- Email: [email protected]
Nonlinear H∞ filtering for interconnected Markovian jump systems1
Journal of Systems Engineering and Electronics, Vol .17, No. 1 , 2006, p p . 138
- 146
Nonlinear H , filtering for interconnected Markovian jump systems* Z h n g Xiaomei1'2 8z Zheng Yufan2 1. School of Science, Nantong Univ., Natong 226007, P. R. China; 2. Dept. of Computer Science and Technology, East China Normal Univ. , Shanghai 200062, P. R. China (Received Decembex 29, 2004) The problem of nonlinear H , filtering for interconnected Markovian jump systems is discussed. The aim of this note is the design of a nonlinear Markovian jump filter such that the resulting error system is exponentially meansquare stable and ensures a prescribed H , performance. A sufficient condition for the solvability of this problem is given in terms of linear matrix inequalities( LMIs) . A simulation example is presented to demonstrate the effectiveness of the propxed design approach.
&tract:
Key words: nonlinear H , filtering, Markovian jump systems, interconnected systems, linear matrix inequalities
1. INTRODUCTION State estimation or observer design has been an important research topic and has been widely used in control and signal processing in the last few One of the estimation approaches is Kalman filtering, which is based on the assumption that the system model under consideration is available and its disturbances are Gaussian noises with known statistics. However, the noise sources may not be exactly known and uncertainties may appear. Therefore, an alternative estimation approach called the H , filtering has been introduced. In the H , setting, the noise sources are arbitrary signals with bounded energy or average power, and no exact statistics are required to be known. Recently, considerable attention has been devoted to the study of Markovian jump linear systems. This class of systems has two components in the state vector. The first one which varies continuously is known as the continuous state of the system and the second one which varies discretely is referred to be the mode of the system. The family of systems was introduced by Krasovskii and Lidskii in 196lL4].For this class of systems the problems of stability, filtering, and optimal control have been analyzed in the recent literature, see, Refs. [ 5 81. However, to the best of our knowledge, to date, the nonlinear H , filtering problem for interconnected Markovian jump systems
-
has not been drawn much attention yet. In this paper, we are concerned with the problem of nonlinear H m filtering for interconnected Markovian jump systems. The problem we address is to design a nonlinear Markovian jump filter such that the resulting error system is exponentially meansquare stable and ensures a prescribed H , performance. Notation llmughout in this paper, R" and R" denote respectively the n-dimensional Euclidean space and the set of all n X rn real matrices; the supersuipt 5 represents the trans-; X> Y (or X > Y respxtivdy) where X and Y aresymmetricmatrices, meansthat XY is pmitive semi-definite (or positive definite respectively); I is the identity matrix with compatible dimension (without mnfusion) ; ( 0 ,F ,P ) is a complete probability space with 0 the sample space and F the o-algebra of suhsetsofthemnplespace; %-{At isthetraceofamatrixA; E l * \ denotes theexpectationoperator; Lz[O, m) is the space of squareintegrable vector functions over [O,m);
11 11
is the Euclidean vector norm; denotesthe Lz[O,m) f l ~ r mover [O,m) and denotesthe L 2 ( 0 , F , P ) nonnover
A&(
0
)
and
11 11 L, II II E ,
[ o , ~ ) that , is
Am( .) denote the minimum and maxi-
mum eigenvalue of a symmetric matrix respectively. Matrices, if not explicitly stated, are assumed to have
* This project was supported by the National Natural Science Foundation ofChina (60474076) ; NSF (04KJB510105) from the Jiansu Provincial Department for Education and NSF from Nantong University (052001).
Nonlinear H , filtering for interconnected Markovian jump systems
139
compatible dimensions.
Consider a class of nonlinear interconnected Markovian jump systems which is composed of N subsystems described as follows.
Ai(~(t))~i(t) + Bi( +( t ) I f ; ( t xi( t 1 +( t 1) + =
(&):ii(t)
N
E i ( + ( t ) ) w i ( t )+
C
[ A i j ( r ( t ) ) ~ j (+ t)
j = 1,j f i
Bij (
r (t )> f i j ( t x j ( t
r ( t )) 1
(1)
y i ( t > = C i ( r ( t ) > ~ j (+t )Di(r(t))~i(t) (2) 2;( t
x i ( O ) = 0,
)=
Hi
( r ( t >xi( t )
r(O) = ro,
(3)
i = 1 , 2 , * - . , N(4)
where x;( t ) € R"i is the state of the i th subsystem,
w;€ RPi is the disturbance input which belongs to L2 10, +a),y ; ( t ) € R q iand z i ( t ) € R m iare themeasurement and the signal to be estimated of the ith subsystem respectively, i = 1,2, * - . ,N. Let r ( t ) be a continuous-time Markovian process with right continuous trajectories and taking values in a finite set
S = { 1,2,
,s 1
with transition probability matrix
& ( t ) = H ; ( r ( t ) ) G i ( t ) , i = 1,2,...,N ( 6 ) where & ( t ) € R"i is the state estimate, Z i ( t ) is an estimate of zi ( t ) . For each r ( t ) € S , the matrices L i ( r ( t ) ) ,i = 1 , 2 ; - - , N , are to be determined.
Let the error state be defined by e i ( t ) : = x;(t) - & ( t ) then the filtering error dynamics is givenb by
N
C
[ A i j ( r ( t ) ) e j ( t )+ BG(r(t>>(fG(t,xj(t>,r(t)) - f ; j ( t , G j ( t > , r ( t > > >+I
j=l,j#i
[ E i ( r (t ) ) - Li ( r (t )>oi( r ( t ) ) I wi ( t ) Denote i i ( t ) = z i ( t ) - Z i ( t ) , then
ii( t )
=
Hi ( r ( t ) lei ( t )
(8)
The objective of this paper is to determine the filter (2') of the form ( 5 ) , ( 6 ) , i = 1 , 2 , . - . , N , such that, for a given scalar Y >0,
(7)
(1) the filtering error dynamics of the form (7) ( i = 1 , 2 , * - * , N ) is exponentially mean-square stable, which means when w; ( t ) = O , i = 1,2, ,N , for all initial mode roE S , the following condition is satisfied
140
Zhang X i a o m ' & Zheng Yufan Lemma lL9]For any x , y E R " , 2xTy
T T formreamstants b>O and c>O, wheree = (el,eZ,
.-*,e;)T, e ( t ,e ( O ) ,q), demtes the trajectory of the ermr state e ( t ) fiamthe initial state (e(O),ro).
< x T u - l x + yTUy
holds for any U >O
.
(2) if ei(O)=O andw = ( w ~ , w ~ , . . * , then
In this section, we first present a sufficient condition for the exponential mean-square stability of the interconnected Markovian jump system of the forms ( 1 ) ( 3 ) with wi(t ) - O , i = 1,2, , N . 'Iheorem 1 The interconnected Markovian jump systmof the form ( 1 ) ( 3 ) with w i( t ) = O , i = 1 , 2 ; - - , N is exponentially mean-square stable if there exist matrices Pi ( 1 ) > 0 , Pi ( 2 ) > 0 , ..-, Pi(s)>O,R,j(l) > O , R, ( 2 ) > O , *..,R,j ( S) >O and positive Scalars ei ( 1 1 , ei ( 2 ) ,* - * , .ci (s) ,aG( I ) , 8, ( 2 1 , ..*,Sij (s) , i ,j = 1 , 2 , -.-,N , j # i , such that the following LMIs hold for a = 1 ,2, ,s ,i = 1,2, ,N
-
---
-
In this case, the system of the form (7) - ( 8 ) ( i = 1 , 2 , - * - , N ) is said to be exponentially meansquare stable with disturbance attenuation Y . The following matrix inequality is essential for the proofs in next sections.
.--
--.
where
Proof
Let the mode at time t be a ; that is,
r ( t ) = a E S . Consider the stochastic Lyapunov function candidate for the system of the form ( 1 ) ( 3 ) with wi ( t)-O, i = 1 , 2 , N
-
- a * ,
wherex = ( ~ T , x $ , - * - , x ; )Let ~ . A be the weak generator of { ( s ( t ) ,r ( t ) ) 1 ,t 2 0 defined by
A V ( s ( t ) , r ( t ) )= l i m 1d [ E { V ( x ( t+ A ) , A-0
Using Schur complement, it is easy to check that ( 1 0 ) implies
S;(a) < 0 , i Hence, for x#O
= 1,2,..*,N
(15)
where
=
A1
oES
N
o€S
=
l
Pi ( a ) 1 , which implies the system
max { maxA,(
,
mini min ( A - ( P i ( a > > 1 , A Z
lSiSN
( X i with ) w i ( t ) = 0 ( i = l , 2 , * * . , N ) is exponentially mean-square stable. The proof is thus completed. Now we are in a position to give our main result. Theorem 2 Consider the nonlinear interconnected Markovian jump system of the form ( 1 ) (3) i = 1 , 2 , * - * , N ,andlet y > O beagivenscalar. Then, there exists a nonlinear Markovian jump filter ( C { ) ( i = 1 , 2 , * - - , N in ) the formof ( 5 ) and ( 6 ) such that thesystemof the form ( 7 ) - ( 8 ) ( i = 1 , 2 , , N ) is exponentially mean-square stable with disturbance attenuation r , if for each r ( t ) = a € S , there exist positive-definite matrices Xi( 1 ) > 0 , X i ( 2 ) >o, * * * , Xi(S)>o, Y i ( 1 )>o, Y i ( 2 )>o, ..-,
-
---
Yi(s)>O and Rij(l)>O, R i j ( 2 ) > 0 , R i j ( s ) >0, and matrices Zi( 1 1 , Zi( 2 1 , 2; ( s ) , and positive scalars E ; ( 1 1 , E ; ( 2 1 , *..,~i ( s I , & +( 1 ) , 6+(2),-.*,a+(s ) ,i ,j = 1 , 2 , N , j # i , such that the following LMIs hold for a = 1 , 2 , s, i = 1 , 2 , ***, N *-a,
.a*,
. a * ,
*
*
*
(19)
Zhang Xiaomei 8z Zheng Yufan
142
In this case, a suitable nonlinear H , filter (2:) ( i = 1 , 2 , - - - , N )in the fonnof ( 5 ) and ( 6 ) has parameters as follows
Li(a) = ~ ; l ( a ) ~ i ( a )
(Zi)$t)
a
E S, i
= 1,2,*.*,N
(20)
Proof Define ~ ~ (=t ()x T ( t ) , e T ( t ) ) T , then, for each r ( t ) = a E S , the filtering error dy= 1,2, namics from the systems (2i)and ($),i ---,N, is described by
= A i ( a ) T i ( t ) + Bi(.>zi(t,si(t),,i(t),a)
+&(a)q(t) +
F i J a ) = ( 0 H i ( a ) ) , i,j = 1 , 2 , - - * , N , # j i Now, we establish the exponential mean-square stability of the filtering error system of the form (21) and (22),
which implies the filtering error system of the form (7) is exponentially man-square stable. To this end, we mn-
Nonlinear H , filtering fbr interconnected Markovian jump systems sider (21) with w i ( t ) = O , i = 1 , 2 , * - - , N .
143
the parameters in ( 2 0 ) , the LMI in (19) is equiva-
where
It is easy to see that ( 2 3 ) implies that
Define a stochastic Lyapunov function candidate for the system of the fonn ( 2 1 ) and ( 2 2 ) as follows N
V(
t ) r( t )) =
C qT( t >Pi( r ( t ) ) qi( t ) ( 2 5 ) i=l
where q = ( q ~ , q ~ , - * - , q ~ ) T .
Then, b y noticing ( 2 4 ) and following the similar line as in the proof of Theorem 1 , we can deduce that the filtering error system (2i) ( i = 1 , 2 , - * . ,N ) is exponentially mean-square stable. Next, we will show ( 9 ) holds for any nonzero w . Notice forany r ( t ) = a E S
144
Zhang Xiaomei & Zheng Yufan
where
Pi ( a )Ei( a ) -
]
4. NUMERICAL. SFMULATION In this section, we shall give a numerical simulation example to illustrate the applicability of the proposed approach. Consider the nonlinear interconnected Markovim jump system in the form ( 1 ) - ( 4 ) with two modes and N = 2. The transition probability matrix
y2z
Using the Schur complement, it follows from (23) that
&&> < 0
(28)
Therefore, J ( T )<0, which implies that (9) holds for any nonzero w . The proof is thus completed.
A=
( -y*5 y:). For mode 1, the system (1)-
(3) with parameters as follows -5 1 0.1 0.2 0.1 0 = (0.2 -3)Y = (0.3 0 . 6 ) , ~ 1 2 ( 1 )= o.l) 0.1 sin 5 1 2 0.1 sin 5 2 1 0 . 1 sin 511 0.2 0 B12(1) = o . l ) , El(1) = i0l6) c1(1) = (2 3 ) , Dl(1) = 0.01, Hi(1) = ( 0 . 5 - 0 . 1 ) 0.1 2 0 . 1 0.1 - 1.5 -2.5
(
(
0.5 C2(1) = (2 O),D,(l) = O.O2,H2(1) = (0 1) F1(1) = 0.11, F12(1) = 0.31, F2(1) = 0.21, F21(1) = 0.11 For mode 2, the system (1)-(3) with parameters is given by
(
0 0.1 0.1 o . l ) , A12(2) = 0 0.2 0 . 1 sin 5 2 1 f12(t,52,2) = 0.1 sin 5 2 2
- 2 0.5 0.1 Ai(2) = (0.2 - 5 ) . Bi(2) = fl(t,51,2)
(0.1
=
1
0.1 cos 5 1 2 - 0.2 sin 511
("),
9
(
)
0.1 o . l ) , Ei(2) = C1(2) = (2 l ) , 01(2) = 0.02,H1(2) = (0 1) A2(2> = '*'), B2(2) = 0.9 0 . 1 A21(2) = 0 . 1 0 -5 0.1 0.2 0 0.1 0.01 cos 5 2 1 0.1 0 s 511 f2(t,52,2) = ,f21(t,51,2) = 0.1 sin( 5 2 1 + 522) 0.5 cos(511 + 512) 0.1 0 0.5 = o . l ) , E2(2> = (o.8), C2(2>= (0.8 0.2), D2(2) = 0.01, H2(2) = (1 0)
B12(2) =
iOal
(-;*'
(
1
),
(
)
1
Nonlinear H , filtering for interconnected Markovian j u m p systems
where X I = (xll T
145
Fi(2) = 0.21, F 1 2 = 0.11, F21(2) = 0.51, F2(2) = 0.11 xT2)*,x2= (x& 512)T
Inthisexample, weassume Y = 1 . 5 and R12(l)=R21(1)=R12(2)=R21(2)=1. UsingMatlab LMI Control Toolbox to solve the LMIs (19) in Theorem 2, we obtain
x2(1) =
1.192 5 (0.079 2
),
0.079 2 Yl(1) = 1.743 5 1.444 5 - 0.200 2 ) , Y2(1> = 1.975 2 0.200 2 2.183 7 0.072 3 = ) , YI(2) = 0.072 3 1.243 5 1.737 5 0.094 8 0.094 8 1.199 5
Xl(1) =
(-
(
1.283 6 0.238 7 (0.238 7 2.021 6 1 1.856 7 3 ( -0.289 3 0.289 2.041 1 8 0.144 5 (2.360 0.144 5 1.324 5 1 -
2.271 5 0.124 1 0.124 1 1.195 7
€1(1) = 4.996 9,€2(1) = 5.579 4,€1(2) = 4.960 1 , ~ 2 ( 2 )= 5.102 5 612(1> = 4.919 8,&1(l) = 4.919 1,812(2) = 4.927,621(2) = 5.274 9 Therefore, by Theorem 2, the corresponding parameters of a suitable nonlinear Markovian jump H , filter can be chosen as
- 0.220 4
-
0.594 1
- 0.127
Denote the error states eil= xi1 , ei2 = xi2 - & ( i = 1 , 2 ) . The simulation results for the error dy3n
2
-
namics mode 1 and mode 2 are shown in Figures 14, respectively. I
I
-1
b
1
2
3
4
-:ell; --:e12
5
k
1.077 9 0.081 8
-
2
b
1 2 3 4 5 8 -:eZI; -.--:eZ2
Fig. 1 Responses of the fust error Fig.2 Responses of the first error subsystem to initial condition(2,3) subsystem to initial condition(4,-1)
5 . CONCLUSION In this paper, the problem of nonlinear H , filtering is studied for interconnected Markovian jump systems. An LMI approach has been developed to design a nonlinear Markovian jump filter, which ensures that the resulting error system is exponentially stable in the mean square and guarantees a prescribed H , performance. The desired filter can be constructed by means of coupled LMIs if they are solvable. The effectiveness of the proposed design method has been illustrated by a simulation example.
REFERENCES [ l ] Gao 2 W. The class of all reduced-order state observers.
0
1
2 3 4 5 -:el,; -.--:e12
6
'
-2i
1
2 3 4 5 -.e . 21'. -.-.:e22
6
Fig.3 Responses of the first error Fig.4 Responses of the first error subsystem to initial condition(4,3) subsystem to initial condition(2,-2)
Systems Analysis Malelling Simulation ,2002,42 :13091317. [ 21 Gao Z W , A T P So. On parameterization design for linear observers. Journal of the Frankin Institute, 2003,340: 87-102. [3] Lu G P, Daniel W C Ho. Robust H , observer for a class of nonlinear discrete systems. The Proceedings of the 5th Asian Control Conference , 2004 :262- 267. [ 4 ] Kramvskii N N, Lidskii E A. Analytical design of controllers in systems with random attributes. Automation
-
and Remote Control, 1961, 22 (1-111) : 1021 1025.
[ 5 ] Boukas E K, Shi P, Benjelloun K. On stabilization of uncertain linear systems with parameters. International Journal of Control, 1999, 72: 842-850. [6] Boukas E K, Liu Z K. Robust H , filtering for pdytopic uncertain time-delay systems with Markovian jumps.
Zhang Xiaomei & Zheng Yufan
146 Proc. of the American Control Confmence, 2000: 3795
-3799. [71 Gta 0 L V, Tuesta E F. Finite horizon quadratic optimal control and a separation principle for Markovian jump linear system. IEEE Tran on Automatic Control, 2003,
48: 1836-1842. [S] Ji Y, Chizeck H J. Controllability, stability, and umtinuoustime Markovian jump linear quadratic control. IEEE Tran on Automatic Control, 1990, 35: 777-788. [9] Wang Y, Xie L, De Souza C E. Robust control of a class of uncertain nonlinear systems. Systems @ Control Letters, 1992, 19: 139-149.
[ 101 Davis M H A. Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models. Juurrial of the Row1 Statistical Society, 1984, B46: 353 -
388. [ 111 Boyd S, El Ghaoui L, Feron E, et al. Linear matrix in-
equalities in systems and control theory. SL4M studies in applied mathematics, Philadephia, P A : Society fw industrial and applied mathematics , 1994.
Zhang Xiaomei was born in 1964. She is an associated professor of Department of Applied Mathematics of Nantong University. Presently she is studying for Ph. D in Department of Computer Science and Technology of East China Normal University. E-mail: [email protected]. jsinfo. net Zheng Yufan was born in 1941. He is a Ph. D student supervisor of Department of Computer Science and Technology of East China Normal University. His major research interest is the theory of nonlinear control.
- 146
Nonlinear H , filtering for interconnected Markovian jump systems* Z h n g Xiaomei1'2 8z Zheng Yufan2 1. School of Science, Nantong Univ., Natong 226007, P. R. China; 2. Dept. of Computer Science and Technology, East China Normal Univ. , Shanghai 200062, P. R. China (Received Decembex 29, 2004) The problem of nonlinear H , filtering for interconnected Markovian jump systems is discussed. The aim of this note is the design of a nonlinear Markovian jump filter such that the resulting error system is exponentially meansquare stable and ensures a prescribed H , performance. A sufficient condition for the solvability of this problem is given in terms of linear matrix inequalities( LMIs) . A simulation example is presented to demonstrate the effectiveness of the propxed design approach.
&tract:
Key words: nonlinear H , filtering, Markovian jump systems, interconnected systems, linear matrix inequalities
1. INTRODUCTION State estimation or observer design has been an important research topic and has been widely used in control and signal processing in the last few One of the estimation approaches is Kalman filtering, which is based on the assumption that the system model under consideration is available and its disturbances are Gaussian noises with known statistics. However, the noise sources may not be exactly known and uncertainties may appear. Therefore, an alternative estimation approach called the H , filtering has been introduced. In the H , setting, the noise sources are arbitrary signals with bounded energy or average power, and no exact statistics are required to be known. Recently, considerable attention has been devoted to the study of Markovian jump linear systems. This class of systems has two components in the state vector. The first one which varies continuously is known as the continuous state of the system and the second one which varies discretely is referred to be the mode of the system. The family of systems was introduced by Krasovskii and Lidskii in 196lL4].For this class of systems the problems of stability, filtering, and optimal control have been analyzed in the recent literature, see, Refs. [ 5 81. However, to the best of our knowledge, to date, the nonlinear H , filtering problem for interconnected Markovian jump systems
-
has not been drawn much attention yet. In this paper, we are concerned with the problem of nonlinear H m filtering for interconnected Markovian jump systems. The problem we address is to design a nonlinear Markovian jump filter such that the resulting error system is exponentially meansquare stable and ensures a prescribed H , performance. Notation llmughout in this paper, R" and R" denote respectively the n-dimensional Euclidean space and the set of all n X rn real matrices; the supersuipt 5 represents the trans-; X> Y (or X > Y respxtivdy) where X and Y aresymmetricmatrices, meansthat XY is pmitive semi-definite (or positive definite respectively); I is the identity matrix with compatible dimension (without mnfusion) ; ( 0 ,F ,P ) is a complete probability space with 0 the sample space and F the o-algebra of suhsetsofthemnplespace; %-{At isthetraceofamatrixA; E l * \ denotes theexpectationoperator; Lz[O, m) is the space of squareintegrable vector functions over [O,m);
11 11
is the Euclidean vector norm; denotesthe Lz[O,m) f l ~ r mover [O,m) and denotesthe L 2 ( 0 , F , P ) nonnover
A&(
0
)
and
11 11 L, II II E ,
[ o , ~ ) that , is
Am( .) denote the minimum and maxi-
mum eigenvalue of a symmetric matrix respectively. Matrices, if not explicitly stated, are assumed to have
* This project was supported by the National Natural Science Foundation ofChina (60474076) ; NSF (04KJB510105) from the Jiansu Provincial Department for Education and NSF from Nantong University (052001).
Nonlinear H , filtering for interconnected Markovian jump systems
139
compatible dimensions.
Consider a class of nonlinear interconnected Markovian jump systems which is composed of N subsystems described as follows.
Ai(~(t))~i(t) + Bi( +( t ) I f ; ( t xi( t 1 +( t 1) + =
(&):ii(t)
N
E i ( + ( t ) ) w i ( t )+
C
[ A i j ( r ( t ) ) ~ j (+ t)
j = 1,j f i
Bij (
r (t )> f i j ( t x j ( t
r ( t )) 1
(1)
y i ( t > = C i ( r ( t ) > ~ j (+t )Di(r(t))~i(t) (2) 2;( t
x i ( O ) = 0,
)=
Hi
( r ( t >xi( t )
r(O) = ro,
(3)
i = 1 , 2 , * - . , N(4)
where x;( t ) € R"i is the state of the i th subsystem,
w;€ RPi is the disturbance input which belongs to L2 10, +a),y ; ( t ) € R q iand z i ( t ) € R m iare themeasurement and the signal to be estimated of the ith subsystem respectively, i = 1,2, * - . ,N. Let r ( t ) be a continuous-time Markovian process with right continuous trajectories and taking values in a finite set
S = { 1,2,
,s 1
with transition probability matrix
& ( t ) = H ; ( r ( t ) ) G i ( t ) , i = 1,2,...,N ( 6 ) where & ( t ) € R"i is the state estimate, Z i ( t ) is an estimate of zi ( t ) . For each r ( t ) € S , the matrices L i ( r ( t ) ) ,i = 1 , 2 ; - - , N , are to be determined.
Let the error state be defined by e i ( t ) : = x;(t) - & ( t ) then the filtering error dynamics is givenb by
N
C
[ A i j ( r ( t ) ) e j ( t )+ BG(r(t>>(fG(t,xj(t>,r(t)) - f ; j ( t , G j ( t > , r ( t > > >+I
j=l,j#i
[ E i ( r (t ) ) - Li ( r (t )>oi( r ( t ) ) I wi ( t ) Denote i i ( t ) = z i ( t ) - Z i ( t ) , then
ii( t )
=
Hi ( r ( t ) lei ( t )
(8)
The objective of this paper is to determine the filter (2') of the form ( 5 ) , ( 6 ) , i = 1 , 2 , . - . , N , such that, for a given scalar Y >0,
(7)
(1) the filtering error dynamics of the form (7) ( i = 1 , 2 , * - * , N ) is exponentially mean-square stable, which means when w; ( t ) = O , i = 1,2, ,N , for all initial mode roE S , the following condition is satisfied
140
Zhang X i a o m ' & Zheng Yufan Lemma lL9]For any x , y E R " , 2xTy
T T formreamstants b>O and c>O, wheree = (el,eZ,
.-*,e;)T, e ( t ,e ( O ) ,q), demtes the trajectory of the ermr state e ( t ) fiamthe initial state (e(O),ro).
< x T u - l x + yTUy
holds for any U >O
.
(2) if ei(O)=O andw = ( w ~ , w ~ , . . * , then
In this section, we first present a sufficient condition for the exponential mean-square stability of the interconnected Markovian jump system of the forms ( 1 ) ( 3 ) with wi(t ) - O , i = 1,2, , N . 'Iheorem 1 The interconnected Markovian jump systmof the form ( 1 ) ( 3 ) with w i( t ) = O , i = 1 , 2 ; - - , N is exponentially mean-square stable if there exist matrices Pi ( 1 ) > 0 , Pi ( 2 ) > 0 , ..-, Pi(s)>O,R,j(l) > O , R, ( 2 ) > O , *..,R,j ( S) >O and positive Scalars ei ( 1 1 , ei ( 2 ) ,* - * , .ci (s) ,aG( I ) , 8, ( 2 1 , ..*,Sij (s) , i ,j = 1 , 2 , -.-,N , j # i , such that the following LMIs hold for a = 1 ,2, ,s ,i = 1,2, ,N
-
---
-
In this case, the system of the form (7) - ( 8 ) ( i = 1 , 2 , - * - , N ) is said to be exponentially meansquare stable with disturbance attenuation Y . The following matrix inequality is essential for the proofs in next sections.
.--
--.
where
Proof
Let the mode at time t be a ; that is,
r ( t ) = a E S . Consider the stochastic Lyapunov function candidate for the system of the form ( 1 ) ( 3 ) with wi ( t)-O, i = 1 , 2 , N
-
- a * ,
wherex = ( ~ T , x $ , - * - , x ; )Let ~ . A be the weak generator of { ( s ( t ) ,r ( t ) ) 1 ,t 2 0 defined by
A V ( s ( t ) , r ( t ) )= l i m 1d [ E { V ( x ( t+ A ) , A-0
Using Schur complement, it is easy to check that ( 1 0 ) implies
S;(a) < 0 , i Hence, for x#O
= 1,2,..*,N
(15)
where
=
A1
oES
N
o€S
=
l
Pi ( a ) 1 , which implies the system
max { maxA,(
,
mini min ( A - ( P i ( a > > 1 , A Z
lSiSN
( X i with ) w i ( t ) = 0 ( i = l , 2 , * * . , N ) is exponentially mean-square stable. The proof is thus completed. Now we are in a position to give our main result. Theorem 2 Consider the nonlinear interconnected Markovian jump system of the form ( 1 ) (3) i = 1 , 2 , * - * , N ,andlet y > O beagivenscalar. Then, there exists a nonlinear Markovian jump filter ( C { ) ( i = 1 , 2 , * - - , N in ) the formof ( 5 ) and ( 6 ) such that thesystemof the form ( 7 ) - ( 8 ) ( i = 1 , 2 , , N ) is exponentially mean-square stable with disturbance attenuation r , if for each r ( t ) = a € S , there exist positive-definite matrices Xi( 1 ) > 0 , X i ( 2 ) >o, * * * , Xi(S)>o, Y i ( 1 )>o, Y i ( 2 )>o, ..-,
-
---
Yi(s)>O and Rij(l)>O, R i j ( 2 ) > 0 , R i j ( s ) >0, and matrices Zi( 1 1 , Zi( 2 1 , 2; ( s ) , and positive scalars E ; ( 1 1 , E ; ( 2 1 , *..,~i ( s I , & +( 1 ) , 6+(2),-.*,a+(s ) ,i ,j = 1 , 2 , N , j # i , such that the following LMIs hold for a = 1 , 2 , s, i = 1 , 2 , ***, N *-a,
.a*,
. a * ,
*
*
*
(19)
Zhang Xiaomei 8z Zheng Yufan
142
In this case, a suitable nonlinear H , filter (2:) ( i = 1 , 2 , - - - , N )in the fonnof ( 5 ) and ( 6 ) has parameters as follows
Li(a) = ~ ; l ( a ) ~ i ( a )
(Zi)$t)
a
E S, i
= 1,2,*.*,N
(20)
Proof Define ~ ~ (=t ()x T ( t ) , e T ( t ) ) T , then, for each r ( t ) = a E S , the filtering error dy= 1,2, namics from the systems (2i)and ($),i ---,N, is described by
= A i ( a ) T i ( t ) + Bi(.>zi(t,si(t),,i(t),a)
+&(a)q(t) +
F i J a ) = ( 0 H i ( a ) ) , i,j = 1 , 2 , - - * , N , # j i Now, we establish the exponential mean-square stability of the filtering error system of the form (21) and (22),
which implies the filtering error system of the form (7) is exponentially man-square stable. To this end, we mn-
Nonlinear H , filtering fbr interconnected Markovian jump systems sider (21) with w i ( t ) = O , i = 1 , 2 , * - - , N .
143
the parameters in ( 2 0 ) , the LMI in (19) is equiva-
where
It is easy to see that ( 2 3 ) implies that
Define a stochastic Lyapunov function candidate for the system of the fonn ( 2 1 ) and ( 2 2 ) as follows N
V(
t ) r( t )) =
C qT( t >Pi( r ( t ) ) qi( t ) ( 2 5 ) i=l
where q = ( q ~ , q ~ , - * - , q ~ ) T .
Then, b y noticing ( 2 4 ) and following the similar line as in the proof of Theorem 1 , we can deduce that the filtering error system (2i) ( i = 1 , 2 , - * . ,N ) is exponentially mean-square stable. Next, we will show ( 9 ) holds for any nonzero w . Notice forany r ( t ) = a E S
144
Zhang Xiaomei & Zheng Yufan
where
Pi ( a )Ei( a ) -
]
4. NUMERICAL. SFMULATION In this section, we shall give a numerical simulation example to illustrate the applicability of the proposed approach. Consider the nonlinear interconnected Markovim jump system in the form ( 1 ) - ( 4 ) with two modes and N = 2. The transition probability matrix
y2z
Using the Schur complement, it follows from (23) that
&&> < 0
(28)
Therefore, J ( T )<0, which implies that (9) holds for any nonzero w . The proof is thus completed.
A=
( -y*5 y:). For mode 1, the system (1)-
(3) with parameters as follows -5 1 0.1 0.2 0.1 0 = (0.2 -3)Y = (0.3 0 . 6 ) , ~ 1 2 ( 1 )= o.l) 0.1 sin 5 1 2 0.1 sin 5 2 1 0 . 1 sin 511 0.2 0 B12(1) = o . l ) , El(1) = i0l6) c1(1) = (2 3 ) , Dl(1) = 0.01, Hi(1) = ( 0 . 5 - 0 . 1 ) 0.1 2 0 . 1 0.1 - 1.5 -2.5
(
(
0.5 C2(1) = (2 O),D,(l) = O.O2,H2(1) = (0 1) F1(1) = 0.11, F12(1) = 0.31, F2(1) = 0.21, F21(1) = 0.11 For mode 2, the system (1)-(3) with parameters is given by
(
0 0.1 0.1 o . l ) , A12(2) = 0 0.2 0 . 1 sin 5 2 1 f12(t,52,2) = 0.1 sin 5 2 2
- 2 0.5 0.1 Ai(2) = (0.2 - 5 ) . Bi(2) = fl(t,51,2)
(0.1
=
1
0.1 cos 5 1 2 - 0.2 sin 511
("),
9
(
)
0.1 o . l ) , Ei(2) = C1(2) = (2 l ) , 01(2) = 0.02,H1(2) = (0 1) A2(2> = '*'), B2(2) = 0.9 0 . 1 A21(2) = 0 . 1 0 -5 0.1 0.2 0 0.1 0.01 cos 5 2 1 0.1 0 s 511 f2(t,52,2) = ,f21(t,51,2) = 0.1 sin( 5 2 1 + 522) 0.5 cos(511 + 512) 0.1 0 0.5 = o . l ) , E2(2> = (o.8), C2(2>= (0.8 0.2), D2(2) = 0.01, H2(2) = (1 0)
B12(2) =
iOal
(-;*'
(
1
),
(
)
1
Nonlinear H , filtering for interconnected Markovian j u m p systems
where X I = (xll T
145
Fi(2) = 0.21, F 1 2 = 0.11, F21(2) = 0.51, F2(2) = 0.11 xT2)*,x2= (x& 512)T
Inthisexample, weassume Y = 1 . 5 and R12(l)=R21(1)=R12(2)=R21(2)=1. UsingMatlab LMI Control Toolbox to solve the LMIs (19) in Theorem 2, we obtain
x2(1) =
1.192 5 (0.079 2
),
0.079 2 Yl(1) = 1.743 5 1.444 5 - 0.200 2 ) , Y2(1> = 1.975 2 0.200 2 2.183 7 0.072 3 = ) , YI(2) = 0.072 3 1.243 5 1.737 5 0.094 8 0.094 8 1.199 5
Xl(1) =
(-
(
1.283 6 0.238 7 (0.238 7 2.021 6 1 1.856 7 3 ( -0.289 3 0.289 2.041 1 8 0.144 5 (2.360 0.144 5 1.324 5 1 -
2.271 5 0.124 1 0.124 1 1.195 7
€1(1) = 4.996 9,€2(1) = 5.579 4,€1(2) = 4.960 1 , ~ 2 ( 2 )= 5.102 5 612(1> = 4.919 8,&1(l) = 4.919 1,812(2) = 4.927,621(2) = 5.274 9 Therefore, by Theorem 2, the corresponding parameters of a suitable nonlinear Markovian jump H , filter can be chosen as
- 0.220 4
-
0.594 1
- 0.127
Denote the error states eil= xi1 , ei2 = xi2 - & ( i = 1 , 2 ) . The simulation results for the error dy3n
2
-
namics mode 1 and mode 2 are shown in Figures 14, respectively. I
I
-1
b
1
2
3
4
-:ell; --:e12
5
k
1.077 9 0.081 8
-
2
b
1 2 3 4 5 8 -:eZI; -.--:eZ2
Fig. 1 Responses of the fust error Fig.2 Responses of the first error subsystem to initial condition(2,3) subsystem to initial condition(4,-1)
5 . CONCLUSION In this paper, the problem of nonlinear H , filtering is studied for interconnected Markovian jump systems. An LMI approach has been developed to design a nonlinear Markovian jump filter, which ensures that the resulting error system is exponentially stable in the mean square and guarantees a prescribed H , performance. The desired filter can be constructed by means of coupled LMIs if they are solvable. The effectiveness of the proposed design method has been illustrated by a simulation example.
REFERENCES [ l ] Gao 2 W. The class of all reduced-order state observers.
0
1
2 3 4 5 -:el,; -.--:e12
6
'
-2i
1
2 3 4 5 -.e . 21'. -.-.:e22
6
Fig.3 Responses of the first error Fig.4 Responses of the first error subsystem to initial condition(4,3) subsystem to initial condition(2,-2)
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and Remote Control, 1961, 22 (1-111) : 1021 1025.
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Zhang Xiaomei & Zheng Yufan
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Zhang Xiaomei was born in 1964. She is an associated professor of Department of Applied Mathematics of Nantong University. Presently she is studying for Ph. D in Department of Computer Science and Technology of East China Normal University. E-mail: [email protected]. jsinfo. net Zheng Yufan was born in 1941. He is a Ph. D student supervisor of Department of Computer Science and Technology of East China Normal University. His major research interest is the theory of nonlinear control.