Robust Kalman filter of discrete-time Markovian jump system with ...

Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007

99

Robust Kalman filter of discrete-time Markovian jump system with parameter and noise uncertainty Jin Zhu, Junhong Park, Kwan-Soo Lee HanYang University Department of Mechanical Engineering Seoul,133-791, Republic of Korea {jinzhu,parkj,ksleehy}@hanyang.ac.kr Abstract: Robust Kalman filtering problems for discrete-time Markovian jump systems with parameter and noise uncertainty were investigated. Because of the existence of stochastic Markovian switching, the covariance matrices of system state noise and observation noise are time-varying or unmeasurable instead of stationary, meanwhile the system suffers from structure parameter uncertainty as well. By view of robust estimation, maximum admissible upper bound of the disturbance to noise covariance matrix was given based on the estimation performance, and an optimal state estimator was therefore adopted under the worst situation. Not only can this method minimize the worst performance function of uncertainty, but also the estimation error performance can be guaranteed to be within the given precision. A numerical example shows the validity of the method. Key–Words: Markovian jump systems;robust Kalman filter; uncertainty;

1

Introduction

Optimal filtering problem has been a hot topic in past decades, among which Kalman filtering is one of the most popular estimation approaches and considerable effort has been devoted to its theory and applications. The applications of Kalman filtering theory may be found in a large spectrum of different fields ranging from various engineering problems to biology, geoscience, economics, and management etc[1]. On the other hard, Markovian jump systems, which are convenient tools for representing many realworld systems[2]have aroused much attention in recent years. And fruitful achievements have been made in the last three decades on stability analysis[3, 4], filtering[5, 6]and control design[7, 8]. In the efforts towards filtering, Boukas[9]and Mahmoud [10]gave Kalman filtering equations for continuous-time and discrete-time Markovian jump linear systems with structure uncertainty respectively. However, in above referred contributions, all the research work was carried out based on one assumption: both the state equation and output measurement are subjected to STATIONARY Gaussian noises so that an optimal filtering gain is obtained based on the stationary noise covariance matrix. But this is not the case for Markovian jump systems. In practical environment, because of the stochastic switching in Markovian jump systems,which is usually accompanied by sudden change of working environment, the statistical characteristics(covariance matrix) of noise may be time-varying

instead of stationary, and in some cases it is impossible to get the exact value of noise covariance matrix,in such cases the noise is so-called ”uncertain”. Thus the pre-designed filter may fail resulting from the change of noise covariance matrix and the controller using the estimation of system state will be incorrect. Thus the noise uncertainty will in worst case lead to system instability and a practical problem occurs: Could the pre-designed Kalman filter still efficient with the presentence of uncertain noise and parameter? How to achieve this? In this paper, robust Kalman filtering for discretetime Markovian jump systems under uncertain noise and parameter is considered. Firstly we give some assumptions to obtain estimation performance. Secondly we seek the maximum admissible upper bound of non-structure disturbance to noise such that the deviation of estimation performance can be within a prescribed precision. Then we discuss about the sub-optimal analytical solution by using Lagrange method. Finally we prove the establishment of saddle inequality and show that our filter design is a minimax robust filter. At the end of the paper, an illustrative example is used to show the validity of our method.

2 Problem Description Throughout the paper,unless otherwise specified,we denote by (Ω, F, {Ft }t≥0 , P ), a complete probabil-

Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007

i ∈ S, parameter(structure)uncertainty ∆A(i), ∆C(i) satisfy

ity space with a filtration {Ft }t≥0 satisfying the usual conditions(i.e. it is right continuous and F0 contains all p-null sets). Let |x| stand for the usual Euclidean norm for a vector x , and |X| denote the Frobenius

∆A(i) = H1 (i)F (i)E(i) ∆C(i) = H2 (i)F (i)E(i)

1 2

norm of a matrix X defined by |X| = λmax (XX T ), where T r(·) denotes the matrix trace and the superscript T denotes transpose.Matrix X > 0(≥ 0) denotes X is positive definite(semi-positive definite). Let {rk , k ≥ 0} is a discrete Markov chain on the probability space taking values in finite state space S = {1, 2, . . . , N } with P = [pij ] the chain generator,an N × N matrix.The entries pij , i, j ∈ S are interpreted as transition rates such that

Theorem 1 Consider stochastically stable Markovian jump system (1)and assume the noise is stationary, which means ∆W = ∆V = 0, we have the following extended Kalman filter([10]):

Here pij ≥ 0 is the transition probability from i to j . Notice that the total probability axiom imposes

x ˆ˙ = Aˆi x ˆ + Ki [y − Cˆi x ˆ]

j=1

1 T Aˆi = Ai + ( H1i H1i + W )Ψ−1 i ²i 1 Cˆi = Ci + H2i H1i Ψ−1 i ²i

Consider the following discrete-time Markovian jump system with uncertain noise and parameter: xk+1 = [A(rk ) + ∆A(rk )]xk + ω 0 yk = [C(rk ) + ∆C(rk )]xk + υ 0

(1)

Ψi = Ai (

Rm

where xk ∈ is state vector, yk ∈ is measurement output. A(·) ∈ Rn×n , C(·) ∈ Rn×m are known matrices. ω 0 , υ 0 are n−dimensional and m−dimensional white noise and satisfy the following assumption: Assumption 1. For any given time s, τ ≥ 0, there is (1) (2) (3)

(4)

E[ωτ0 ]

=0 =0 Cov[ωs0 , ωτ0 ] = W 0 δs,τ = (W + ∆W )δs,τ , W ≥ 0, ∆W ≥ 0 Cov[υs0 , υτ0 ] = V 0 δs,τ = (V + ∆V )δs,τ , V > 0, ∆V ≥ 0

W0

"

υτ0T )]

=

Rn×n , V 0

W 0 δτ,s 0 0 0 V δs,τ Rm×m consist

In Assumption 1, ∈ ∈ of two parts,where W, V denote the stationary noise covariance matrix and the values are exactly known. ∆W, ∆V denote the uncertainty caused by disturbance or time-varying,they are unknown but bounded. δ(·, ·) is a Dirac function taking values in {0, 1}. For the deduction of Kalman filter,we introduce the following assumption([10]): Assumption 2. For any fixed system mode rk =

N X

pij Ψj )ATi + ²i Ψi EiT Ei Ψi

j=1

1 T +W + H1i H1i ²i 1 T T ¯ i CˆiT + 1 H1i H2i Ki = (Aˆi Q )( H2i H2i + V )−1 ²i ²i ¯ i (Aˆi − Ki Cˆi )T Qi = (Aˆi − Ki Cˆi )Q +Ki V KiT + W

E[υτ0 ]

w0 E[( 0s ) · (wτ0T υs

(3)

where matrix Qi , Ki are given by the following coupled Riccati equations:

pij = 1, pij ≥ 0 ∀i ∈ S

Rn

(2)

where H1 (i), H2 (i), E(i), i ∈ S are known constant matrix and F (i), i ∈ S is unknown matrix satisfying F T (i)F (i) ≤ I. For simplification, we denote A(rk = i), C(rk = i), H1 (rk = i), H2 (rk = i), E(rk = i), ∆A(rk = i), ∆C(rk = i) by Ai ,Ci , H1i , H2i , Ei , ∆Ai , ∆Ci .

pij = P r(rk+1 = j|rk = i)

N X

100

¯i = Q

N X

pij Qj

(4)

j=1

#

Here parameter ²i is chosen such that tr(Qi ) reaches the minimum. With the above standard Kalman filter gain (3) adopted, the state estimation error satisfies: E{(x − x ˆ)(x − x ˆT )} ≤ max tr(Qj ) j∈S

(5)

Define the estimation error performance as J(K1 , K2 , . . . , KN , W, V ) = max tr(Qj ) j∈S

(6)

According to Theorem 1 and quality of Kalman filtering, if the noise is stationary (∆W = ∆V = 0),the estimation error performance could achieve the minimum value by adopting standard Kalman filtering (3).

Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007

However in practical world, the standard Kalman filter may fail with uncertain noise :∆W 6= 0, ∆V 6= 0, thus the new covariance matrix of noise is W 0 , V 0 . If we still adopt the former pre-designed Kalman filter gain Ki ,the new state estimation error should be Q0i , which satisfies: ¯ 0 (Aˆi − Ki Cˆi )T Q0i = (Aˆi − Ki Cˆi )Q i +Ki V 0 KiT + W 0

(7)

Therefore the new estimation performance is J(K1 , K2 , . . . , KN , W 0 , V 0 ) = max tr(Q0j ) j∈S

(8)

According to (6) and (8), the deviation of estimation performance yielded by noise uncertainty (∆W, ∆V ) can be written as: = =

∆J(K1 , K2 , . . . , KN , ∆W, ∆V ) J(K1 , . . . , KN , W 0 , V 0 ) − J(K1 , . . . , KN , W, V ) max tr(Q0j ) − max tr(Qj ) ≤ r (9) j∈S

j∈S

Thus our purpose is: if we want the pre-designed Kalman filter still efficient under uncertain noise (∆W, ∆V ), we should limit noise uncertainty to a certain bound. As long as the (∆W, ∆V ) is within this bound, the admissible deviation of estimation performance ∆J(K1 , K2 , . . . , KN , ∆W, ∆V ) ≤ r where r > 0 is a design parameter according to practical requirement. In the following work, we set out to find corresponding equation between (∆W, ∆V ) and r.

Fact 2. Define the maximum admissible deviation of estimation performance r as r=

∆J(K1 , K2 , . . . , KN , ∆W, ∆V )

tr(Q0i ) ≤ r + max tr(Qj )

(11)

j∈S

Therefore for each mode i ∈ S,there is tr(∆Qi ) = tr(Q0i ) − tr(Qi ) ≤ r + max tr(Qj ) − tr(Qi ) (12) j∈S

Let |∆W | ≤ a, |∆V | ≤ b, thus we have 0 ≤ ∆W ≤ aIn

0 ≤ ∆V ≤ bIm

According to Fact.2, if the noise disturbance reaches the maximum aIn , bIm , the deviation of estimation performance will reach the maximum value r. According to (10) and (12), we have ∀i ∈ S atr(Di ) + btr(Gi ) ≤ r + max tr(Qj ) − tr(Qi ) (13) j∈S

where matrix Di , Gi > 0, i ∈ S satisfy the following equations: ¯ i (Aˆi − Ki Cˆi )T + In Di = (Aˆi − Ki Cˆi )D ¯ i (Aˆi − Ki Cˆi )T + Ki K T Gi = (Aˆi − Ki Cˆi )G i

¯i = D

3.1 Expression of upper bound

N X

pij Dj

j=1

According to (4), (7), we have

¯i = G

¯ 0 (Aˆi − Ki Cˆi )T ∆Q0i = (Aˆi − Ki Cˆi )∆Q i

N X

pij Gj

j=1

(10)

where ∆Qi = Q0i − Qi , from (10), it is easy to see that tr(∆Qi ) is a linear mapping of (∆W, ∆V ). Define a compact convex set as Ξ = {(∆W, ∆V ) : 0 ≤ ∆W ≤ ∆W ∗ , 0 ≤ ∆V ≤ ∆V ∗ }, thus ∆J(K1 , K2 , . . . , KN , ∆W, ∆V ) is a mapping from Ξ to R1 , and it has following facts: Fact 1. For any given (∆Wj , ∆Vj ) ∈ Ξ, j = 1, 2, if ∆W1 ≤ ∆W2 , ∆V1 ≤ ∆V2 , we have ∆J(K1 , K2 , . . . , KN , ∆W1 , ∆V1 ) ≤ ∆J(K1 , K2 , . . . , KN , ∆W2 , ∆V2 )

max

(∆W,∆V )∈Ξ

Thus r could be achieved only by maximum matrix pair (∆W ∗ , ∆V ∗ ). The purpose of following work is to construct a maximal conpact convex set Ξ∗ , for any (∆W, ∆V ) ∈ Ξ∗ ,(9) is sure to establish, and a mini-max robust filtering is applied to minimize the worst performance under the noise uncertainty. According to the finity of mode S, (9) is equivalent to

3 Upper bound of nonstructural disturbance

+Ki ∆V KiT + ∆W

101

By above analysis, the seeking of admissible maximum bound of (∆W, ∆V ) is equal to get the optimal solution of a, b such that satisfying the inequalities: s.t.

max a·b a · tr(Di ) + b · tr(Gi ) ≤ r + max tr(Qj ) − tr(Qi ) j∈S

a≥0

b≥0

i∈S

Thus the the seeking of admissible maximum bound of (∆W, ∆V ) is transformed to be a nonlinear programming problem with linear inequalities constraints. Now we discuss about the solution of this problem.

(14)

Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007

3.2

Analytical solution

Since Ξ = {(∆W, ∆V )} is a compact convex set and the inequalities in (14) compose a compact closed set on which a·b is defined as a continuous function.Thus there must exist the optimal solution of a, b. Decompose the original nonlinear program problem Eq. (14) into N sub-problems:

s.t.

max a1 · b1 a1 · tr(D1 ) + b1 · tr(G1 ) ≤ r + max tr(Qj ) − tr(Q1 )

s.t.

max a2 · b2 a2 · tr(D2 ) + b2 · tr(G2 ) ≤ r + max tr(Qj ) − tr(Q2 )

(4), there exist a maximum admissible compact set Ξ. When the disturbance of noise covariance matrix (∆W, ∆V ) ∈ Ξ, the deviation of system state estimation performance is within a given bound r.

4 Mini-max robust filter ∗ denotes the standard extended Let K1∗ , K2∗ , . . . , KN Kalman filtering gain corresponding to the maximum admissible noise disturbance (∆W ∗ , ∆V ∗ ), according to the quality of Kalman filtering, we have

j∈S

On the other hand, according to Fact 1, we have

.. . max aN · bN aN · tr(DN ) + bN · tr(GN ) ≤ r + max tr(Qj ) − tr(QN )

∗ ∆J(K1∗ , K2∗ , . . . , KN , ∆W, ∆V ) ∗ ∗ ∗ ∆J(K1 , K2 , . . . , KN , ∆W ∗ , ∆V ∗ )

≤

Thus we have the following saddle point inequality:

j∈S

∗ ∆J(K1∗ , K2∗ , . . . , KN , ∆W, ∆V ) ∗ ∗ ∗ ≤ ∆J(K1 , K2 , . . . , KN , ∆W ∗ , ∆V ∗ ) ≤ ∆J(K1 , K2 , . . . , KN , ∆W ∗ , ∆V ∗ )

By using Lagrange method, we have the optimal analytical solution for each sub-problem as: r + maxj∈S tr(Qj ) − tr(Qi ) 2tr(Di ) r + maxj∈S tr(Qj ) − tr(Qi ) b∗i = 2tr(Gi )

∗ ∆J(K1∗ , K2∗ , . . . , KN , ∆W ∗ , ∆V ∗ ) ∆J(K1 , K2 , . . . , KN , ∆W ∗ , ∆V ∗ )

≤

j∈S

s.t.

102

By Game theory, the optimal estimator under the worst situation is the mini-max estimator:

a∗i =

min =

Thus the analytical solution for the original nonlinear program problem Eq.(14) is taken as r + maxj∈S tr(Qj ) − tr(Qi ) a∗ = min a∗i = min{ } i∈S i∈S 2tr(Di ) r + maxj∈S tr(Qj ) − tr(Qi ) b∗ = min b∗i = min{ } i∈S i∈S 2tr(Gi )

max

Ki (∆W,∆V )∈Ξ

(15)

max

∆J(K1 , K2 , . . . , KN , ∆W, ∆V )

min ∆J(K1 , K2 , . . . , KN , ∆W, ∆V )

(∆W,∆V )∈Ξ Ki

5 Simulation Consider the following two-mode discrete-time Markovian jump system: Let the system mode rk = 1 be given by "

(16) Remark: The analytical solution of the nonlinear programming problem is given by above analysis, however, it is only an optimal solution for each subproblem. This analytical solution in Eq.(16) is local optimal but global sub-optimal. For the global optimal solution, we could only get the numerical solution by using ”fmincon” function in Matlab software.The optimal analytical solution of such nonlinear programming problem is still an open problem in mathematics for further exploration. Theorem 2 Consider Markovian jump system (1),if we adopt state estimator (3) and Kalman filter gain

A1 = H1 (1) = [0.1

#

h

0.7 0

0 1.1

0.1]T

H2 (1) = 0.1

C1 =

i

1.5

0

E1 = [0.3 0.2]

Let the system mode 2 be given by "

A2 = H1 (2) = [0.15

0.9 0

0.15]T

0 1

#

H2 (2) = 0.1

h

C2 =

i

1

0

E2 = [0.2 0.2]

Stationary noise covariance matrix and mode transition matrix is # " " # 1 0 0.5 0.5 W = ,V = 1, P = 0 1 0.7 0.3

Proceedings of the 7th WSEAS International Conference on Simulation, Modelling and Optimization, Beijing, China, September 15-17, 2007

The admissible bound of performance deviation is r = 0.3 1)Solve " the equation (4),get Q # 1 , Q2 and " K1 , K2 :# 2.4188 1.2012 0.6590 Q1 = , K1 = 1.2012 2.7563 1.0087 "

#

"

#

0.6754 2.6753 1.2531 Q2 = , K2 = 1.2531 2.8966 0.5096 2)Substitute the result to (14), by using Lagrange method, the upper bound of noise uncertainty are given as: a∗ = 0.1493, b∗ = 0.1677 3)Let the noise covariance matrix correspond to the maximum perturbation: W ∗ = W + ∆W = W + a∗ · I2 , V ∗ = V + ∆V = V + b∗ · I1 4)Repeat step 2), and we have the correspondent Q∗1 , Q∗2 ,K1∗ , K2∗ for new noise covariance matrix (W ∗ , V" ∗ ): # " # 2.4351 1.2235 0.6903 Q∗1 = , K1∗ = 1.2235 2.7898 1.1458 "

#

"

2.7858 1.2658 0.7012 = , K2∗ = 1.2658 3.0177 0.5241 5)With the robust Kalman filtering applying: ∆J(K1∗ , K2∗ , ∆W ∗ , ∆V ∗ ) = 0.2316 < 0.3

[6]

[7]

[8]

#

Q∗2

6

[5]

Conclusion

In this paper, the robust Kalman filter for discretetime Markovian jump system with uncertain noise and parameter is considered. A new method is given to obtain the maximum admissible bound of the disturbance to noise so that the deviation of estimation performance is guaranteed to be within a given precision. The seeking of bound to noise uncertainty could be transformed to a nonlinear programming problem, and the analytical solution of this problem is also discussed in this paper, which is a sub-optimal and conservative solution via Lagrange method. The simulation show the validity of this design method. References: [1] B.D.O.Anderson and J.B.Moore, ”Optimal filtering,” Englehood Cliffs,NJ:PrenticeHall,1979. [2] Mariton M. ”Jump Linear Systems in Automatic Control,” Marcel-Dekker , New York, 1990. [3] Y.Ji, H.J.Chizeck. ”Controllability,stability, and continuous-time Markovian jump linear quadratic control,” IEEE Trans. Automat. Contr. 35,1990,pp. 777–788. [4] X.Feng, K.A.Loparo,Y.Ji, and H.J.Chizeck. ”Stochastic stability properties of jump

[9]

[10]

103

linear systems,” IEEE Trans. Automat. Contr. 37,1992,pp. 38–53. F.Dufour and P.Bertrand, ”The filtering problem for continuous-time linear systems with Markovian switching coefficients,” Syst.Contr.Lett., 23,1994,pp. 453–461. de Souza CE, Trofino A and Barbosa KA, ”Mode-independent H-infinity filters for Markovian jump linear systems,” IEEE Trans. Automat. Contr. 51,2006,pp. 1837–1841. Nguang SK and Shi P,”Robust H-infinity output feedback control design for Takagi-Sugeno systems with Markovian jumps: A linear matrix inequality approach,” Journal of dynamic systems measurement and control-Trans. of ASME 128,2006,pp. 617–625. Zhu J,Xi.H, Ji.H et al.”A robust adaptive controller for Markovian jump uncertain nonlinear systems with Wiener noises of unknown covariance,” International Journal of Control Automation and Systems 5,2007,pp. 128–137. Peng Shi , EL-K´ ebir Boukas ,Ramesh K.Agarwal. ”Kalman Filtering for Continuoustime Uncertain Systems with Markovian Jumping Parameters,” IEEE Trans. Automat. Contr. 44,1999,pp. 1592–1597. Magdi S.Mahmoud , Peng Shi, Abdulla Ismail. ”Robust Kalman Filtering for Discretetime Markovian Jump Systems with Parameter Uncertainty”. Journal of computational and applied mathematics 169,2004, pp. 53–69.