Theta-D Filter Versus Nonlinear Filtering Techniques ...

AIAA 2010-3521

AIAA Infotech@Aerospace 2010 20 - 22 April 2010, Atlanta, Georgia

AIAA 2010 InfoTech Conference

-D FILTER VS NONLINEAR FILTERING TECHNIQUES IN NONLINEAR TARGET TRACKING APPLICATIONS Quang M. Lam 1 and Bruce Anderson 2 Orbital Sciences Corporation Dulles, VA 20166

Downloaded by AFRL/RWOC Eglin on January 9, 2015 | http://arc.aiaa.org | DOI: 10.2514/6.2010-3521

Ming Xin 3 Mississippi State University Mississippi State, MS 39762

Nonlinear filtering and control techniques (  -D filter and controller) have recently gained a lot of attentions in the estimation and control community, especially in the context of robust and adaptation capabilities. This paper examines the -D filter performance for a nonlinear target tracking problem and evaluates its performance and implementation simplicity versus other nonlinear filtering techniques such as unscented Kalman Filtering (UKF) or Extended Kalman filtering (EKF) techniques. The -D filter is derived by constructing the dual of a new nonlinear regulator control technique,   D approximation which involves approximate solutions to the Hamilton Jacobi Bellman (HJB) equation. The structure of this filter is similar to the State Dependent Riccati Equation Filter (SDREF). However, this method does not require an online computationally intensive solving of the algebraic Riccati equation at each sample time as compared with the SDREF. A complex highly nonlinear tracking problem is employed to evaluate this new class of nonlinear filter vs other nonlinear filtering contenders such as UKF and/or EKF. Current evaluation of the   D filter compared with EKF and UKF via Monte Carlos simulation for this particular target tracking problem has illustrated a very attractive and encouraging result toward nonlinear estimation problems. I. INTRODUCTION

NASA embarked on its landmark Apollo program, which culminated in the first lunar landing in human history on July 20, 1969. The crucial navigation computer onboard the Apollo spacecraft was entrusted to provide critical mission trajectories, which allow the safe launch and re-entry of the spacecraft and its astronauts. At the heart its flight computer was a recursive state estimation algorithm known as the Kalman filter [1], which is well known today in many applications involving the real-time estimation of the state vector of a system that evolves over time from a sequence of noisy observation vectors. Examples of these problems span multiple engineering disciplines, ranging from aerospace applications (e.g., [2]-[6]) to biomedical systems (e.g., [7][8]), to name a few. Arguably, the most intuitively appealing example is that of target tracking. Here, the state vector contains the kinematic characteristics of the target under surveillance, and the observation vector is a noise corrupted sensor measurement such as a radar return or optical imaging frame. A 1

Chief Scientist, Email: [email protected], Senior Member AIAA Program Director, Email: [email protected] 3 Assistant Professor of Aerospace Engineering, Email: [email protected] 2

1 American Institute of Aeronautics and Astronautics Copyright © 2010 by Q. Lam, B. Anderson, and M. Xin. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.



recursive Bayesian filtering approach such as the Kalman filter allows an estimate to be made each time a new observation is received without having to reprocess past observations. The sensor may receive false measurements not originating from any targets, while existing targets may not be detected. As a result, the observation at each time step is a collection of indistinguishable measurements, only some of which are generated by existing targets while the remainders are false alarms or clutter. Nonlinear filtering for target tracking poses significant technical challenges and is an established area of research with a plethora of techniques and applications, see for example classical texts such as [9] to [11]. Although driven in the early 1970s primarily by traditional applications such as radar, sonar, guidance, navigation, and air traffic control [12] ], today nonlinear estimation techniques are continuously sought to further improve the performance of EKF based approaches. Traditional methods that serve as the work-horses of nonlinear filtering are based on the idea of Extended Kalman Filtering (EKF)/Unscented Kalman Filtering (UKF) [13] or Multiple Model Adaptive Estimation (MMAE) [14] and/or Interacting Multiple Model (IMM) estimation (see [9] or [15].) This paper investigates a different nonlinear filtering scheme for a nonlinear target tracking problem, called -D estimator [16], which is derived from the State Dependent Riccati Equation (SDRE) approach [17]. Our goal is to evaluate the effectiveness of a -D estimator with respect to the quality and accuracy of a mainstream tracking system’s performance (i.e., EKF based). If one can come up with an adaptive nonlinear estimator which allows the filtering system to bypass the linearization process of the EKF for the nonlinear equation of motion used in the EKF predictor and the nonlinear measurement process for the EKF (measurement) update, that nonlinear estimator definitely would yield some great performance features from the theoretical perspective and offer the GN&C community a better set of nonlinear estimation algorithm serving future missions. II. NONLINEAR ESTIMATION FORMULATION 2.1 SUBOPTIMAL CONTROL OF A CLASS OF NONLINEAR SYSTEMS,   D APPROXIMATION TECHNIQUE Consider the state feedback control problem for the class of nonlinear time-invariant systems described by

x  f ( x )  B ( x )u

(1)

with the cost function: J 

1 2



 0

[ x T Q x  u T R u ]d t nm

(2) nn

mm

where x R , f : R , BR ,u : R ,QR , RR positive definite matrix; g is a constant matrix and f(0)=0. n

n

m

. Q is semi-definite matrix and R is

To ensure that the control problem is well posed we assume that a solution to the optimal control problem (1), (2) exists. We also assume that f ( x) is locally Lipschitz in x on a set  and zero state observable through Q. The optimal solution of the infinite-horizon nonlinear regulator problem can be obtained by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation [7]: V T 1 V T V 1 T f ( x)  B( x)R1B(x)T  x Qx  0 (3) x 2 x x 2 2 American Institute of Aeronautics and Astronautics

AIAA 2010 InfoTech Conference with V (0)  0 The optimal control is given by

V x

u   R 1 B T

(4)

and V(x) is the optimal cost , i.e.

V ( x )  min



1 2

u

 (x

T

Qx  u T Ru ) dt

(5)

0

The HJB equation is extremely difficult to solve in general, rendering optimal control techniques of limited use for nonlinear systems.


Now consider perturbations added to the cost function:

J

 1  T x Q  Di i ) x  u T Ru ]dt [ (   0 2 i 1

 where  and Di are chosen such that  D i i

(6)

is a small number compared to Q 2 .

i 1

2

Write the original state equation as:

x  f ( x )  B ( x )u  [ A0   (

A( x )



)] x  ( g 0  

g ( x)



)u

(7)

where A0 is a constant matrix such that (A0,g0) is a stabilizable pair.

 

Define

V x

(8)

By using (8) in (3), we have 1 2



1 2

T f ( x)  T B( x)R1B( x)T   xT (Q   Di i ) x  0

(9)

i 1

Assume a power series expansion of  as

 



 T ( x ) i0

i

i

x

(10)

Here we assume that Ti s are symmetric. Substitute (7) and (10) into the HJB equation (3) and equate the coefficients of powers of  to zero to get the following equations:

T0 A0  A0 T 0  T0 g 0 R  1 g 0 T0  Q  0 T

T

T1 ( A0  g 0 R 1 g0T T0 )  ( A0  T0 g 0 R 1 g 0T )T1   T

 T0

g



R 1 g 0 T T0  D1





A ( x)T0



 T0 g 0 R 1

gT



(11)

T0

(12)

T2 (A0  g0R1g0TT0 )  (A0 T0g0R1g0T )T2  T

T0 A( x)

T

g 1 gT gT g T1A(x) AT (x)T1 gT g T T0  T1g0 R1g0T T1  T1g0 R1 T0  T1 R1g0T T0  D2  T0g0R1 T1 T0 R1g0 T1 T0 R 











(13) 

3 American Institute of Aeronautics and Astronautics





AIAA 2010 InfoTech Conference T Tn1A(x) AT (x)Tn1 n1 g T 1 g  Tj (g0R  R1g0 )Tn1 j Tn (A0  g R g T ) (A T g R g )Tn  1 T 0 0 0

n2

1 T 0 0 0

T 0





n 1

  T j gR g Tn  2  j   T j g 0 R 1 g 0T Tn  j  Dn 1

 

j 0

T

j 0

j 1

(14)

The expression for control can be obtained in terms of the power series for  as

u   R B ( x)    R B ( x) 1

1

T



T

 T ( x)


i0

i

i

x

(15)

It is easy to find that the equation (11) is an algebraic Riccati equation. The rest of equations are Lyapunov equations which are linear in terms of Tn. In the rest of this paper we will call it the  -D approximation technique. The algorithm without D i term is called the  approximation. The algorithm [8] would result in the  approximation. One of the problems with  approximation is that large initial conditions may give rise to large control. In order to deal with this problem, we construct the following expression for Di: D1  k1e l1t [

T0 A( x)





AT ( x)T0



(16)

]

D2  k2 e l2t [

T1 A( x)





AT ( x)T1



]

(17)  D n  k n e  ln t [ 

Tn 1 A ( x )





AT ( x )Tn 1



(18)

]

The idea in constructing Di in this manner is that large control results from the state dependent term A(x) on the right hand side of the equations (11)-(14). So we choose Di such that T A( x) AT ( x)Ti 1 T A( x) AT ( x)Ti 1 (19)  i 1   Di   i [  i 1  ] 







where  i is a small number chosen to satisfy some conditions required in the proof of convergence and l t

stability of the above algorithm. On the other hand, the exponential term e i is used to let the perturbation terms in the cost function and HJB equation diminish as the time evolves. This will guarantee the HJB equations to be solved asymptotically. Since the solution of HJB equation is the sufficient condition for optimality, the suboptimal control (15) will approximate the optimal control asymptotically. In   D technique, Di terms play a crucial role in both guaranteeing stability and modulating the performance. 2.2 FORMULATION OF   D FILTER Considering that the steady state linear Kalman filter is the dual of the steady state linear regulator, we can obtain the filtering counterpart of the   D controller by taking the dual of the coefficient matrices. Consider the stochastic nonlinear system x  f ( x)  w

(20) y  h( x )  v (21) where w is a Gaussian zero-mean white process noise with E[ w(t ) wT (t   )]  W (t ) ( ) and v is a Gaussian zero-mean white measurement noise with E[v(t )vT (t   )]  V (t ) ( ) . The W(t) and V(t) are power spectral densities.


AIAA 2010 InfoTech Conference In the SDREF formulation, it brings the original system (20),(21) to the state dependent coefficient form:

x  F ( x) x  w

(22)

y  H ( x) x  v The SDREF is given by

xˆ  F ( xˆ ) xˆ  K f ( xˆ )[ y ( x)  H ( xˆ ) xˆ ]

K f ( xˆ )  P( xˆ ) H ( xˆ )V where and P is the positive definite solution to T

(23)

1

(24)


F ( xˆ)P( xˆ)  P( xˆ)F T ( xˆ)  P( xˆ)H T ( xˆ)V 1H ( xˆ)P( xˆ)  TW   0 (25) Now in   D method, if we consider (10) and assume

 



 T ( x )

i

i

i0

x  P ( x) x

(26)

we can obtain the state dependent Riccati equation by substituting (26) into the HJB equation (3):

AT ( x ) P ( x )  P ( x ) A( x)  P ( x) B ( x) R 1 BT ( x ) P ( x)  Q  0 (27) So the solution to   D method



 T ( x)

i

i

i 0

can be used as the solution to the state dependent Riccati

equation. Rewrite (20) (21) as: x  f ( x )   w  [ A0   (

y  (H0  

H ( x)

A( x)



)] x   w

(28)

)x  v

(29)

 The   D filter is given by

A( xˆ ) H ( xˆ ) xˆ  ( A0   ) xˆ  K f ( xˆ )[ y ( x)  ( H 0   ) xˆ ]



(30)



K f ( xˆ )  P( xˆ )( H 0  H T ( xˆ ))V 1 T

where

(31)



P ( xˆ )   Ti ( xˆ ) i

(32)

i 0

and Ti ( xˆ ) is the solution to the following equations (33)-(36) in which we take the dual of A0, A( x), g 0, and g ( x) in (11)-(14), i.e. A0T , AT ( xˆ ), H 0T , and H T ( xˆ ) . (33) T 0 A 0T  A 0 T 0  T 0 H 0T R  1 H 0 T 0  Q  0 T1 ( A0T  H 0T R 1 H 0T0 )  ( A0  T0 H 0T R 1 H 0 )T1  

T0 A( xˆ )





AT ( xˆ )T0



 T0 H T R 1

H



T0

(34) TA(xˆ) AT (xˆ)T1 H HT T2 (A0T  H0T R1H0T0 ) (A0 T0H0T R1H0 )T2  1  T0H0T R1 T1 T0 R1H0T1



T0

H

T



R1

H



T0  T1H0T R1H0T1  T1H0T R1



H

 

T0  T1



H

T





R1H0T0  D2 (35)


 T0

HT



R 1 H 0T0  D1

AIAA 2010 InfoTech Conference Tn1A(xˆ) AT (xˆ)Tn1 n1 H HT T 1 Tn(A0T H0T R1HT  Tj (H0T R1  R1H0)Tn1j 0 0 ) (A0 TH 0 0 R H0 )Tn 



n2

n 1

j 0

j 1





j0

  T j H T R 1 HTn  2  j   T j H 0T R 1 H 0Tn  j  Dn



(36)

III Reentry Object Tracking Problem The reentry target tracking model is described as follows,


x1 (t )  x3 (t ) x 2 (t )  x 4 (t )

x 3 (t )  D(t ) x3 (t )  G (t ) x1 (t )  w1 (t ) x 4 (t )  D (t ) x 4 (t )  G (t ) x 2 (t )  w2 (t )

(37)

x 5 (t )  w3 (t ) Where D(t) is the drag-related force term, G(t) is the gravity-related force term, and w(t) is the process noise vector. The force terms are given by

D(t )    (t )e Gm G (t )   3 0 r (t )

[ R0  R ( t )] V (t ) H0

 (t )   0 e x (t ) 5

Where R(t) is the distance from the center of the Earth and V(t) is the velocity of the object. The complete model of this reentry object is captured in Table 3-1. along with its associated parameters. The state dependent (SD) state space formulation of equation (37) is described as follows:

x1 (t )  x3 (t ) x 2 (t )  x 4 (t )

x 3 (t )  [0.25G 0.25G

x1 x2

0.5 D

0.5G

x1 x4

 x1  x  2 x3   0.5 D ] x3   w1 (t ) x5    x4   x5    (38)




x 4 (t )  [0.25G

x2 x 0.25G 0.5G 2 x1 x3

 x1  x  2 x4    0.5D 0.5 D ] x3   w2 (t ) x5    x4   x5   

x 5 (t )  c1e c 2t x5 (t )  w3 (t ) Where the aerodynamic parameter x5(t) is being “remodeled” as a correlated noise process so that it can support the “stable” SD formulation for that particular element. The value of c1 and c2 are traded to reach an optimal performance for the -D estimator. The system of equation (38) is now compiled as the A(x) matrix as follows,     0 1 0 0  0   0  0 0 1 0   x3 (t )  x1 (t ) x1 (t )  A( x)  0.25G 0.25G 0.5 D 0.5G 0.5D (39) x 4 (t ) x5 (t )  x 2 (t )   x 2 (t ) x 2 (t ) x 4 (t )   0.5 D 0.5D 0.5G 0.25G x (t ) 0.25G x3 (t ) x5 (t )  1    0  0 0 0 c1e c 2t The nonlinear measurement state dependent matrix is written as follows, r (t ) r (t )   0 0 0 0.5 x (t ) 0.5 x (t ) 1 2  h(x) =  (40)  (t )  (t )   0 0 0 0.5 x (t ) 0.5 x (t ) 1 2   where the range, r(t) and bearing, (t) are computed as follows,

r (t )  ( x1 (t )  x r ) 2  ( x 2 (t )  y r ) 2  v1 (t ) (41) ( x 2 (t )  y r )  v 2 (t ) ( x1 (t )  x r ) Where (xr,yr) is the location of the ground based radar and their values are defined in Table 3-1..

 (t )  tan 1

Table 3-1: Reentry Object Dynamic Model % The parameters for re-entry mechanics. %



beta0 = -0.59783; H0 = 13.406; Gm0 = 3.9860e5; R0 = 6374; dt = 0.1; sim_iter = 1; xr = R0; yr = 0;


vr = (1e-3).^2; va = (.17e-3).^2; % See Julier's paper, section VI true_m0 = [6500.4; 349.14; -1.8093; -6.7967; 0.6932]; %true_P0 = diag([1e-6 1e-6 1e-6 1e-6 0]); true_P0 = diag([1e-6 1e-6 1e-6 1e-6 0]); true_Qc = diag([2.4064e-5 2.4064e-5 0]) / 0.1; L = [0 0 1 0 0

0 0 0 1 0

0; 0; 0; 0; 1];

m0 = 1.0*[6500.4; 349.14; -1.8093; -6.7967; 0]; P0 = 50*diag([1e-6 1e-6 1e-6 1e-6 1]);% Assuming poor initial knowledge of covariance error Qc = diag([2.4064e-5 2.4064e-5 1e-6]) / 0.1; % Parameters needed by the dynamic model as cell array d_param = {dt,beta0,H0,Gm0,R0,L}; % Parameters needed by the measurement model as cell array h_param = {xr,yr};


AIAA 2010 InfoTech Conference IV. Performance Evaluation of the -D Estimator Using a Nonlinear Tracking Problem

600

500

400

300 x2(km)


Consider the problem that is illustrated in Figure 4.1. A vehicle enters the atmosphere at high altitude and at a very high speed. The position of the body is tracked by a radar which measures range and bearing. This type of problem has been identified by a number of authors [18], [19]&[20] as being particularly stressful for filters and trackers because of the strong nonlinearities exhibited by the forces which act on the vehicle. There are three types of forces in effect. The most dominant is aerodynamic drag, which is a function of vehicle speed and has a substantial nonlinear variation in altitude. The second type of force is gravity, which accelerates the vehicle toward the center of the earth. The final forces are random buffeting terms. The effect of these forces gives a trajectory of the form shown in Figure 4-1: initially, the trajectory is almost ballistic; but as the density of the atmosphere increases, drag effects become important and the vehicle rapidly decelerates until its motion is almost vertical. The tracking problem is made more difficult by the fact that the drag properties of the vehicle might be only very crudely known.

200

100

0

-100

-200 6340

6360

6380

6400

6420 6440 x1(km)

6460

6480

6500

6520

Figure 4.1: Reentry Tracking Problem: The dashed line is the sample vehicle trajectory and the solid line is a portion of the earth’s surface. The position of the radar is marked by a blue o



EKF, x, truth vs estimate 6600 6400 6200

0

20

40

60

0

20

40

0

20

40

0

0

80 100 120 EKF, y, truth vs estimate

140

160

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200

60

80 100 120 140 EKF, xdot, truth vs estimate

160

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80 100 120 140 EKF, ydot, truth vs estimate

160

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200

20

40 60 80 100 120 140 160 EKF, x5-aerodynamic parameter, truth vs estimate

180

200

20

40

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200

400 200 0


0 -1 -2 10 0 -10 10 0 -10

60

80

100

120

140

160

Figure 4.2: Reentry Object State Estimation Using EKF



SDREF Performance Truth (solid blue) vs Estimate (red dash) 6600 x

6400 6200

0

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0

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0

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0

20

40

60

80

100

120

140

160

180

200

y

400 200 0

xdot

-2 -4

ydot

10

aerodyn. parameter, x5


0

0 -10 1 0 -1

Figure 4.3: SDREF Performance



2

EKF, UKF, and SDREF, Squared Estimate Error Response of the Aero. Dyn. Parameter

10

EKF SEx5 UKF SEx5 SDREF SEx5

0

10

-2

10

-4


10

-6

10

-8

10

-10

10

-12

10

0

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200

Figure 4.4: State 5 Performance Comparison of EKF, UKF, and SDREF (SDREF outperformed the EKF and UKF for one run out of 20 Monte Carlo simulation) The current performance of the -D (or SDREF) estimator before fine-tuning is presented in Figure 4.3. We are currently investigating the pair [f(x),h(x)] for other forms of factorization to further improve the overall performance of the -D tracking filter. The nonlinear expression of the -D estimator, coded in Matlab, is presented in Tables 4-1 and 4-2, respectively for the nonlinear dynamic predictor and the nonlinear measurement equation. Table 4-1: -D Estimator (Dynamic Model ) % SDRE F matrix for Theta-D Estimator function f_sdre = reentry_df_sdre(x,param) dt b0 H0 Gm0 R0

= = = = =

param{1}; param{2}; param{3}; param{4}; param{5};




f_sdre = zeros(5,5); for j=1:5 if x(j,:)==0 x(j,:)=1e-2; else x(j,:)=x(j,:); end end R = sqrt(x(1,:).^2 + x(2,:).^2); V = sqrt(x(3,:).^2 + x(4,:).^2); b = b0 * exp(x(5,:)); D = b .* exp((R0-R)/H0) * V; G = -Gm0 ./ R.^3; % % G31=-Gm0./((x(1,:)^2+x(2,:)^2).^2*R); D33=b .* exp((R0-R)/H0) * V; % G42=G31; D44=D33; f_sdre(1,3) = 1; % f_sdre(2,4) = 1; % f_sdre(3,1) = 0.25*G31; f_sdre(3,2) = 0.25*G31*(x(1,:)/x(2,:)); f_sdre(3,3)=0.5*D33; f_sdre(3,4)=0.5*G31*(x(1,:)/x(4,:)); f_sdre(3,5)=0.5*D33*(x(3,:)/x(5,:)); % f_sdre(4,1) = 0.25*G31*(x(2,:)/x(1,:)); f_sdre(4,2) = 0.25*G42; f_sdre(4,3) = 0.5*G31*(x(2,:)/x(3,:)); f_sdre(4,4) = 0.5*D44; f_sdre(4,5) = 0.5*D44*(x(4,:)/x(5,:)); % f_sdre(5,5) = .695*exp(-.95*dt);% correlated noise model

Table 4-2 : Nonlinear Measurement Process of the -D Estimator function dy = reentry_h_sdre(x,param) xr = param{1};% x radar location yr = param{2};% y radar location y = zeros(2,size(x,2)); dy = zeros(2,size(x,1)); range=sqrt((x(1,:) - xr).^2 + (x(2,:) - yr).^2); % matrix as state dependent dy(1,1)=(0.5/x(1,:))*range; dy(1,2)=(0.5/x(2,:))*range; % Similarly, splitting the bearing information so that other state can be % brought in like x(3,:) and x(4,:)


AIAA 2010 InfoTech Conference theta = atan2(x(2,:) - yr, x(1,:) - xr); dy(2,1)=(0.5/x(1,:))*theta; dy(2,2)=(0.5/x(2,:))*theta;


IV. Conclusion Despite its single case performance for state 5 (shown in Figures 4.3 and 4.4) which beats the EKF performance at the present moment (out of a 20 multiple Monte Carlo Run), the SDREF or -D filter presents a promising aspect going forward due to the elimination of the linearization step as required by the EKF and UKF. Without relying on the linearized model, if properly and carefully factored out for the SDRE formulation, the -D filter may lead to a new class of robust nonlinear filter. References: [1] R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME J. Basic Eng., vol. 82, pp. 34–45, Mar. 1960. [2] J. Leonard and H. Durrant-Whyte, “Application of multi-target tracking to sonar based mobile robot navigation,” in Proc. 29th Conf. Decision & Control, Hawaii, USA, pp. 3118– 3123, 1990. [3] R. K. Mehra, “A comparison of several nonlinear filters for reentry vehicle tracking,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 307–319, Aug. 1971. [4] D. Lerro and Y. K. Bar-Shalom, “Tracking with debiased consistent converted measurements vs. EKF,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-29, pp. 1015–1022, July 1993. [5] S. J. Julier and J. K. Uhlmann, “A new extension of the Kalman filter to nonlinear systems,” in Proc. AeroSense: 11th Int. Symp. Aerospace/Defense Sensing, Simulation and Controls, 1997, pp. 182–193. [6] R. Merwe and E. Wan, “The square-root unscented Kalman filter for state and parameter-estimation,” in Proc. IEEE Int. Conf. Acoustics, Speech, and Signal Processing (ICASSP), vol. 6, 2001, pp. 3461–3464. [7] G. Kitigawa and G. Gersh, Smoothness Priors Analysis of Time Series. Springer-Verlag, New York, 1996. [8] D. Kocak, N. Lobo, and E.Widder, “Computer vision techniques for quantifying, tracking, and identifying bioluminescent plankton,” in IEEE Jnl. Ocean. Engineering, vol. 24, pp. 81–95, 1999 [9] Y. Bar-Shalom and T. Fortmann, Tracking and Data Association. Academic Press, San Diego, 1988. [10] Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Principles and Techniques. Storrs, CT: YBS Publishing, 1995. [11] S. Blackman, Multiple Target Tracking with Radar Applications. Artech House, Norwood, 1986. [12] M. Skolnik, Radar handbook (2nd Ed.). McGraw-Hill, 1990 [13] Julier, S. and Uhlmann, J.,”Unscented Filtering and Nonlinear Estimation,” Proceedings Of The Ieee, Vol. 92, No. 3, March 2004



[14] Lam, Q. M. and Crassidis, J. L., “Evaluation of a Multiple Model Adaptive Estimation Scheme for Space Vehicle’s Enhanced Navigation Solution,” Presented at the AIAA GN&C August 2007, Hilton Head, SC [15] Blackman, S. and Popoli, R. (1999). Design and Analysis of Modern Tracking Systems. Artech House Radar Library [16] Ming Xin, S.N. Balakrishnan, “A New Filtering Technique For A Class Of Nonlinear Systems,” Proceedings of the CDC 2002


[17] C.P. Mracek, J.R. Cloutier, and C.N. D’Souza. “ A New Technique for Nonlinear Estimation,” Proceedings of the IEEE Conference on Control Applications, Dearborn, MI, September 1996. [18] P. J. Costa, “Adaptive model architecture and extended Kalman–Bucy filters,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, pp.525–533, Apr. 1994. [19] M. Athans, R. P. Wishner, and A. Bertolini, “Suboptimal state estimation for continuous-time nonlinear systems from discrete noisy measurements,” IEEE Trans. Automat. Contr., vol. AC-13, pp. 504–518, Oct. 1968 [20] R. K. Mehra, “A comparison of several nonlinear filters for reentry vehicle tracking,” IEEE Trans. Automat. Contr., vol. AC-16, pp. 307–319, Aug. 1971.


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