H1 Filtering for Markovian Jump Linear Systemsy Marcelo D. Fragoso1 and Carlos E. de Souza2 1 Department of Research and Development National Laboratory for Scienti c Computing - LNCC/CNPq Rua Lauro Muller 455, 22290-160 Rio de Janeiro, RJ, Brazil. 2 Department of Electrical and Computer Engineering The University of Newcastle, NSW 2308, Australia. Fax: (49) 216 993 - email:
[email protected] Fax: (21) 2958499 - email:
[email protected]
Abstract The problem of H1 ltering for continuous-time linear systems with Markovian jump is investigated. It is assumed that the jumping parameter is available. We propose a methodology for designing Markovian jump linear lters which ensure a prescribed bound on the L2 -induced gain from the noise signals to the estimation error. The main result is tailored via linear matrix inequalities.
Keywords: H1 ltering, Markovian jump linear systems, state estimation, linear matrix inequalities.
This work was carried out while the rst author was visiting the National Laboratory for Scienti c Computing, LNCC/CNPq, Rio de Janeiro, Brazil. y Accepted for presentation in the 35th Conference on Decision and Control, Kobe, Japan, December 1996.
1 Introduction It is a well known fact that in a great variety of stochastic modelling problems, it is very dicult to know precisely the statistics of the additive noise actuating in the system. This is a particularly important issue when we are dealing with what is known in the specialized literature as the ltering problem . One way to deal with this issue is to use a nowadays very popular measure of performance, the H1-norm, which has been introduced in the robust control setting (see, e.g., [6], and the references therein). In this context, the ltering problem is known in the literature as the H1 ltering problem , and its success can be con rmed, in part, by the amount of available literature on this subject; see, for example, [1], [2], [10], [14][16], [20], [21] and the references therein. Roughly speaking, in the H1 ltering approach the noise sources one considers are arbitrary signals with bounded energy, or bounded average power, and the estimator is designed to guarantee that the L2-induced gain from the noise signals to the estimation error be less than a certain prescribed level. The subject matter of this paper is to study the problem of H1 ltering for a class of linear continuous-time systems whose structures are subject to abrupt parameters changes (jumps), modelled here via a continuous-time nite-state Markov chain (it is also known in the literature as the class of Markovian jump linear systems). These changes may be a consequence of random component failures or repairs, abrupt environmental disturbances, changes in the operating point of a nonlinear plant, etc. This can be found, for instance, in control of solar thermal central receivers, robotic manipulator systems, aircraft control systems, large
exible structures for space stations (such as antenna, solar arrays), etc. Several authors have analyzed di erent aspects of such a class and some successful applications have, in part, spurred a considerable interest on it (see, e.g., [3]-[5], [8], [12], [13], [17]-[19], [22] and the references therein). In particular with regard to the ltering problem, minimum variance ltering schemes for discrete-time systems have been studied in, for instance, [3], [4], [18] and [22]. To the best of the authors's knowledge, to date the problem of H1 ltering for this class of systems has not yet been addressed. The problem we consider in this paper is the design of a Markovian jump linear lter for the above class of Markovian jump linear systems, which provides a mean square stable error dynamics and a prescribed bound on the L2-induced gain from the noise signals to the estimation error. A linear matrix inequality (LMI) approach is proposed for solving this H1 ltering problem.
Notation. Throughout the paper the superscript `T ' stands for matrix transposition, 0; 8 t 2 ; and 8 t 0 is imposed to the ltering problem treated in this paper. This is in contrast with the H1 ltering approaches in [14], [15] and [21] for linear systems without jumps. 4
3 The H1 Markovian Jump Filter Before presenting our lter design, the motivation for the approach we adopted will be discussed. First, recall that since t is accessible, the system () could be seen as a linear time-varying system de ned by matrices A(t); B (t); C (t); D(t) and L(t) which can be computed on-line based on the state of the process ft g at time t. Indeed
A(t) = Ai; B (t) = Bi ; C (t) = Ci; D(t) = Di; L(t) = Li when (t) = i. This implies that well known results on H1 ltering for linear time-varying systems, such as those in [14], could be used to solve the H1 ltering problem for system (). Subject to the assumption that
S (t) = D(t)DT (t) > 0; 8 t 0; it follows from the results in [14] that this H1 ltering problem is solvable via a causal linear lter if and only if there exists a bounded symmetric positive de nite solution P (t) over [0; 1) to the Riccati di erential equation: P_ = (A ? BDT S ?1C )P + P (A ? BDT S ?1C )T + P ( ?2LT L ? C T S ?1C )P + B (I ? DT S ?1D)B T ; P (0) = R?1 (3.1) such that the time-varying system h
i
_ (t) = A ? (PC T + BDT )S ?1C + ?2PLT L (t)
(3.2)
is exponentially stable. Note that, for simplicity of notation, we omitted the time-dependence in the matrices of (3.1) and (3.2). Further, a suitable lter is of the form: x^_ (t) = A(t)^x(t) + K (t)[y(t) ? C (t)^x(t)]; x^(0) = x^0 (3.3) z^(t) = L(t)^x(t) (3.4) where
K (t) = [P (t)C (t) + B (t)DT (t)]S ?1(t): (3.5) Note that in view of the above, the lter gain, K (t), of (3.5) will not be constant while system () remains in a certain operation mode, i.e. the jumping process remains in a certain state, j 2 . Thus, the lter of (3.3)-(3.4) is not a Markovian jump linear system, which is highly undesirable, as the underlying system () is a Markovian jump linear system. Another undesirable feature of the above lter is that since the matrices of system () are not available a priori , the existence of a bounded solution to the Riccati di erential equation (3.1) with the required stability property, i.e. the existence of a lter, cannot be ascertained o -line. Also note that the computation of the lter gain, K (t), if it exists, can only be carried out on-line and would involve the solution of a quadratic matrix di erential equation 5
with time-varying matrix coecients, which is numerically unattractive. The lter design proposed in this paper will not exhibit the above undesirable features. The approach developed here to solve the H1 ltering problem for system () has the following features: (a) The design leads to a Markovian jump linear lter; (b) The existence of a lter can be ascertained o -line; (c) The lter gain, K (t ), can assume N possible values, Ki = K (t ) when t = i, and the matrices Ki; 8 i 2 , can be calculated o -line. The following theorem presents a solution to the H1 ltering problem for the Markovian jumping linear system ().
Theorem 3.1 Consider the system () and let 0 be a given scalar. Let x^0 be an a-priori estimate of the initial state and R > 0 a given initial state error weighting matrix. Then there exists a Markovian jump lter of the form of (2.7)-(2.8) such that the estimation error system is internally mean square stable and h
i 21
jjz ? z^jj2 jjwjj22 + (x0 ? x^0 )T R(x0 ? x^0 ) for all w 2 L2 and x0 2 0 and Yi satisfying the following LMIs: 2 6 4
ATi Xi + XiAi ? CiT YiT ? YiCi + PNj=1 ij Xj + LTi Li
XiBi ? YiDi
BiT Xi ? DiT YiT
? 2 I Xi0 ? 2 R 0
3 7 5 < 0;
8i 2 (3.6) (3.7)
where i0 is the state assumed by ft g at t = 0. Moreover, a suitable lter is given by x^_ (t) = Ai x^(t) + Ki [y(t) ? Cix^(t)] ; x^(0) = x^0 (3.8) z^(t) = Li x^(t) (3.9) for t = i; i 2 , where
Ki = K (t = i) = Xi?1Yi:
(3.10)
x ? x^ and considering (2.1)-(2.3) and (3.8)-(3.9), it Proof. First note that by de ning x~ = follows that a state-space model for the estimation error, z ? z^, is given by x~_ (t) = A~(t )~x(t) + B~ (t )w(t); x~(0) = x0 ? x^0 (3.11) z(t) ? z^(t) = L(t )~x(t) (3.12)
6
where
A~(t ) = A(t ) ? K (t )C (t );
B~ (t ) = B (t ) ? K (t )D(t ):
(3.13)
In view of (3.6) and taking into account (3.10), we have that (Ai ? KiCi)T
Xi + Xi (Ai ? KiCi) +
N X j =1
ij Xj < 0; 8 i 2 :
By Theorem 3.1 and Proposition 3.5 in [7], this implies that the error system of (3.11) is internally mean square stable. Now for T > 0, de ne the following cost function:
J (T ) = E
(Z h T
0
i
)
(z ? z^)T (z ? z^) ? 2wT w dt :
(3.14)
Note that by Proposition A.1 in [5], fx~(t); t g is a Markov process with in nitesimal operator given by h
i
T V~ g(~x(t); t) = gt(~x(t); t) + A~(t )~x + B~ (t )w(t) gx~ (~x(t); t )
+
N X j =1
j g(~x(t); j )
(3.15)
t
where g(~x(t); t ) is a real continuous, bounded, functional with partial derivatives: #
"
T @g ; @g ; ; @g gt @g and g = x~ @t @ x~1 @ x~2 @ x~n where x~j denotes the j -th component of x~. Adding and subtracting V~ [~xT (t)X (t )~x(t)] to (3.14), where X (t ) = X T (t ) > 0; 8 t 2 , and considering (3.12), (3.13) and (3.15), we have:
=
J (T ) =
(Z n T T h E x~ (t) (A(
0
+ LT (t )L(t ) +
N X j =1
T t ) ? K (t )C (t )) X (t ) + X (t ) (A(t ) ? K (t )C (t ))
j Xj ] x~(t) ? 2wT (t)w(t) + x~T (t)X (t )[B (t ) ? K (t )D(t)]w(t) t
o
o
+ wT (t)[B (t ) ? K (t )D(t )]T X (t )~x(t) ? V~ [ x~T (t)X (t )~x(t)] dt :
(3.16)
It was shown in [9] that, since the system of (3.11) is internally mean square stable and w 2 L2, it follows that h i T (T )X ( )~ lim E x ~ x ( T ) = 0: (3.17) T T !1 This implies that J (T ) of (3.14) is well de ned as T ! 1. 7
Using Dynkin's formula (see, e.g., [11]) together with (3.17) and de ning
Y (t ) = X (t )K (t ); 8 t 2 it results from (3.16) that h i jjz ? z^jj22 ? 2 jjwjj22 + x~T (0)Rx~(0) =
(Z 1 T E [~x
0
(3.18)
wT ] (t )
"
#
)
x~ w dt
+ x~T (0) Xi0 ? 2R x~(0)
(3.19)
where 2 (t ) = 4
M (t ) B T (t )X (t ) ? DT (t )Y T (t )
3
X (t )B (t ) ? Y (t )D(t ) 5 2 ? I
M (t ) = AT (t )X (t ) + X (t )A(t ) ? C T (t )Y T (t ) ? Y (t )C (t ) +
N X
j =1
j Xj + LT (t )L(t ): t
Finally, considering the inequalities (3.6) and (3.7), the desired result follows from (3.18) and (3.19). rrr
Remark 3.1 Theorem 3.1 provides a method for designing H1 Markovian jump linear
lters for linear systems subject to Markovian jumping parameters. The proposed design is given in terms of linear matrix inequalities, which has the advantage that it can be solved numerically very eciently using recently developed algorithms for solving LMIs. We observe that the problem of designing an optimal H1 Markovian jump linear lter, i.e. for the smallest possible 0, can be easily solved via the following linear programming problem in ; Xi and Yi; 8 i 2 : minimize subject to 0; Xi > 0; (3:6) and (3:7):
When the e ect of the initial state is ignored, without loss of generality, x0 and x^0 can be set to zero. Thus, the inequality of (3.7) will no longer be required as this case corresponds to choosing a sucient large R (in the sense that its smallest eigenvalue approaches in nity). In such situation, Theorem 3.1 specializes as follows:
8
Corollary 3.1 Consider the system () with x(0) = 0 and let 0 be a given scalar.
Then there exists a Markovian jump lter of the form of (2.7)-(2.8) such that the estimation error system is internally mean square stable and
kz ? z^k2 kwk2
for all w 2 L2 , if for all i 2 there exist matrices Xi > 0 and Yi such that the LMI (3.6) is satis ed. Moreover, a suitable lter is given by (3.8)-(3.10) with x^0 = 0.
In the case of one mode operation, i.e. there no jumps in system (), we have N = 1; = f1g and 11 = 0. Denoting the matrices of system () by A; B; C ; D and L, Theorem 3.1 reduces to the following result:
Corollary 3.2 Consider the system () with no jumps and let R > 0 be a given initial state weighting matrix and > 0 a given scalar. Then there exists a causal linear lter such that the estimation error dynamics is asymptotically stable and
kz ? z^k2 kwk2 + xT0 Rx0
21
for all w 2 L2 and x0 2 0 and Y satisfying the following LMIs: 2 T 3 A X + XA ? C T Y T ? Y C XB ? Y D 4 5
