target distance on infrared search and tracking system. The algorithm used the strong calculation capacity and many measurement parameters of the carrier[2]. Shiyong Liu researched the passive ranging with line-of-sight angle or line-of-sight rate measurement for exoatmosphere interceptor[3]. Xiaohu Wang and Minglan Zhang introduced a distance estimation algorithm based on the distance information included in the inductor voltage of the infrared guide head[4]. But the algorithm depends on the model of the target maneuvering. However, the computing capacity of the built-in computer is limited, due to the special requirements for the volume and the weight. What’s more, the distance estimation is a secondary problem compared with angular coordinate

estimation and target identification in guidance system. So the rapidity of the distance estimation algorithm is important. Because most of guidance laws are robust to the value of distance estimation, the high precision of the distance estimation is not necessary[5,6,7].

In this paper, according to the infrared intensity characteristic of the target, a simple algorithm of the infrared system distance estimation is proposed based on the optimal estimation theory of structure stochastic jump system.

II. THE MODEL OF THE INFRARED SYSTEM DISTANCE ESTIMATION

It is assumed that the moving target radiates infrared energy, and the infrared radiation receiver builds in motionless observer or in the infrared guide head, designed to measure the infrared radiation of the target. The state equation and observation equation can be described as follows:

+⋅=

+Δ⋅−=−

+

kkkk

kkkk

DIE

tVDD

ζ

ξα

1 (1)

Where kD is the relative distance between the missile and the target; kV is the relative velocity; kE is the infrared intensity which the infrared receiver measured, kI is the infrared intensity; α is the atmosphere transmission coefficient; kξ , kζ are centralized Gauss white noise with intensity kG and kQ respectively.

III. THE INFRARED SYSTEM DISTANCE ESTIMATION ALGORITHM BASED ON THE OPTIMAL ESTIMATION THEORY OF

STRUCTURE STOCHASTIC JUMP SYSTEM

Regard the infrared intensity kI as some states, )( kk sI ,

sk ns ,,2,1= ; ks is a sn states Markov chain with transition probability )|( 1 kk ssq + . For the nonlinear function

αψ −= kkkkk DIsD ),( , use the Taylor series expansion at the

point )(~ sDk , the conditional one-step predictor of the state

kD . And only the linear terms are reserved.

))(~)((~)()(~)(),( )1( sDDsDsIsDsIsD kkkkkkkk −−= +−− αα αψ (2)

According to the optimal estimation theory of structure

This work was supported by the National Natural Science Foundation of CHINA ( NO: 60674031 ) General Armaments Department Foundation of CHINA( NO: 6140529) and the Doctorate Foundation of the Engineering College, Air Force Engineering University ( NO.BC06004).

847

1-4244-0737-0/07/$20.00 c©2007 IEEE

stochastic jump system[8,9], the infrared system distance estimation algorithm can be described as follows: Identification function:

=+++ =

s

k

n

skkkk sssp

1111 )(/)()(ˆ αα (3)

=++ =

s

k

n

skkkk ssqspsp

111 )|()(ˆ)(~ (4)

State estimator:

)))(~)()((~

)(~)(~)()(~)(ˆ

11111

1)1(

1111

sDsIEs

sRsDsIsDsD

kkkk

kkkkkα

α

θ

α−+++

−+

++−

++++

−×

−= (5)

)(ˆ)(ˆ)(~)(~

1

111 sDspspsD k

n

skkkk

s

k =

−++ = (6)

Covariance estimator: 2)1(

1111

2111 ))(~)()((~)(~)(~)(ˆ sDsIssRsRsR kkkkkk

+−++

−++++ −= ααθ (7)

)(ˆ)|(}))(ˆ)(~(

)(ˆ{)(~)(~

2)1

1

111

spssqtVsDsD

GsRspsR

kkkkk

n

skkkk

s

k

Δ+−+

+=

+

=

−++ (8)

The optimal state estimation:

=+++ =

sn

skkk sDspD

1111 )(ˆ)(ˆˆ (9)

Where ))(

~exp()(

~)(~)( 1

2/1111 shssps kkkk +

−+++ −= θα

)(~

))(~)((21)(

~ 11

21111 ssDsIEsh kkkkk

−+

−++++ −= θα

11

2111 )(~))(~)(()(

~++

−+++ += kkkkk QsRsDsIs ααθ

sns ,,2,1=

IV. THE SIMPLIFIED ALGORITHM

When the structure state ks is more than three, the multichannel algorithm will become very complex in the practical application. Considered some specific situations (for example, the infrared homing missile), it should do best to reduce the complexity and lessen the internal storage capacity of the algorithm under the condition of satisfying the precision requirement. When ),( kkk sDψ is replaced with

)( kk Dψ (the mean value of the structure state set body), the singlechannel state estimation algorithm is achieved[10].

For ),( kkk sDψ , use Taylor series expansion at point

kk DD ~= and only the linear terms are reserved.

)~(~)(~)()( )1(kkkkkkkk DDDsIDsID −−= +−− αα αψ (10)

Where =

=sn

skkk spsDD

1)(~)(

~~ .

Calculate the mean value of formula(10) about one-step prediction:

)~(~~~~)(~ )1(kkkkkkkk DDDIDID −−= +−− αα αψ (11)

Where =

=sn

skkk spsII

1)(ˆ)(

~ .

For easy expression, three states situations are considered. The three states of kI are as follows:

min)1( II k = , )(21)2( maxmin IIIk += , max)3( II k = (12)

Where ks is a three states Markov chain, and its transition

probability is 3

4 tΔ= πλ

So the simplified infrared system distance estimation algorithm can be described as follows:

Identification function:

+

+++ =3

1111

)(/)()(ˆks

kkk sssp αα 2,1=s (13)

)2(ˆ)1(ˆ1)3(ˆ 111 +++ −−= kkk ppp (14)

State estimator:

))ˆ(

~(

~)ˆ(

)ˆ(~

)ˆ(ˆ

1111

)1(11

α

α

θ

α−

++−+

+−++

Δ−−+×

Δ−−Δ−=

tVDIEGR

tVDItVDD

kkkkkkk

kkkkkk (15)

Covariance estimator: 1

111~

)ˆ(ˆ −+++ += kkkkk QGRR θ (16)

Where ))(

~exp()(

~))(ˆ()( 1

2/111 shsqsps kkkk +

−++ −+= θα

))(~

2/())ˆ)((()(~ 1

12

111 stVDsIEsh kkkkkk−+

−+++ Δ−−= θα

12)1(

11 )ˆ())ˆ)((()(~

++−

++ ++Δ−= kkkkkkk QGRtVDsIs ααθ

12)1(

11 )ˆ())ˆ(~(~

++−

++ ++Δ−= kkkkkkk QGRtVDI ααθ The infrared system distance estimation algorithm only

includes four functions, and the amount of calculation reduces obviously. And it needs less internal storage capacity than that of the original algorithm needs. So the algorithm can satisfy the system which computing speed is slow and the internal storage capacity is small.

V. SIMULATION RESULTS

In this section, the estimation performance of the proposed algorithms are investigated.

Simulation initial conditions 333.0)(ˆ 0 =sp

00~ˆ DD =

20

ˆDR σ=

848 2007 Second IEEE Conference on Industrial Electronics and Applications

min)1( II k =

)(21)2( maxmin III k +=

max)3( II k =3,2,1=s .

Simulation parameters mD 50000 =

280max =I80min =I

smV /5000 =72 10−=Eσ

5.2=α

0 1 2 3 4 5 6

1000

2000

3000

4000

5000

time(sec)

Dis

tanc

e(m

)

Distance valueCommon AlgorithmSimplif ied Algorithm

Fig.1 The infrared distance estimation results of the common algorithm and simplified algorithm

2.95 3 3.05 3.1200

225

250

275

time(sec)

Dis

tacn

e(m

)

Fig.2 The close-up view of Fig.1 Fig.1 indicates that the common algorithm and the

simplified algorithm can both accurately estimate the distance between the infrared tracker and the target. When the initial deviation is not large, the estimation error is very small and the estimation algorithms have satisfactory precision. Fig.2 shows the different precision of the two algorithms. The common algorithm achieves higher precision than the simplified algorithm, but its amount of calculation is more than the other. The simplified algorithm which only includes four functions obviously reduces the complexity and the required internal storage capacity, but its estimation error becomes a little larger. So the suitable algorithm should be made a choice in practical application.

In some situations, the deviation of the initial distance may be larger, which requires that the algorithm has the ability in quickly correcting the error. For example, during the guidance law of the intermediate range air-to-air missile from

midcourse guidance switching to terminal guidance, the distance between the missile and the target which the radar or the inertial navigation system offered to the terminal guidance system maybe have some deviation. In order to check this performance, the algorithm is simulated in the above-mentioned situation. The initial distance deviation is 500m. Fig.3 indicates that the infrared system distance estimation algorithm can correct the initial error in 1~2 seconds. The correct speed is very fast.

0 2 4 6 80

1000

2000

3000

4000

5000

time(sec)

Dis

tanc

e(m

)

Distance Value

Esitmation

Fig.3 Infrared distance estimation initial error correct

VI. CONCLUSION

The infrared system distance estimation algorithm based on the optimal estimation theory of structure stochastic jump system and its simplified algorithm can both accurately estimate the distance between the infrared tracker and the infrared target. The two algorithms have different calculating complexity and the computational accuracy. And they are designed to different system. So the suitable algorithm should be chosen in practical application. What’s more, the algorithms have strong correct ability for the initial distance deviation which can correct the initial distance deviation in 1~2 seconds. The simplified algorithm has the merits of simple and small internal storage capacity needed. With all these advantages, it is both practical and beneficial to see its application in engineering practices.

REFERENCES

[1] Shi Xiaoping, Wang Zicai, Ke Qihong. Estimation of the Range Between the Intercepter and the Target During Infrared Terminal Guidance of Space Interception [J]. Acta Aeronoutica et Astronautica Sinica, 1995,16(3): 291-298.

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[3] LIU Shi-yong, WU Rui-lin ZHOU Buo-zhao. Bearings-only Passive Ranging for Exe-atmospheric Interceptor [J]. Journal of Astronautics, 2005,26(3): 307-313.

[4] WANG Xiao-hu, ZHANG Ming-lian. Researching and Acocomplishing For Homing Missile Attacking Maneuvering Target [J]. Acta Aeronoutica et Astronautica Sinica, 2000, 21(1): 30-33.

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[7] Hsin -Yuan Chen. 3D Nonlinear ∞HH 2 Guidance Law With

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2007 Second IEEE Conference on Industrial Electronics and Applications 849

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Tisinghua University press, 2005. [10] , .

[ ]. , 1980.

850 2007 Second IEEE Conference on Industrial Electronics and Applications