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Design of reduced-order H∞ filtering for Markovian jump systems with mode-dependent time delays
ARTICLE IN PRESS Signal Processing 89 (2009) 187–196
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Design of reduced-order H1 filtering for Markovian jump systems with mode-dependent time delays Guoliang Wang a,b, Qingling Zhang a,b,, Victor Sreeram c a b c
Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110004, PR China Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, Liaoning 110004, PR China Department of Electrical and Electronic Engineering, University of Western Australia, 35 Stirling Highway Crawley, Western Australia 6009, Australia
a r t i c l e in fo
abstract
Article history: Received 22 April 2008 Received in revised form 23 June 2008 Accepted 20 August 2008 Available online 29 August 2008
This paper is concerned with the reduced-order H1 filtering problem for Markovian jump systems with time delays, which are time-varying and depend on the system mode. Sufficient conditions for the existence of the reduced-order H1 filtering are provided in terms of linear matrix inequalities (LMIs) with equality constraints. It is shown that the conditions for the existence of zeroth-order H1 filtering can be expressed by LMIs without any equality constraints. A globally convergent algorithm involving LMIs based on the sequential linear programming matrix is suggested to solve the matrix inequalities characterizing the filtering solutions. Using the numerical solutions of the matrix inequalities, a reduced-order filtering with error less than some prescribed scalar can be computed. Finally, an illustrative example is presented to show the effectiveness of the proposed approach. & 2008 Elsevier B.V. All rights reserved.
Keywords: Markovian jump systems Reduced-order filtering H1 filtering Linear matrix inequalities Mode-dependent time delays
1. Introduction During the past several years, the H1 filtering has received a lot of attention due to its advantages over the traditional Kalman filtering [1]. The purpose of the filtering problem is to estimate the unavailable state variables of a given system, which is useful in control system analysis and synthesis. Therefore, the filtering problem has been extensively investigated in the past decades, see [2–6]. Markovian jump systems were firstly introduced by Krasovskii and Lidskii in 1961. Since then, a great deal of attention has been devoted to the study of this class of systems in recent years. Markovian jump systems can
Corresponding author at: Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110004, PR China. Tel.: +86 24 83689929. E-mail addresses: [email protected] (G. Wang), [email protected] (Q. Zhang).
0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.08.004
be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes, which may result from abrupt phenomena such as random failures and repairs of the components, changes in interconnections of subsystems, sudden environmental changes, modification of the operating point of a linearized model of a nonlinear systems, and so on. The studies of Markovian jump systems are important in practical applications such as manufacturing systems, aircraft control, target tracking, robotics, solar receiver control, and power systems [7]. In the past decades, the H1 filtering design problem of the Markovian jump systems has been reported in the literature, such as [8–12]. In addition, time delays are common in chemical processes, heating systems, biological systems, network systems, and so on. They are often the sources of instability or poor performance. Therefore, time-delay systems have been studied in the past years and various research topics on delay systems have been investigated
ARTICLE IN PRESS 188
G. Wang et al. / Signal Processing 89 (2009) 187–196
[2,6,8–10,12–16]. In [2,8,10,12], the H1 filtering problem for Markovian jump systems with time delays was discussed. On the other hand, it is well known that reduced-order filtering design, i.e., the design of a filtering whose order is lower than that of the system, is a very important issue in many applications, especially when the fast data processing is necessary with a process of limited power. Therefore, it is very meaningful and suitable to construct a stable reduced-order filtering such that the associated reduced-order filtering meets a prescribed H1 norm bound constraint, and considerable attention has been devoted to the study of reduced-order filtering design over the past few years. For instance, [17] reported H1 reduced-order filtering for linear systems with Markovian jump parameters, and the reduced-order H1 filtering for singular systems was considered in [3], while the same author has discussed the reducedorder H1 filtering for stochastic systems in [18]. The reduced-order H1 filtering for Markovian jump systems with time delays was also investigated in [19]. The robust H1 control for discrete-time Markovian jump systems with mode-dependent time delays has been addressed in [13] while [15] researched the guaranteed cost control for continuous-time Markovian jump systems with mode-dependent time delays. Very recently, [9,14,20] proposed the design of robust H1 filtering for stochastic systems with mode-dependent time delays, and [21] dealt with the H1 model reduction for discretetime Markovian jump systems with mode-dependent time delays. However, to the best knowledge of the authors, there is no result for reduced-order H1 filtering for Markovian jump systems with mode-dependent time delays. In this paper, we present an approach for reduced-order filtering design for Markovian jump linear systems with mode-dependent time delays. Conditions for the existence of reduced-order H1 filtering are established through linear matrix inequalities (LMIs) together with equality constraints, and the designed filtering with a lower order can guarantee the error gain satisfying a prescribed scalar. An effective algorithm involving optimization is used to construct the reducedorder filtering. Finally, a numerical example is presented to illustrate the usefulness of the developed approach in this paper. The notations used throughout this paper are quite standard. R denotes the set of real numbers, Rn and Rmn denote, respectively, the n-dimensional Euclidean space and the set of all m n real matrices. The superscript T denotes matrix transposition. For a matrix M 2 Rnm with rank r, the orthogonal complement M? is defined as a (possibly non-unique) ðn rÞ n matrix such that M ? M ¼ 0 and M ? M?T 40. k k refers to the Euclidean vector norm or spectral matrix norm. Efg denotes the expectation operator with respect to some probability measure. In symmetric block matrices, we use ‘‘’’ as an ellipsis for the terms that are introduced by symmetry, that is
L
N
NT
R
¼
L
N
R
.
2. Preliminaries Given a complete probability space ðO; F; PÞ where O is the sample space, F is the algebra of events and P is the probability measure defined on F. fZt ; tX0g is a homogeneous finite-state Markovian process with right continuous trajectories and taking values in a finite set S ¼ f1; 2; . . . ; Ng with generator L ¼ ðlij Þ. The transition probability from mode i to mode j at time t þ Dt can be described as ( lij Dt þ oðDtÞ; iaj; PrðZtþDt ¼ jjZt ¼ iÞ ¼ (1) 1 þ lii Dt þ oðDtÞ; i ¼ j; where Dt40; limDt!0 ðoðDtÞ=DtÞ ¼ 0 and the transition probability rates satisfy lij X0, for i; j 2 S; iaj and lii ¼ P N j¼1;jai lij : Consider the following Markovian jump linear systems with mode-dependent time delays described by 8 x_ ðtÞ ¼ AðZt ÞxðtÞ þ Ad ðZt Þxðt tZt ðtÞÞ þ BðZt ÞwðtÞ; > > > > < yðtÞ ¼ LðZ ÞxðtÞ þ Ld ðZ Þxðt tZ ðtÞÞ þ EðZ ÞwðtÞ; t t t t ðSÞ : zðtÞ ¼ CðZt ÞxðtÞ þ C d ðZt Þxðt tZt ðtÞÞ þ DðZt ÞwðtÞ; > > > > : xðtÞ ¼ fðtÞ; 8t 2 ½m; 0; (2) where xðtÞ 2 Rn is the state vector; wðtÞ 2 Rm is the disturbance input; yðtÞ 2 Rp is the output; zðtÞ 2 Rq is the signal to be estimated. AðZt Þ; Ad ðZt Þ 2 Rnn , BðZt Þ 2 Rnm , LðZt Þ; Ld ðZt Þ 2 Rpn , CðZt Þ; C d ðZt Þ 2 Rqn , EðZt Þ 2 Rpm , DðZt Þ 2 Rqm . In the system ðSÞ; tZt ðtÞ denotes the time-varying delay when the mode is in Zt and satisfies 0oti ðtÞpmi o1;
t_ i ðtÞphi o1;
8i 2 S,
(3)
where mi and hi are real constant numbers for any i 2 S. In the system ðSÞ; fðtÞ is a vector-valued initial continuous function defined on the interval ½m; 0, where m ¼ maxfmi ; i 2 Sg. For notation simplification, in the sequel for each possible fZt ¼ i; i 2 Sg, a matrix MðZt Þ will be denoted by M i ; for example, AðZt Þ is denoted by Ai ; Ad ðZt Þ by Adi, and so on. Definition 1 (Xu et al. [9], Feng et al. [22]). The Markovian jump system ðSÞ is said to achieve exponential meansquare stability if, when wðtÞ ¼ 0, for any finite fðtÞ 2 Rn defined on ½m; 0, and initial mode Z0 2 S, there exist constant scalars a40 and b40 such that
Efkxðt; f; Z0 Þk2 gpa
sup kfðyÞk2 ebt ,
(4)
mpyp0
where xðt; f; Z0 Þ denotes the solution of the system ðSÞ at time t under the initial condition fðtÞ and Z0 . Definition 2. For a given real number g40, the Markovian jump system ðSÞ is said to be exponentially meansquare stable with g-disturbance attenuation if for any initial mode Z0 ðtÞ 2 S, it is exponentially mean-square stable with wðtÞ ¼ 0 and in the case of x0 ¼ 0, the zðtÞ satisfies Z 1 Z 1 E kzðtÞk2 dt og2 kwðtÞk2 dt. (5) 0
0
ARTICLE IN PRESS G. Wang et al. / Signal Processing 89 (2009) 187–196
X i P i ¼ In~ ;
In this paper, we are concerned with the design of an exponentially mean-square stable reduced-order filtering Þ with mode-dependent time delays such that ðS 8 _ > > < xðtÞ ¼ AðZt ÞxðtÞ þ Ad ðZt Þxðt tZt ðtÞÞ þ BðZt ÞyðtÞ; Þ : z ðtÞ ¼ Cð Z Þx ðtÞ þ C ðZ Þx ðt tZ ðtÞÞ þ Dð Z ÞyðtÞ; ðS d t t t t > > : x ðtÞ ¼ jðtÞ; 8t 2 ½m; 0;
189
QZ ¼ In~ ,
(10)
for all i 2 S, where
Si ðXÞ ¼ ½
Ni ¼ f½Li Ldi Ei T g? ,
n~ ¼ n þ n;
I1 ¼ ½In 0n;n ;
(11)
qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi li1 X i liði1Þ X i liðiþ1Þ X i liN X i , (12)
(6) Ri ðXÞ ¼ diagfX 1 ; . . . ; X i1 ; X iþ1 ; . . . ; X N g,
n Z Þ, A ðZ Þ 2 Rn Z Þ, where x ðtÞ 2 Rn , z 2 Rq , Að , Cð d t t t Z Þ 2 Rqp with non; Zt Þ 2 Rnp , Dð Z ðtÞ is C d ðZt Þ 2 Rqn , Bð t the same Markovian chain as in ðSÞ. The H1 norm of kog which denotes the error system satisfies kS S R1 R1 ðEf 0 kzðtÞ z ðtÞk2 dtgÞ1=2 ogð 0 kwðtÞk2 dtÞ1=2 under the zero initial condition.
A¯ i ¼
2 PN
T j¼1 lij P j þ Ai P i þ P i Ai þ ð1 þ ZmÞQ
Gi ¼
P i Adi
P i Bi
C Ti C Tdi DTi
ð1 hi ÞQ
0
g2 I
6 6 Oi ¼ 6 6 6 4
3 7 7 7o0, 7 5
T
I1 A¯ di
I1 B¯ i
I1 Si ðXÞ
I1 X i
ð1 hi ÞQ
0
0
0
" F¼
2
g I
0 Ri ðXÞ
0 0
1þ1Zm Z
0
PN
j¼1 lij P j
0
0
0
;
B¯ i ¼
Bi 0
,
(14)
¯ i ¼ Di . D
(15)
T
(16)
(17)
di
T
þ A¯ i P i þ P i A¯ i þ ð1 þ ZmÞQ
P i A¯ di
P i B¯ i
ð1 hi ÞQ
0
g2 I
T C¯ i
3
7 T 7 C¯ di 7, 7 7 ¯ Ti 5 D I
3
0n;q
0n;n
0n;q
In
Fi ¼ ½Hi K i Ni 02nþp;q ,
2
#
Li
6 Hi ¼ 4 0n;n 0n;n
;
0p;n
(19)
3
In 7 5; 0n; n
2
Ldi
6 K i ¼ 4 0n;n 0n;n
0p;n
3
7 0n; n 5, In
(20) 3
Ei 6 7 Ni ¼ 4 0n;m 5; 0n;m
3 7 7 7 7 7o0, 7 7 5
þ A¯ i P i þ P i A¯ i þ ð1 þ ZmÞQ ÞIT1
i
N j¼1 lij P j
2
J ¼ ½Iq 0q;n .
(21)
Proof. From (2) and (6), the state space representation of can be written as the error system S S 8 ~ Z Þx~ ðtÞ þ A~ x~ ðt tZ ðtÞÞ þ Bð ~ Z ÞwðtÞ; < x_~ ðtÞ ¼ Að d t t t ~Þ : (22) ðS ~ Z Þx~ ðtÞ þ C~ x~ ðt tZ ðtÞÞ þ Dð ~ Z ÞwðtÞ; : z~ ðtÞ ¼ Cð d
t
(8)
Ni
Adi
# C di , A
C i A
Pi F 6 7 6 0n; 7 ~ nþq 7; Ci ¼ 6 6 0m;nþq 7 4 5 J
Theorem 1. There exists a nth reduced-order filtering kog if there exist matrices Þ such that kS S ðS P 1 ; P2 ; . . . ; P N with P i 40 2 RðnþnÞðnþnÞ , X 1 ; X 2 ; . . . ; X N with ðnþnÞðnþ nÞ ðnþnÞðnþ nÞ X i 40 2 R , Q 40 2 R and Z40 2 ðnþnÞðnþ nÞ R satisfying
"
(18)
Proof. The proof can be carried out by the similar lines as in the proof of [9] and by the Schur complement, thus it is omitted. &
I1 ð
A¯ di ¼
C¯ di ¼ ½C di 0;
2
I
I1 ðlii X i þ A¯ i X i þ X i A¯ i ÞIT1
i D B i
2P
where Z ¼ maxfjlii j; i 2 Sg:
6 #6 0 6 6 6 I 6 6 4
0
where "
(7)
2
0
;
Oi þ Ci Gi Fi þ ðCi Gi Fi ÞT o0,
Proposition 1. The Markovian jump system ðSÞ is exponentially mean-square stable with g-disturbance attenuation, if there exist matrices P1 40; P 2 40; . . . ; P N 40 and Q 40 such that the following LMIs hold for i ¼ 1; 2; . . . ; N
6 6 6 6 6 6 6 4
0
Furthermore, if Pi 40; X i 40; Q 40 and Z40 are the solutions of (8)–(10), then a reduced-order filtering can be computed by substituting the solutions of (8)–(10) into
In this section, we consider the design of reducedorder filtering for ðSÞ. Before presenting the main results, we first give the following proposition which will be used in the proof of our main results.
2
Ai
C¯ i ¼ ½C i 0;
3. Main results
6 6 6 6 4
(13)
I1 Pi A¯ di IT1 ð1
hi ÞI1 Q IT1
I1 P i B¯ i 0
g2 I
T
I1 C¯ i
t
t
3
7 7" T 7 N ¯ i I1 C di 7 7 T 7 0 ¯i 7 D 5 I
0 I
#T o0,
(9)
ARTICLE IN PRESS 190
G. Wang et al. / Signal Processing 89 (2009) 187–196
where x~ ¼ ½xT ðtÞ x T ðtÞT ; z~ ðtÞ ¼ zðtÞ z ðtÞ, ~ ZðtÞÞ ¼ A~ ; A~ ðZðtÞÞ ¼ A~ ; Bð ~ ZðtÞÞ ¼ B~ i ; Að i d di C~ d ðZðtÞÞ ¼ C~ di ; for each Zt " Ai A~ i ¼ Bi Li
~ ZðtÞÞ ¼ C~ i , Cð
~ ZðtÞÞ ¼ D ~i Dð
¼ i 2 S; we have # " Adi 0 ~ ¼ ; A di A Bi Ldi i
0 A
#
" B~ i ¼
;
di
Bi
From LMIs (8), (9) and inverse condition (10), if there exist matrices Pi 40; X i 40, for all i 2 S; Q 40 and Z40 satisfying (26), there exists matrix Gi such that (25) holds. All the parameters of the approximation model satisfying (25) can be obtained by substituting the solutions of (8)–(10) into (25). This completes the proof. &
#
, B i Ei
i Li C i , C~ i ¼ ½C i D ~C ¼ ½C D i Ldi C di ; D ~ i ¼ Di D i Ei . di di From (14), (15), (19)–(21), we obtain that A~ i ¼ A¯ i þ FGi Hi ;
A~ di ¼ A¯ di þ FGi K i ;
B~ i ¼ B¯ i þ FGi N i ,
(23)
C~ i ¼ C¯ i þ JGi Hi ;
C~ di ¼ C¯ di þ JGi K i ;
~i ¼ D ¯ i þ JGi Ni . D
(24)
kog and Þ such that kS S We see that there exists ðS exponentially mean-square stable if there exist matrices P i 40; X i 40; Q 40 and Z40 for (22) where i 2 S, such that (7) holds. Furthermore, noticing that (17)–(21), (7) can be rewritten as
Oi þ Ci Gi Fi þ ðCi Gi Fi ÞT o0,
(25)
where Oi ; Ci and Fi are given in (18) and (19). From [23,24], (25) is equivalent to ?T C? i Oi Ci o0;
For
T?T FT? o0 for all i 2 S. i Oi Fi
(26)
C? i ,
we have 2 h i In 0n;n 6 6h i 6 ? 0n;n 0n; ~ ~ n Ci ¼ 6 6 6h i 4 0m;n 0m;n 2
P 1 6 i 60 ~ n~ 6 n; 6 60 6 m;n~ 4 0q;n~
0n;n~
0n;m
0n;q
In~
0n;m ~
0n;q ~
0m;n~
Im
0m;q
0n; ~ n~
0n;m ~
In~
0n;m ~
0m;n~
Im
0q;n~
0q;m
0n;q ~
3 7 7 7 7 7 7 5
Remark 1. Theorem 1 provides sufficient conditions for the solvability of the reduced-order H1 filtering. It is worth pointing out that when Adi ¼ 0; Ldi ¼ 0; C di ¼ 0; A di ¼ 0; C di ¼ 0, we can get the reduced-order filtering for Markovian jump system without time delays, which was also discussed in [17,18] whose results have rank constraints due to partition of positive definite matrix. In this paper, the partition of positive definite matrix is avoided instead of converse constraint. In addition, via [25,26], our result is easy to implement, in the following section, we will discuss this topic particularly. Remark 2. For a fixed system mode, if there exist ^ Z Þ; A^ ðt; Z Þ; Bðt; ^ ZÞ uncertainties in the matrices Aðt; d t t t such that ^ Z Þ ¼ AðZ Þ þ DAðt; Z Þ, Aðt; t t t A^ d ðt; Zt Þ ¼ Ad ðZt Þ þ DAd ðt; Zt Þ, ^ Z Þ ¼ BðZ Þ þ DBðt; Z Þ, Bðt; t
t
t
where AðZt Þ; Ad ðZt Þ; BðZt Þ are known real constant matrices representing the nominal system for each Zt 2 S. The admissible uncertainties are assumed to be modeled in the form
3
7 7 0n;q ~ 7 7. 0m;q 7 7 5 Iq
P 1 1 1 ¯ T ¯ 1 where Xi ¼ I1 ð N j¼1 lij P i P j P i þ P i Ai þ Ai P i þ ð1 þ ZmÞ T 1 1 1 1 P i QP i ÞI1 . Let X i ¼ P i and Q ¼ Z, by Schur complement and conditions (11)–(14), (29) becomes to (8). For T?T FT? o0, we get (9) similarly. i Oi Fi
½DAðt; Zt Þ DAd ðt; Zt Þ DBðt; Zt Þ (27)
¼ MðZt ÞFðt; Zt Þ½N1 ðZt Þ N2 ðZt Þ N 3 ðZt Þ,
(30)
On the other hand, denote
where MðZt Þ; N1 ðZt Þ; N 2 ðZt Þ and N3 ðZt Þ are known real constant matrices for all Zt 2 S. The uncertain timevarying matrix satisfies
Ni ¼ f½Li Ldi Ei T g? ¼ ½U i V i W i ,
F T ðt; Zt ÞFðt; Zt ÞpI;
U i LTi
V i LTdi
þ þ ¼ 0. We select which implies " # Ui 0 V i 0 W i 0 FT? ¼ i 0 0 0 0 0 I 2 3 ½I 0 ½0 0 0 0 7 " #6 7 Ni 0 6 6 ½0 0 ½I 0 0 0 7 ¼ 6 7. 7 0 I 6 4 ½0 0 ½0 0 I 0 5 ½0 0 ½0 0 0 I By calculation, we obtain 2 Xi I1 A¯ di ? ?T Ci Oi Ci ¼ 6 4 ð1 hi ÞQ
I1 B¯ i 0n;m ~ g2 I
(31)
0 0
;
DB¯ i ¼
DBi 0
. (32)
Similar to (23), we have (28)
3 7 5o0,
Zt 2 S.
Then, for each Zt ¼ i 2 S, define DAi 0 DAdi ; DA¯ di ¼ DA¯ i ¼ 0 0 0
W i ETi
(29)
A~ i ¼ A¯ i þ DA¯ i þ FGi Hi , A~ di ¼ A¯ di þ DA¯ di þ FGi K i , B~ i ¼ B¯ i þ DB¯ i þ FGi Ni .
(33)
The other matrices are same to Theorem 1. Then by the similar methods as in the above proof of Theorem 1 and in [9, Lemma 2], we have the similar result about uncertain Markovian jump system.
ARTICLE IN PRESS G. Wang et al. / Signal Processing 89 (2009) 187–196
From Theorem 1, we see that the desired reduced-order filtering involves time delays. However, in some applications, it is desirable to approximate the system ðSÞ with a reduced-order filtering which has no time delays, and the associated model error also achieves a prescribed H1 norm condition. In view of this, we now consider the reduced-order filtering with no time delays. In the following, we formulate the reduced-order filtering problem for the time-delay system ðSÞ as finding an exponentially mean-square stable nth reduced-order filtering model given by 8 _ > > < xðtÞ ¼ AðZt ÞxðtÞ þ BðZt ÞyðtÞ; 0 Z ÞyðtÞ; (34) ðS Þ : zðtÞ ¼ CðZt ÞxðtÞ þ Dð t > > : x ðtÞ ¼ jðtÞ; 8t 2 ½m; 0; where x ðtÞ 2 Rn ; z 2 Rq , such that
(35)
0 Theorem 2. There exists a nth reduced-order filtering S 0 kog if there exist matrices P 1 ; P 2 ; . . . ; P N such that kS S with P i 40 2 RðnþnÞðnþnÞ , X 1 ; X 2 ; . . . ; X N with X i 40 2 ðnþnÞðnþ nÞ ðnþnÞðnþ nÞ R , Q 40 2 R and Z40 2 RðnþnÞðnþnÞ satisfying 6 6 6 6 6 6 6 4
T
I1 ðlii X i þ A¯ i X i þ X i A¯ i ÞIT1
I1 A¯ di
I1 B¯ i
I1 Si ðXÞ
I1 X i
ð1 hi ÞQ
0 g2 I
0 0
0 0
Ri ðXÞ
0
1þ1Zm Z
Ni 0
I1 ð
6 0 6 6 6 I 6 4
X i P i ¼ In~ ;
PN
j¼1 lij P j
7 7 7 7 7o0, 7 7 5
T
þ A¯ i Pi þ Pi A¯ i þ ð1 þ ZmÞQ ÞIT1
0
0
0
C¯ i ¼ ½C i 0;
;
A¯ di ¼
0
0
0
;
ð1 hi ÞQ
0
g2 I
3 Pi F 6 7 6 0n; 7 ~ nþq 7; Ci ¼ 6 6 0m;nþq 7 4 5 J
F¼
0n;q
0n;n
0n;q
In
Fi ¼ ½Hi K i Ni 0nþp;q ,
#
" Hi ¼
;
" Ni ¼
0n;m
0p;n
0n;n
In
#
" ;
Ki ¼
Ldi
0p;n
0n;n
0n; n
# ,
(49)
Proof. From (2) and (34), the state space representation 0 can be written as of the error system S S 8 ~ Z Þx~ ðtÞ þ A~ x~ ðt tZ ðtÞÞ þ Bð ~ Z ÞwðtÞ; < x_~ ðtÞ ¼ Að d 0 t t t ~ ðS Þ : ~ Z Þx~ ðtÞ þ C~ x~ ðt tZ ðtÞÞ þ Dð ~ Z ÞwðtÞ; (50) : z~ ðtÞ ¼ Cð d
t
T
I
(39)
(41)
0
Li
(47)
J ¼ ½Iq 0q;n .
;
,
I
#
Ei
Bi
7 T 7 C¯ di 7 7, 7 ¯ Ti 5 D
(48)
3
2
¯ Ti D
B¯ i ¼
T C¯ i
(46)
g2 I
¯ i ¼ Di . C¯ di ¼ ½C di 0; D
(42) (43)
Furthermore, if P i 40; X i 40; Q 40 and Z40 are the solutions of (36)–(38), then a reduced-order filtering can be computed by substituting the solutions of (36)–(38) into
Oi þ Ci Gi Fi þ ðCi Gi Fi ÞT o0,
Ni ¼ f½Li Ldi Ei T g? ,
Adi
P i B¯ i
qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi li1 X i liði1Þ X i liðiþ1Þ X i liN X i ,
Ai
P i A¯ di
0
Ri ðXÞ ¼ diagfX 1 ; . . . ; X i1 ; X iþ1 ; . . . ; X N g,
þ A¯ i P i þ P i A¯ i þ ð1 þ ZmÞQ
ð1 hi ÞI1 Q IT1
(38)
n~ ¼ n þ n;
T
N j¼1 lij P j
(40)
A¯ i ¼
6 6 Oi ¼ 6 6 6 4
I1 P i B¯ i
for all i 2 S, where
Si ðXÞ ¼ ½
2P
(45)
i
I1 Pi A¯ di IT1
QZ ¼ In~
I1 ¼ ½In 0n;n~ ;
# C i , A
i D Bi
3
(36) 2
Gi ¼
"
0 kog. kS S
2
where "
191
(44)
I1 C¯ i
t
t
3 7
T 7 I1 C¯ di 7 Ni 0
7 7 5
0
T
I
o0,
(37)
where x~ ¼ ½xT ðtÞ x T ðtÞT ; z~ ðtÞ ¼ zðtÞ z ðtÞ, A~ i ¼ A¯ i þ FGi Hi ; A~ di ¼ A¯ di þ FGi K i ; B~ i ¼ B¯ i þ FGi N i , C~ i ¼ C¯ i þ JGi Hi ; C~ di ¼ C¯ di þ JGi K i , " # Ai 0 ~ ¯ i þ JGi Ni ; A¯ i ¼ , Di ¼ D 0 0 " # " # Adi 0 Bi A¯ di ¼ ; B¯ i ¼ , 0 0 0 C¯ i ¼ ½C i 0;
C¯ di ¼ ½C di 0;
¯ i ¼ Di . D
Similar to the proof of Theorem 1, we conclude that under the conditions of Theorem 2 there exists Gi such that (7) holds. Then there exists a reduced-order filtering system (34). Using Schur complement, the desired results follow immediately. & Remark 3. From Theorems 1 and 2, we can obtain that when the N ¼ 1, the Markovian jump linear systems with
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methods are promising, such as, the alternating projections method [28], the min–max algorithm [29], XY-centering algorithm [30], and cone complementarity linearization (CCL) algorithm [25]. In Ref. [31], the alternating projections method, the min–max algorithm, and the CCL algorithm were compared, and numerical experiments show that the CCL algorithm is the best one. Very recently, a new algorithm which was named sequential linear programming matrix method (SLPMM) was proposed to solve such non-convex feasibility problems. It is seen as an improved version of the CCL algorithm. As is indicated in [26], the SLPMM algorithm is superior to the CCL algorithm in that it always generates a sequence of iteration with strictly decreasing objective function values, and it is also globally convergent. In order to solve the non-convex feasibility problem appeared in this paper, we will employ the SLPMM algorithm. To utilize the SLPMM algorithm to solve the reducedorder H1 filtering problem, we firstly define a convex set
mode-dependent time delays are specialized to deterministic systems with time-varying delays. And in [27], it discussed robust reduced-order filtering for deterministic system without time delays. Therefore, from Theorems 1 and 2, we get the reduced-order H1 filtering for deterministic systems with time delays, and we may say that our results can be viewed as extension results on the reduced-order H1 filtering problem from deterministic systems to stochastic systems. From Theorems 1 and 2, we obtain the strict LMIs conditions for the solvability of the zeroth-order H1 filtering. Corollary 1. If there exist matrices P1 ; P2 ; . . . ; P N , with P i 40 2 Rnn and Q 40 2 Rnn satisfying 2P 6 6 4
N j¼1 lij P j
þ ATi P i þ P i Ai þ ð1 þ ZmÞQ
P i Adi
ð1 hi ÞQ
P i Bi
3
7 0 7 5o0, g2 I
(51)
2 PN "
Ni 0
#6 0 6 6 6 6 I 6 4
j¼1 lij P j
þ ATi P i þ P i Ai þ ð1 þ ZmÞQ
P i Adi
P i Bi
ð1 hi ÞQ
0
g2 I
for all i 2 S, where
Ni ¼ f½Li Ldi Ei T g? .
(53)
C Ti
3
7" 7 N i C Tdi 7 7 7 T 7 0 Di 5 I
0 I
#T o0
(52)
of all the feasible solutions of LMIs (8) and (9) as following:
i to solve the zeroth-order H1 Then, there exists constant D filtering problem. In this case, the desired zeroth-order H1 i corresponding to feasible matrix can be filtering solution D computed by substituting the solutions of (51) and (52) into
S:¼fXjX satisfies LMIs ð8Þ and ð9Þg,
Oi þ CD i Fi þ ðCD i Fi ÞT o0,
From Theorem 1, the reduced-order filtering for Markovian jump linear systems with mode-dependent time delays can be obtained if there exist X i 40, P i 40, Q 40, Z40, such that
2 PN 6 6 Oi ¼ 6 6 4
j¼1 lij P j
þ ATi P i þ P i Ai þ ð1 þ ZmÞQ
(54) P i Adi
P i Bi
ð1 hi ÞQ
0
g2 I
C Ti
3
7 C Tdi 7 7, 7 DTi 5 I
(55) 2
0n;q
3
60 7 6 n;q 7 7; 7 4 0m;q 5
C¼6 6
Fi ¼ ½Li Ldi Ei 0p;q .
(56)
Iq
Remark 4. From Corollary 1, we can see that all the conditions are strict LMIs, which are convex. So we could solve them efficiently by Matlab LMI toolbox. From Theorem 1, we see that the construction of the reduced-order filtering resides in solving LMIs (8), (9) and the inverse constraints in (10). This non-convex condition makes it difficult find the numerical solutions, and there are many numerical approaches proposed to deal with such a problem. Among those approaches, the LMI-based
(57)
where
X:¼fX i 40; Pi 40; Q 40; Z40; i 2 Sg.
X i ; P i 2 S;
X i Pi ¼ I
for all i 2 S; QZ ¼ I.
(58)
(59)
From [25,26], we know fact about the algorithm that for any matrices X40 and P40, if the LMI X I X0 (60) I P is feasible, then traceðXPÞXn. And traceðXPÞ ¼ n if and only if XP ¼ I. Define the set as (" # ) Xi I Q I T:¼ X0; (61) X0 for all i 2 S . I Pi I Z By the SLPMM approach, the above non-convex problem of (57)–(59) is equivalent to the following minimization problem: ( ! !) N X . (62) X i P i þ QZ min trace
X2S\T
i¼1
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We see that the optimal solution of problem (62) is ðN þ 1Þn~ satisfying traceðX i P i Þ ¼ traceðQZÞ ¼ n~
for all i 2 S.
(63)
Hence, the reduced-order filtering problem is changed to a problem of finding a global solution of the minimization problem (62). Based on the above analysis, an algorithm HinfRFP (H1 Reduced-order Filtering Problem) for Markovian jump systems with mode-dependent time delays is presented to solve this problem. HinfRFP Algorithm: Step 1: For the system ðSÞ with given reduced-order g and error accuracy d. n; Step 2: Find any initial solution ðX i0 ; Pi0 ; Q 0 ; Z 0 ; i 2 SÞ 2 S, and set k ¼ 0. Step 3: Define function f k ðXÞ ¼ trace
!
N X
ðX i P ik þ P i X ik Þ þ QZ k þ ZQ k .
Pi
# X0;
Q I
) I X0 for i 2 S . Z
r2½0;1
0:3
3:8
Ad1
C d1 ¼ ½0:2 0:3 0:6;
Ld1 ¼ ½1 1 1;
0:1
0:2 6 ¼6 4 0:5 0
1 1
0:6
3
7 0:8 7 5, 2:5
D1 ¼ 0:2,
E1 ¼ 1.
2
0:5
2:5
6 A2 ¼ 6 4 0:1
0:1
3
2
7 0:3 7 5; 2
3:5
0:1 1 2 3 0:6 6 7 7 B2 ¼ 6 4 0:5 5, 0
0
0:3
0:6
0:5
0
1
0:8
6 Ad2 ¼ 6 4 0:1 0:6
C d2 ¼ ½0:4 0:6 0:5;
Ld2 ¼ ½1 1 1;
3 7 7, 5
D2 ¼ 0:5,
E2 ¼ 1.
(66)
Step 6: Let X iðkþ1Þ ¼ X ik þ r ðX i X ik Þ; Piðkþ1Þ ¼ Pik þ r ðPi Pik Þ; Q kþ1 ¼ Q k þ r ðQ Q k Þ; Z kþ1 ¼ Z k þ r ðZ Z k Þ,
The time-varying delay is t2 ðtÞ ¼ 0:2 sinð0:6tÞ satisfying (3) with m2 ¼ 0:2; h2 ¼ 0:12. Suppose the transition probability matrix is given by
and k ¼ k þ 1. If kokmax then go to step 3, else exit. Pr ¼ Remark 5. This algorithm is similar to that developed in [26]. From the HinfRFP algorithm, we can see that when r 1, the HinfRFP algorithm will become to the CCL algorithm [25]. So we may say that HinfRFP contains CCL algorithm. As explained in [26], HinfRFP algorithm is globally convergent since the sequence of function (64) generated by HinfRFP algorithm always converges to some ~ while the alternating projection methf ðX ÞX2ðN þ 1Þn, od, which was used to construct the reduced-order filtering for Markovian jump systems in [17,19] is guaranteed to converge only locally. HinfRFP algorithm can also be applied to Theorem 2 similarly. Noting from HinfRFP algorithm, we can find a suboptimal g for some determined reduced-order n for Þ that (57)–(59) hold. For g, we can propose a system ðS minimization problem:
subjects to constraints (57)–(59).
1
C 1 ¼ ½0:8 0:3 0;
2
7 7; 5
2:5
L2 ¼ ½1 1 1;
min g
3
0:6
C 2 ¼ ½0:5 0:2 0:3;
!
þðQ k þ rðQ Q k ÞÞðZ k þ rðZ Z k ÞÞ .
1
3 6 A1 ¼ 6 4 0:3 0:1 2 3 1 6 7 7 B1 ¼ 6 4 0 5, 1
(65)
N X ðX ik þ rðX i X ik ÞÞðP ik þ rðP i P ik ÞÞ i¼1
2
The time-varying delay is t1 ðtÞ ¼ 0:3 sinð0:3tÞ satisfying (3) with m1 ¼ 0:3; h1 ¼ 0:09: For mode 2, the dynamics of the system are described as
~ d, then construct a Step 4: If jf k ðXÞ 2ðN þ 1Þnjp reduced-order filtering in the form of (6), and exit; otherwise go to the step 5. Step 5: Compute r 2 ½0; 1 by solving min trace
In this section, we present an illustrative example to demonstrate the effectiveness of this approach. Consider a linear Markovian jump system with mode-dependent time delays in the form of ðSÞ with two modes. For mode 1, the dynamics of the system are described as
(64)
Find X solving the following convex programming: I
4. Numerical example
L1 ¼ ½1 1 1;
i¼1
" ( Xi min f k ðXÞ I X2S
193
(67)
0:5
0:5
0:3
0:3
.
Our purpose is to find a first-order model. We first choose 2
1 60 6 6 60 Ni ¼ 6 60 6 6 40 0
3
1
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0 7 7 7 0 7 7 0 7 7 7 0 5
0
0
0
0
1
for i ¼ 1; 2.
1
Via the HinfRFP algorithm, we solve the matrix inequalities (8) and (9) with inverse constraints in (10) for i ¼ 1; 2 and get 2
0:2764
6 6 0:0875 6 P1 ¼ 6 6 0:0563 4 0:1549
0:0875
0:0563
1:0217
0:1141
0:1141
0:9914
0:3226
0:5917
0:1549
3
7 0:3226 7 7 7, 0:5917 7 5 1:0081
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2
0:9569
0:1885
0:3942
1:3395
3
A d2 ¼ 0:8467, B 2 ¼ 0:1073, C 2 ¼ 1:5301, C d2 ¼ 2 ¼ 0:5557. The last optimal r is r ¼ 0:3295, 1:3418, D and the reduced-order filtering level of the system ðSÞ is g ¼ 0:394. In the following, we will show the approximation performance of the obtained reduced-order H1 filtering. With the initial state of x~ ð0Þ ¼ ½0 0 0 0T , Fig. 1b is outputs trajectories of the original system and the first-order system subject to disturbance input uðtÞ ¼ et=8 sinð2tÞ under the systems jump mode Fig. 1a, while Fig. 2b is subject to disturbance input uðtÞ ¼ sinð2tÞ under the system jumps mode Fig. 2a. We can see the obtained reduced-order model approximates the original model very well.
6 7 6 0:1885 0:7920 0:2425 0:9164 7 6 7 P2 ¼ 6 7, 6 0:3942 0:2425 1:3660 1:3099 7 4 5 1:3395 0:9164 1:3099 3:0306 2 3 6:0983 1:6232 2:1582 2:7232 6 7 6 1:6232 1:7214 1:1819 1:4939 7 6 7 X1 ¼ 6 7, 6 2:1582 1:1819 2:6026 2:2374 7 4 5 3:2017 3 3:8733 6 7 6 2:7098 3:4661 1:2913 2:8038 7 6 7 X2 ¼ 6 7, 6 1:7263 1:2913 1:8970 1:9733 7 4 5 3:8733 2:8038 1:9733 3:7426 3 2 0:6166 0:3461 0:1885 0:5396 6 7 6 0:3461 2:8414 0:6090 1:2000 7 6 7 Q ¼6 7, 6 0:1885 0:6090 2:7983 1:8427 7 4 5 0:5396 1:2000 1:8427 3:4330 2 3 3:4386 1:3381 1:1190 1:6088 6 7 6 1:3381 1:1642 0:8811 1:0902 7 6 7 Z¼6 7. 6 1:1190 0:8811 1:2256 1:1417 7 4 5 2
2:7232 5:2215
1:6088
1:4939 2:7098
1:0902
2:2374 1:7263
1:1417
5. Conclusions This paper has presented an approach for the reducedorder filtering design for Markovian jump systems with mode-dependent time-varying delays. The designed filtering can guarantee the error gain satisfying a prescribed scalar. Sufficient conditions for the existence of the reduced-order H1 filtering have been proposed in terms of LMIs with equality constraints. The HinfRFP algorithm is used to handle the non-convex constraints. A numerical example has been given to demonstrate the validity of the theoretical results. The results obtained in this paper may be extended to Markovian jump systems with uncertainties.
1:5381
systems jump mode
Then the parameters of the first-order model can be computed as A 1 ¼ 4:6461, A d1 ¼ 3:5469, B 1 ¼ 0:7479, 1 ¼ 0:1506; A 2 ¼ 2:4290, C 1 ¼ 0:8624, C d1 ¼ 0:4812, D
2.5 2 1.5 1 0.5 0
5
10
15
20
25
30
35
t 0.5
outputs
original filtering reduced order filtering
0
−0.5 0
5
10
15
20
25
t Fig. 1. Systems jump mode and outputs under uðtÞ ¼ et=8 sinð2tÞ.
30
35
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195
systems jump mode
2.5 2 1.5 1 0.5 0
5
10
15
20
25
30
35
t 1.5 original filtering reduced order filtering
outputs
1 0.5 0 −0.5 −1 0
5
10
15
20
25
30
35
t Fig. 2. Systems jump mode and outputs under uðtÞ ¼ sinð2tÞ.
Acknowledgments This work was supported by National Science Foundation of China under Grant no. 60574011. The authors gratefully thank the anonymous authors whose work largely constitutes this sample file. References [1] P. Shi, E.K. Boukas, R.K. Agarwal, Kalman filtering for continuoustime uncertain systems with Markovian jumping parameters, IEEE Trans. Automat. Control 44 (8) (1999) 1592–1597. [2] H.J. Gao, C.H. Wang, Delay-dependent robust H1 and L2 L1 filtering for a class of uncertain nonlinear time-delay systems, IEEE Trans. Automat. Control 48 (9) (2003) 1661–1666. [3] S.Y. Xu, J. Lam, Reduced-order H1 filtering for singular systems, Systems and Control Lett. 56 (2007) 48–57. [4] S.Y. Xu, T.W. Chen, An LMI approach to the H1 filter design for uncertain systems with distributed delays, IEEE Trans. Circuits Systems 51 (4) (2004) 195–201. [5] K.P. Sun, A. Packard, Robust H2 and H1 filters for uncertain LFT systems, IEEE Trans. Automat. Control 50 (5) (2005) 715–720. [6] X.M. Zhang, Q.L. Han, Delay-dependent robust H1 filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality, IEEE Trans. Circuits Systems 53 (12) (2006) 1466–1470. [7] Y. Ji, H.J. Chizeck, Jump linear quadratic Gaussian control in continuous time, IEEE Trans. Automat. Control 37 (12) (1992) 1884–1892. [8] S.Y. Xu, J. Lam, Delay-dependent H1 control and filtering for uncertain Markovian jump systems with time-varying delays, IEEE Trans. Circuits Systems 54 (9) (2007) 2070–2077. [9] S.Y. Xu, T.W. Chen, J. Lam, Robust H1 filtering for uncertain Markovian jump systems with mode-dependent time delays, IEEE Trans. Automat. Control 48 (5) (2003) 900–907. [10] Z.D. Wang, J. Lam, X.H. Liu, Exponential filtering for uncertain Markovian jump time-delay systems with nonlinear disturbances, IEEE Trans. Circuits Systems 51 (5) (2004) 262–268. [11] J.L. Xiong, J. Lam, Fixed-order robust H1 filter design for Markovian jump systems with uncertain switching probabilities, IEEE Trans. Automat. Signal Process. 54 (4) (2006) 1421–1430.
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Contents lists available at ScienceDirect
Signal Processing journal homepage: www.elsevier.com/locate/sigpro
Design of reduced-order H1 filtering for Markovian jump systems with mode-dependent time delays Guoliang Wang a,b, Qingling Zhang a,b,, Victor Sreeram c a b c
Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110004, PR China Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang, Liaoning 110004, PR China Department of Electrical and Electronic Engineering, University of Western Australia, 35 Stirling Highway Crawley, Western Australia 6009, Australia
a r t i c l e in fo
abstract
Article history: Received 22 April 2008 Received in revised form 23 June 2008 Accepted 20 August 2008 Available online 29 August 2008
This paper is concerned with the reduced-order H1 filtering problem for Markovian jump systems with time delays, which are time-varying and depend on the system mode. Sufficient conditions for the existence of the reduced-order H1 filtering are provided in terms of linear matrix inequalities (LMIs) with equality constraints. It is shown that the conditions for the existence of zeroth-order H1 filtering can be expressed by LMIs without any equality constraints. A globally convergent algorithm involving LMIs based on the sequential linear programming matrix is suggested to solve the matrix inequalities characterizing the filtering solutions. Using the numerical solutions of the matrix inequalities, a reduced-order filtering with error less than some prescribed scalar can be computed. Finally, an illustrative example is presented to show the effectiveness of the proposed approach. & 2008 Elsevier B.V. All rights reserved.
Keywords: Markovian jump systems Reduced-order filtering H1 filtering Linear matrix inequalities Mode-dependent time delays
1. Introduction During the past several years, the H1 filtering has received a lot of attention due to its advantages over the traditional Kalman filtering [1]. The purpose of the filtering problem is to estimate the unavailable state variables of a given system, which is useful in control system analysis and synthesis. Therefore, the filtering problem has been extensively investigated in the past decades, see [2–6]. Markovian jump systems were firstly introduced by Krasovskii and Lidskii in 1961. Since then, a great deal of attention has been devoted to the study of this class of systems in recent years. Markovian jump systems can
Corresponding author at: Institute of Systems Science, Northeastern University, Shenyang, Liaoning 110004, PR China. Tel.: +86 24 83689929. E-mail addresses: [email protected] (G. Wang), [email protected] (Q. Zhang).
0165-1684/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.sigpro.2008.08.004
be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes, which may result from abrupt phenomena such as random failures and repairs of the components, changes in interconnections of subsystems, sudden environmental changes, modification of the operating point of a linearized model of a nonlinear systems, and so on. The studies of Markovian jump systems are important in practical applications such as manufacturing systems, aircraft control, target tracking, robotics, solar receiver control, and power systems [7]. In the past decades, the H1 filtering design problem of the Markovian jump systems has been reported in the literature, such as [8–12]. In addition, time delays are common in chemical processes, heating systems, biological systems, network systems, and so on. They are often the sources of instability or poor performance. Therefore, time-delay systems have been studied in the past years and various research topics on delay systems have been investigated
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[2,6,8–10,12–16]. In [2,8,10,12], the H1 filtering problem for Markovian jump systems with time delays was discussed. On the other hand, it is well known that reduced-order filtering design, i.e., the design of a filtering whose order is lower than that of the system, is a very important issue in many applications, especially when the fast data processing is necessary with a process of limited power. Therefore, it is very meaningful and suitable to construct a stable reduced-order filtering such that the associated reduced-order filtering meets a prescribed H1 norm bound constraint, and considerable attention has been devoted to the study of reduced-order filtering design over the past few years. For instance, [17] reported H1 reduced-order filtering for linear systems with Markovian jump parameters, and the reduced-order H1 filtering for singular systems was considered in [3], while the same author has discussed the reducedorder H1 filtering for stochastic systems in [18]. The reduced-order H1 filtering for Markovian jump systems with time delays was also investigated in [19]. The robust H1 control for discrete-time Markovian jump systems with mode-dependent time delays has been addressed in [13] while [15] researched the guaranteed cost control for continuous-time Markovian jump systems with mode-dependent time delays. Very recently, [9,14,20] proposed the design of robust H1 filtering for stochastic systems with mode-dependent time delays, and [21] dealt with the H1 model reduction for discretetime Markovian jump systems with mode-dependent time delays. However, to the best knowledge of the authors, there is no result for reduced-order H1 filtering for Markovian jump systems with mode-dependent time delays. In this paper, we present an approach for reduced-order filtering design for Markovian jump linear systems with mode-dependent time delays. Conditions for the existence of reduced-order H1 filtering are established through linear matrix inequalities (LMIs) together with equality constraints, and the designed filtering with a lower order can guarantee the error gain satisfying a prescribed scalar. An effective algorithm involving optimization is used to construct the reducedorder filtering. Finally, a numerical example is presented to illustrate the usefulness of the developed approach in this paper. The notations used throughout this paper are quite standard. R denotes the set of real numbers, Rn and Rmn denote, respectively, the n-dimensional Euclidean space and the set of all m n real matrices. The superscript T denotes matrix transposition. For a matrix M 2 Rnm with rank r, the orthogonal complement M? is defined as a (possibly non-unique) ðn rÞ n matrix such that M ? M ¼ 0 and M ? M?T 40. k k refers to the Euclidean vector norm or spectral matrix norm. Efg denotes the expectation operator with respect to some probability measure. In symmetric block matrices, we use ‘‘’’ as an ellipsis for the terms that are introduced by symmetry, that is
L
N
NT
R
¼
L
N
R
.
2. Preliminaries Given a complete probability space ðO; F; PÞ where O is the sample space, F is the algebra of events and P is the probability measure defined on F. fZt ; tX0g is a homogeneous finite-state Markovian process with right continuous trajectories and taking values in a finite set S ¼ f1; 2; . . . ; Ng with generator L ¼ ðlij Þ. The transition probability from mode i to mode j at time t þ Dt can be described as ( lij Dt þ oðDtÞ; iaj; PrðZtþDt ¼ jjZt ¼ iÞ ¼ (1) 1 þ lii Dt þ oðDtÞ; i ¼ j; where Dt40; limDt!0 ðoðDtÞ=DtÞ ¼ 0 and the transition probability rates satisfy lij X0, for i; j 2 S; iaj and lii ¼ P N j¼1;jai lij : Consider the following Markovian jump linear systems with mode-dependent time delays described by 8 x_ ðtÞ ¼ AðZt ÞxðtÞ þ Ad ðZt Þxðt tZt ðtÞÞ þ BðZt ÞwðtÞ; > > > > < yðtÞ ¼ LðZ ÞxðtÞ þ Ld ðZ Þxðt tZ ðtÞÞ þ EðZ ÞwðtÞ; t t t t ðSÞ : zðtÞ ¼ CðZt ÞxðtÞ þ C d ðZt Þxðt tZt ðtÞÞ þ DðZt ÞwðtÞ; > > > > : xðtÞ ¼ fðtÞ; 8t 2 ½m; 0; (2) where xðtÞ 2 Rn is the state vector; wðtÞ 2 Rm is the disturbance input; yðtÞ 2 Rp is the output; zðtÞ 2 Rq is the signal to be estimated. AðZt Þ; Ad ðZt Þ 2 Rnn , BðZt Þ 2 Rnm , LðZt Þ; Ld ðZt Þ 2 Rpn , CðZt Þ; C d ðZt Þ 2 Rqn , EðZt Þ 2 Rpm , DðZt Þ 2 Rqm . In the system ðSÞ; tZt ðtÞ denotes the time-varying delay when the mode is in Zt and satisfies 0oti ðtÞpmi o1;
t_ i ðtÞphi o1;
8i 2 S,
(3)
where mi and hi are real constant numbers for any i 2 S. In the system ðSÞ; fðtÞ is a vector-valued initial continuous function defined on the interval ½m; 0, where m ¼ maxfmi ; i 2 Sg. For notation simplification, in the sequel for each possible fZt ¼ i; i 2 Sg, a matrix MðZt Þ will be denoted by M i ; for example, AðZt Þ is denoted by Ai ; Ad ðZt Þ by Adi, and so on. Definition 1 (Xu et al. [9], Feng et al. [22]). The Markovian jump system ðSÞ is said to achieve exponential meansquare stability if, when wðtÞ ¼ 0, for any finite fðtÞ 2 Rn defined on ½m; 0, and initial mode Z0 2 S, there exist constant scalars a40 and b40 such that
Efkxðt; f; Z0 Þk2 gpa
sup kfðyÞk2 ebt ,
(4)
mpyp0
where xðt; f; Z0 Þ denotes the solution of the system ðSÞ at time t under the initial condition fðtÞ and Z0 . Definition 2. For a given real number g40, the Markovian jump system ðSÞ is said to be exponentially meansquare stable with g-disturbance attenuation if for any initial mode Z0 ðtÞ 2 S, it is exponentially mean-square stable with wðtÞ ¼ 0 and in the case of x0 ¼ 0, the zðtÞ satisfies Z 1 Z 1 E kzðtÞk2 dt og2 kwðtÞk2 dt. (5) 0
0
ARTICLE IN PRESS G. Wang et al. / Signal Processing 89 (2009) 187–196
X i P i ¼ In~ ;
In this paper, we are concerned with the design of an exponentially mean-square stable reduced-order filtering Þ with mode-dependent time delays such that ðS 8 _ > > < xðtÞ ¼ AðZt ÞxðtÞ þ Ad ðZt Þxðt tZt ðtÞÞ þ BðZt ÞyðtÞ; Þ : z ðtÞ ¼ Cð Z Þx ðtÞ þ C ðZ Þx ðt tZ ðtÞÞ þ Dð Z ÞyðtÞ; ðS d t t t t > > : x ðtÞ ¼ jðtÞ; 8t 2 ½m; 0;
189
QZ ¼ In~ ,
(10)
for all i 2 S, where
Si ðXÞ ¼ ½
Ni ¼ f½Li Ldi Ei T g? ,
n~ ¼ n þ n;
I1 ¼ ½In 0n;n ;
(11)
qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi li1 X i liði1Þ X i liðiþ1Þ X i liN X i , (12)
(6) Ri ðXÞ ¼ diagfX 1 ; . . . ; X i1 ; X iþ1 ; . . . ; X N g,
n Z Þ, A ðZ Þ 2 Rn Z Þ, where x ðtÞ 2 Rn , z 2 Rq , Að , Cð d t t t Z Þ 2 Rqp with non; Zt Þ 2 Rnp , Dð Z ðtÞ is C d ðZt Þ 2 Rqn , Bð t the same Markovian chain as in ðSÞ. The H1 norm of kog which denotes the error system satisfies kS S R1 R1 ðEf 0 kzðtÞ z ðtÞk2 dtgÞ1=2 ogð 0 kwðtÞk2 dtÞ1=2 under the zero initial condition.
A¯ i ¼
2 PN
T j¼1 lij P j þ Ai P i þ P i Ai þ ð1 þ ZmÞQ
Gi ¼
P i Adi
P i Bi
C Ti C Tdi DTi
ð1 hi ÞQ
0
g2 I
6 6 Oi ¼ 6 6 6 4
3 7 7 7o0, 7 5
T
I1 A¯ di
I1 B¯ i
I1 Si ðXÞ
I1 X i
ð1 hi ÞQ
0
0
0
" F¼
2
g I
0 Ri ðXÞ
0 0
1þ1Zm Z
0
PN
j¼1 lij P j
0
0
0
;
B¯ i ¼
Bi 0
,
(14)
¯ i ¼ Di . D
(15)
T
(16)
(17)
di
T
þ A¯ i P i þ P i A¯ i þ ð1 þ ZmÞQ
P i A¯ di
P i B¯ i
ð1 hi ÞQ
0
g2 I
T C¯ i
3
7 T 7 C¯ di 7, 7 7 ¯ Ti 5 D I
3
0n;q
0n;n
0n;q
In
Fi ¼ ½Hi K i Ni 02nþp;q ,
2
#
Li
6 Hi ¼ 4 0n;n 0n;n
;
0p;n
(19)
3
In 7 5; 0n; n
2
Ldi
6 K i ¼ 4 0n;n 0n;n
0p;n
3
7 0n; n 5, In
(20) 3
Ei 6 7 Ni ¼ 4 0n;m 5; 0n;m
3 7 7 7 7 7o0, 7 7 5
þ A¯ i P i þ P i A¯ i þ ð1 þ ZmÞQ ÞIT1
i
N j¼1 lij P j
2
J ¼ ½Iq 0q;n .
(21)
Proof. From (2) and (6), the state space representation of can be written as the error system S S 8 ~ Z Þx~ ðtÞ þ A~ x~ ðt tZ ðtÞÞ þ Bð ~ Z ÞwðtÞ; < x_~ ðtÞ ¼ Að d t t t ~Þ : (22) ðS ~ Z Þx~ ðtÞ þ C~ x~ ðt tZ ðtÞÞ þ Dð ~ Z ÞwðtÞ; : z~ ðtÞ ¼ Cð d
t
(8)
Ni
Adi
# C di , A
C i A
Pi F 6 7 6 0n; 7 ~ nþq 7; Ci ¼ 6 6 0m;nþq 7 4 5 J
Theorem 1. There exists a nth reduced-order filtering kog if there exist matrices Þ such that kS S ðS P 1 ; P2 ; . . . ; P N with P i 40 2 RðnþnÞðnþnÞ , X 1 ; X 2 ; . . . ; X N with ðnþnÞðnþ nÞ ðnþnÞðnþ nÞ X i 40 2 R , Q 40 2 R and Z40 2 ðnþnÞðnþ nÞ R satisfying
"
(18)
Proof. The proof can be carried out by the similar lines as in the proof of [9] and by the Schur complement, thus it is omitted. &
I1 ð
A¯ di ¼
C¯ di ¼ ½C di 0;
2
I
I1 ðlii X i þ A¯ i X i þ X i A¯ i ÞIT1
i D B i
2P
where Z ¼ maxfjlii j; i 2 Sg:
6 #6 0 6 6 6 I 6 6 4
0
where "
(7)
2
0
;
Oi þ Ci Gi Fi þ ðCi Gi Fi ÞT o0,
Proposition 1. The Markovian jump system ðSÞ is exponentially mean-square stable with g-disturbance attenuation, if there exist matrices P1 40; P 2 40; . . . ; P N 40 and Q 40 such that the following LMIs hold for i ¼ 1; 2; . . . ; N
6 6 6 6 6 6 6 4
0
Furthermore, if Pi 40; X i 40; Q 40 and Z40 are the solutions of (8)–(10), then a reduced-order filtering can be computed by substituting the solutions of (8)–(10) into
In this section, we consider the design of reducedorder filtering for ðSÞ. Before presenting the main results, we first give the following proposition which will be used in the proof of our main results.
2
Ai
C¯ i ¼ ½C i 0;
3. Main results
6 6 6 6 4
(13)
I1 Pi A¯ di IT1 ð1
hi ÞI1 Q IT1
I1 P i B¯ i 0
g2 I
T
I1 C¯ i
t
t
3
7 7" T 7 N ¯ i I1 C di 7 7 T 7 0 ¯i 7 D 5 I
0 I
#T o0,
(9)
ARTICLE IN PRESS 190
G. Wang et al. / Signal Processing 89 (2009) 187–196
where x~ ¼ ½xT ðtÞ x T ðtÞT ; z~ ðtÞ ¼ zðtÞ z ðtÞ, ~ ZðtÞÞ ¼ A~ ; A~ ðZðtÞÞ ¼ A~ ; Bð ~ ZðtÞÞ ¼ B~ i ; Að i d di C~ d ðZðtÞÞ ¼ C~ di ; for each Zt " Ai A~ i ¼ Bi Li
~ ZðtÞÞ ¼ C~ i , Cð
~ ZðtÞÞ ¼ D ~i Dð
¼ i 2 S; we have # " Adi 0 ~ ¼ ; A di A Bi Ldi i
0 A
#
" B~ i ¼
;
di
Bi
From LMIs (8), (9) and inverse condition (10), if there exist matrices Pi 40; X i 40, for all i 2 S; Q 40 and Z40 satisfying (26), there exists matrix Gi such that (25) holds. All the parameters of the approximation model satisfying (25) can be obtained by substituting the solutions of (8)–(10) into (25). This completes the proof. &
#
, B i Ei
i Li C i , C~ i ¼ ½C i D ~C ¼ ½C D i Ldi C di ; D ~ i ¼ Di D i Ei . di di From (14), (15), (19)–(21), we obtain that A~ i ¼ A¯ i þ FGi Hi ;
A~ di ¼ A¯ di þ FGi K i ;
B~ i ¼ B¯ i þ FGi N i ,
(23)
C~ i ¼ C¯ i þ JGi Hi ;
C~ di ¼ C¯ di þ JGi K i ;
~i ¼ D ¯ i þ JGi Ni . D
(24)
kog and Þ such that kS S We see that there exists ðS exponentially mean-square stable if there exist matrices P i 40; X i 40; Q 40 and Z40 for (22) where i 2 S, such that (7) holds. Furthermore, noticing that (17)–(21), (7) can be rewritten as
Oi þ Ci Gi Fi þ ðCi Gi Fi ÞT o0,
(25)
where Oi ; Ci and Fi are given in (18) and (19). From [23,24], (25) is equivalent to ?T C? i Oi Ci o0;
For
T?T FT? o0 for all i 2 S. i Oi Fi
(26)
C? i ,
we have 2 h i In 0n;n 6 6h i 6 ? 0n;n 0n; ~ ~ n Ci ¼ 6 6 6h i 4 0m;n 0m;n 2
P 1 6 i 60 ~ n~ 6 n; 6 60 6 m;n~ 4 0q;n~
0n;n~
0n;m
0n;q
In~
0n;m ~
0n;q ~
0m;n~
Im
0m;q
0n; ~ n~
0n;m ~
In~
0n;m ~
0m;n~
Im
0q;n~
0q;m
0n;q ~
3 7 7 7 7 7 7 5
Remark 1. Theorem 1 provides sufficient conditions for the solvability of the reduced-order H1 filtering. It is worth pointing out that when Adi ¼ 0; Ldi ¼ 0; C di ¼ 0; A di ¼ 0; C di ¼ 0, we can get the reduced-order filtering for Markovian jump system without time delays, which was also discussed in [17,18] whose results have rank constraints due to partition of positive definite matrix. In this paper, the partition of positive definite matrix is avoided instead of converse constraint. In addition, via [25,26], our result is easy to implement, in the following section, we will discuss this topic particularly. Remark 2. For a fixed system mode, if there exist ^ Z Þ; A^ ðt; Z Þ; Bðt; ^ ZÞ uncertainties in the matrices Aðt; d t t t such that ^ Z Þ ¼ AðZ Þ þ DAðt; Z Þ, Aðt; t t t A^ d ðt; Zt Þ ¼ Ad ðZt Þ þ DAd ðt; Zt Þ, ^ Z Þ ¼ BðZ Þ þ DBðt; Z Þ, Bðt; t
t
t
where AðZt Þ; Ad ðZt Þ; BðZt Þ are known real constant matrices representing the nominal system for each Zt 2 S. The admissible uncertainties are assumed to be modeled in the form
3
7 7 0n;q ~ 7 7. 0m;q 7 7 5 Iq
P 1 1 1 ¯ T ¯ 1 where Xi ¼ I1 ð N j¼1 lij P i P j P i þ P i Ai þ Ai P i þ ð1 þ ZmÞ T 1 1 1 1 P i QP i ÞI1 . Let X i ¼ P i and Q ¼ Z, by Schur complement and conditions (11)–(14), (29) becomes to (8). For T?T FT? o0, we get (9) similarly. i Oi Fi
½DAðt; Zt Þ DAd ðt; Zt Þ DBðt; Zt Þ (27)
¼ MðZt ÞFðt; Zt Þ½N1 ðZt Þ N2 ðZt Þ N 3 ðZt Þ,
(30)
On the other hand, denote
where MðZt Þ; N1 ðZt Þ; N 2 ðZt Þ and N3 ðZt Þ are known real constant matrices for all Zt 2 S. The uncertain timevarying matrix satisfies
Ni ¼ f½Li Ldi Ei T g? ¼ ½U i V i W i ,
F T ðt; Zt ÞFðt; Zt ÞpI;
U i LTi
V i LTdi
þ þ ¼ 0. We select which implies " # Ui 0 V i 0 W i 0 FT? ¼ i 0 0 0 0 0 I 2 3 ½I 0 ½0 0 0 0 7 " #6 7 Ni 0 6 6 ½0 0 ½I 0 0 0 7 ¼ 6 7. 7 0 I 6 4 ½0 0 ½0 0 I 0 5 ½0 0 ½0 0 0 I By calculation, we obtain 2 Xi I1 A¯ di ? ?T Ci Oi Ci ¼ 6 4 ð1 hi ÞQ
I1 B¯ i 0n;m ~ g2 I
(31)
0 0
;
DB¯ i ¼
DBi 0
. (32)
Similar to (23), we have (28)
3 7 5o0,
Zt 2 S.
Then, for each Zt ¼ i 2 S, define DAi 0 DAdi ; DA¯ di ¼ DA¯ i ¼ 0 0 0
W i ETi
(29)
A~ i ¼ A¯ i þ DA¯ i þ FGi Hi , A~ di ¼ A¯ di þ DA¯ di þ FGi K i , B~ i ¼ B¯ i þ DB¯ i þ FGi Ni .
(33)
The other matrices are same to Theorem 1. Then by the similar methods as in the above proof of Theorem 1 and in [9, Lemma 2], we have the similar result about uncertain Markovian jump system.
ARTICLE IN PRESS G. Wang et al. / Signal Processing 89 (2009) 187–196
From Theorem 1, we see that the desired reduced-order filtering involves time delays. However, in some applications, it is desirable to approximate the system ðSÞ with a reduced-order filtering which has no time delays, and the associated model error also achieves a prescribed H1 norm condition. In view of this, we now consider the reduced-order filtering with no time delays. In the following, we formulate the reduced-order filtering problem for the time-delay system ðSÞ as finding an exponentially mean-square stable nth reduced-order filtering model given by 8 _ > > < xðtÞ ¼ AðZt ÞxðtÞ þ BðZt ÞyðtÞ; 0 Z ÞyðtÞ; (34) ðS Þ : zðtÞ ¼ CðZt ÞxðtÞ þ Dð t > > : x ðtÞ ¼ jðtÞ; 8t 2 ½m; 0; where x ðtÞ 2 Rn ; z 2 Rq , such that
(35)
0 Theorem 2. There exists a nth reduced-order filtering S 0 kog if there exist matrices P 1 ; P 2 ; . . . ; P N such that kS S with P i 40 2 RðnþnÞðnþnÞ , X 1 ; X 2 ; . . . ; X N with X i 40 2 ðnþnÞðnþ nÞ ðnþnÞðnþ nÞ R , Q 40 2 R and Z40 2 RðnþnÞðnþnÞ satisfying 6 6 6 6 6 6 6 4
T
I1 ðlii X i þ A¯ i X i þ X i A¯ i ÞIT1
I1 A¯ di
I1 B¯ i
I1 Si ðXÞ
I1 X i
ð1 hi ÞQ
0 g2 I
0 0
0 0
Ri ðXÞ
0
1þ1Zm Z
Ni 0
I1 ð
6 0 6 6 6 I 6 4
X i P i ¼ In~ ;
PN
j¼1 lij P j
7 7 7 7 7o0, 7 7 5
T
þ A¯ i Pi þ Pi A¯ i þ ð1 þ ZmÞQ ÞIT1
0
0
0
C¯ i ¼ ½C i 0;
;
A¯ di ¼
0
0
0
;
ð1 hi ÞQ
0
g2 I
3 Pi F 6 7 6 0n; 7 ~ nþq 7; Ci ¼ 6 6 0m;nþq 7 4 5 J
F¼
0n;q
0n;n
0n;q
In
Fi ¼ ½Hi K i Ni 0nþp;q ,
#
" Hi ¼
;
" Ni ¼
0n;m
0p;n
0n;n
In
#
" ;
Ki ¼
Ldi
0p;n
0n;n
0n; n
# ,
(49)
Proof. From (2) and (34), the state space representation 0 can be written as of the error system S S 8 ~ Z Þx~ ðtÞ þ A~ x~ ðt tZ ðtÞÞ þ Bð ~ Z ÞwðtÞ; < x_~ ðtÞ ¼ Að d 0 t t t ~ ðS Þ : ~ Z Þx~ ðtÞ þ C~ x~ ðt tZ ðtÞÞ þ Dð ~ Z ÞwðtÞ; (50) : z~ ðtÞ ¼ Cð d
t
T
I
(39)
(41)
0
Li
(47)
J ¼ ½Iq 0q;n .
;
,
I
#
Ei
Bi
7 T 7 C¯ di 7 7, 7 ¯ Ti 5 D
(48)
3
2
¯ Ti D
B¯ i ¼
T C¯ i
(46)
g2 I
¯ i ¼ Di . C¯ di ¼ ½C di 0; D
(42) (43)
Furthermore, if P i 40; X i 40; Q 40 and Z40 are the solutions of (36)–(38), then a reduced-order filtering can be computed by substituting the solutions of (36)–(38) into
Oi þ Ci Gi Fi þ ðCi Gi Fi ÞT o0,
Ni ¼ f½Li Ldi Ei T g? ,
Adi
P i B¯ i
qffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi pffiffiffiffiffiffiffi li1 X i liði1Þ X i liðiþ1Þ X i liN X i ,
Ai
P i A¯ di
0
Ri ðXÞ ¼ diagfX 1 ; . . . ; X i1 ; X iþ1 ; . . . ; X N g,
þ A¯ i P i þ P i A¯ i þ ð1 þ ZmÞQ
ð1 hi ÞI1 Q IT1
(38)
n~ ¼ n þ n;
T
N j¼1 lij P j
(40)
A¯ i ¼
6 6 Oi ¼ 6 6 6 4
I1 P i B¯ i
for all i 2 S, where
Si ðXÞ ¼ ½
2P
(45)
i
I1 Pi A¯ di IT1
QZ ¼ In~
I1 ¼ ½In 0n;n~ ;
# C i , A
i D Bi
3
(36) 2
Gi ¼
"
0 kog. kS S
2
where "
191
(44)
I1 C¯ i
t
t
3 7
T 7 I1 C¯ di 7 Ni 0
7 7 5
0
T
I
o0,
(37)
where x~ ¼ ½xT ðtÞ x T ðtÞT ; z~ ðtÞ ¼ zðtÞ z ðtÞ, A~ i ¼ A¯ i þ FGi Hi ; A~ di ¼ A¯ di þ FGi K i ; B~ i ¼ B¯ i þ FGi N i , C~ i ¼ C¯ i þ JGi Hi ; C~ di ¼ C¯ di þ JGi K i , " # Ai 0 ~ ¯ i þ JGi Ni ; A¯ i ¼ , Di ¼ D 0 0 " # " # Adi 0 Bi A¯ di ¼ ; B¯ i ¼ , 0 0 0 C¯ i ¼ ½C i 0;
C¯ di ¼ ½C di 0;
¯ i ¼ Di . D
Similar to the proof of Theorem 1, we conclude that under the conditions of Theorem 2 there exists Gi such that (7) holds. Then there exists a reduced-order filtering system (34). Using Schur complement, the desired results follow immediately. & Remark 3. From Theorems 1 and 2, we can obtain that when the N ¼ 1, the Markovian jump linear systems with
ARTICLE IN PRESS 192
G. Wang et al. / Signal Processing 89 (2009) 187–196
methods are promising, such as, the alternating projections method [28], the min–max algorithm [29], XY-centering algorithm [30], and cone complementarity linearization (CCL) algorithm [25]. In Ref. [31], the alternating projections method, the min–max algorithm, and the CCL algorithm were compared, and numerical experiments show that the CCL algorithm is the best one. Very recently, a new algorithm which was named sequential linear programming matrix method (SLPMM) was proposed to solve such non-convex feasibility problems. It is seen as an improved version of the CCL algorithm. As is indicated in [26], the SLPMM algorithm is superior to the CCL algorithm in that it always generates a sequence of iteration with strictly decreasing objective function values, and it is also globally convergent. In order to solve the non-convex feasibility problem appeared in this paper, we will employ the SLPMM algorithm. To utilize the SLPMM algorithm to solve the reducedorder H1 filtering problem, we firstly define a convex set
mode-dependent time delays are specialized to deterministic systems with time-varying delays. And in [27], it discussed robust reduced-order filtering for deterministic system without time delays. Therefore, from Theorems 1 and 2, we get the reduced-order H1 filtering for deterministic systems with time delays, and we may say that our results can be viewed as extension results on the reduced-order H1 filtering problem from deterministic systems to stochastic systems. From Theorems 1 and 2, we obtain the strict LMIs conditions for the solvability of the zeroth-order H1 filtering. Corollary 1. If there exist matrices P1 ; P2 ; . . . ; P N , with P i 40 2 Rnn and Q 40 2 Rnn satisfying 2P 6 6 4
N j¼1 lij P j
þ ATi P i þ P i Ai þ ð1 þ ZmÞQ
P i Adi
ð1 hi ÞQ
P i Bi
3
7 0 7 5o0, g2 I
(51)
2 PN "
Ni 0
#6 0 6 6 6 6 I 6 4
j¼1 lij P j
þ ATi P i þ P i Ai þ ð1 þ ZmÞQ
P i Adi
P i Bi
ð1 hi ÞQ
0
g2 I
for all i 2 S, where
Ni ¼ f½Li Ldi Ei T g? .
(53)
C Ti
3
7" 7 N i C Tdi 7 7 7 T 7 0 Di 5 I
0 I
#T o0
(52)
of all the feasible solutions of LMIs (8) and (9) as following:
i to solve the zeroth-order H1 Then, there exists constant D filtering problem. In this case, the desired zeroth-order H1 i corresponding to feasible matrix can be filtering solution D computed by substituting the solutions of (51) and (52) into
S:¼fXjX satisfies LMIs ð8Þ and ð9Þg,
Oi þ CD i Fi þ ðCD i Fi ÞT o0,
From Theorem 1, the reduced-order filtering for Markovian jump linear systems with mode-dependent time delays can be obtained if there exist X i 40, P i 40, Q 40, Z40, such that
2 PN 6 6 Oi ¼ 6 6 4
j¼1 lij P j
þ ATi P i þ P i Ai þ ð1 þ ZmÞQ
(54) P i Adi
P i Bi
ð1 hi ÞQ
0
g2 I
C Ti
3
7 C Tdi 7 7, 7 DTi 5 I
(55) 2
0n;q
3
60 7 6 n;q 7 7; 7 4 0m;q 5
C¼6 6
Fi ¼ ½Li Ldi Ei 0p;q .
(56)
Iq
Remark 4. From Corollary 1, we can see that all the conditions are strict LMIs, which are convex. So we could solve them efficiently by Matlab LMI toolbox. From Theorem 1, we see that the construction of the reduced-order filtering resides in solving LMIs (8), (9) and the inverse constraints in (10). This non-convex condition makes it difficult find the numerical solutions, and there are many numerical approaches proposed to deal with such a problem. Among those approaches, the LMI-based
(57)
where
X:¼fX i 40; Pi 40; Q 40; Z40; i 2 Sg.
X i ; P i 2 S;
X i Pi ¼ I
for all i 2 S; QZ ¼ I.
(58)
(59)
From [25,26], we know fact about the algorithm that for any matrices X40 and P40, if the LMI X I X0 (60) I P is feasible, then traceðXPÞXn. And traceðXPÞ ¼ n if and only if XP ¼ I. Define the set as (" # ) Xi I Q I T:¼ X0; (61) X0 for all i 2 S . I Pi I Z By the SLPMM approach, the above non-convex problem of (57)–(59) is equivalent to the following minimization problem: ( ! !) N X . (62) X i P i þ QZ min trace
X2S\T
i¼1
ARTICLE IN PRESS G. Wang et al. / Signal Processing 89 (2009) 187–196
We see that the optimal solution of problem (62) is ðN þ 1Þn~ satisfying traceðX i P i Þ ¼ traceðQZÞ ¼ n~
for all i 2 S.
(63)
Hence, the reduced-order filtering problem is changed to a problem of finding a global solution of the minimization problem (62). Based on the above analysis, an algorithm HinfRFP (H1 Reduced-order Filtering Problem) for Markovian jump systems with mode-dependent time delays is presented to solve this problem. HinfRFP Algorithm: Step 1: For the system ðSÞ with given reduced-order g and error accuracy d. n; Step 2: Find any initial solution ðX i0 ; Pi0 ; Q 0 ; Z 0 ; i 2 SÞ 2 S, and set k ¼ 0. Step 3: Define function f k ðXÞ ¼ trace
!
N X
ðX i P ik þ P i X ik Þ þ QZ k þ ZQ k .
Pi
# X0;
Q I
) I X0 for i 2 S . Z
r2½0;1
0:3
3:8
Ad1
C d1 ¼ ½0:2 0:3 0:6;
Ld1 ¼ ½1 1 1;
0:1
0:2 6 ¼6 4 0:5 0
1 1
0:6
3
7 0:8 7 5, 2:5
D1 ¼ 0:2,
E1 ¼ 1.
2
0:5
2:5
6 A2 ¼ 6 4 0:1
0:1
3
2
7 0:3 7 5; 2
3:5
0:1 1 2 3 0:6 6 7 7 B2 ¼ 6 4 0:5 5, 0
0
0:3
0:6
0:5
0
1
0:8
6 Ad2 ¼ 6 4 0:1 0:6
C d2 ¼ ½0:4 0:6 0:5;
Ld2 ¼ ½1 1 1;
3 7 7, 5
D2 ¼ 0:5,
E2 ¼ 1.
(66)
Step 6: Let X iðkþ1Þ ¼ X ik þ r ðX i X ik Þ; Piðkþ1Þ ¼ Pik þ r ðPi Pik Þ; Q kþ1 ¼ Q k þ r ðQ Q k Þ; Z kþ1 ¼ Z k þ r ðZ Z k Þ,
The time-varying delay is t2 ðtÞ ¼ 0:2 sinð0:6tÞ satisfying (3) with m2 ¼ 0:2; h2 ¼ 0:12. Suppose the transition probability matrix is given by
and k ¼ k þ 1. If kokmax then go to step 3, else exit. Pr ¼ Remark 5. This algorithm is similar to that developed in [26]. From the HinfRFP algorithm, we can see that when r 1, the HinfRFP algorithm will become to the CCL algorithm [25]. So we may say that HinfRFP contains CCL algorithm. As explained in [26], HinfRFP algorithm is globally convergent since the sequence of function (64) generated by HinfRFP algorithm always converges to some ~ while the alternating projection methf ðX ÞX2ðN þ 1Þn, od, which was used to construct the reduced-order filtering for Markovian jump systems in [17,19] is guaranteed to converge only locally. HinfRFP algorithm can also be applied to Theorem 2 similarly. Noting from HinfRFP algorithm, we can find a suboptimal g for some determined reduced-order n for Þ that (57)–(59) hold. For g, we can propose a system ðS minimization problem:
subjects to constraints (57)–(59).
1
C 1 ¼ ½0:8 0:3 0;
2
7 7; 5
2:5
L2 ¼ ½1 1 1;
min g
3
0:6
C 2 ¼ ½0:5 0:2 0:3;
!
þðQ k þ rðQ Q k ÞÞðZ k þ rðZ Z k ÞÞ .
1
3 6 A1 ¼ 6 4 0:3 0:1 2 3 1 6 7 7 B1 ¼ 6 4 0 5, 1
(65)
N X ðX ik þ rðX i X ik ÞÞðP ik þ rðP i P ik ÞÞ i¼1
2
The time-varying delay is t1 ðtÞ ¼ 0:3 sinð0:3tÞ satisfying (3) with m1 ¼ 0:3; h1 ¼ 0:09: For mode 2, the dynamics of the system are described as
~ d, then construct a Step 4: If jf k ðXÞ 2ðN þ 1Þnjp reduced-order filtering in the form of (6), and exit; otherwise go to the step 5. Step 5: Compute r 2 ½0; 1 by solving min trace
In this section, we present an illustrative example to demonstrate the effectiveness of this approach. Consider a linear Markovian jump system with mode-dependent time delays in the form of ðSÞ with two modes. For mode 1, the dynamics of the system are described as
(64)
Find X solving the following convex programming: I
4. Numerical example
L1 ¼ ½1 1 1;
i¼1
" ( Xi min f k ðXÞ I X2S
193
(67)
0:5
0:5
0:3
0:3
.
Our purpose is to find a first-order model. We first choose 2
1 60 6 6 60 Ni ¼ 6 60 6 6 40 0
3
1
0
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0
0
0
0
1
1
0 7 7 7 0 7 7 0 7 7 7 0 5
0
0
0
0
1
for i ¼ 1; 2.
1
Via the HinfRFP algorithm, we solve the matrix inequalities (8) and (9) with inverse constraints in (10) for i ¼ 1; 2 and get 2
0:2764
6 6 0:0875 6 P1 ¼ 6 6 0:0563 4 0:1549
0:0875
0:0563
1:0217
0:1141
0:1141
0:9914
0:3226
0:5917
0:1549
3
7 0:3226 7 7 7, 0:5917 7 5 1:0081
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2
0:9569
0:1885
0:3942
1:3395
3
A d2 ¼ 0:8467, B 2 ¼ 0:1073, C 2 ¼ 1:5301, C d2 ¼ 2 ¼ 0:5557. The last optimal r is r ¼ 0:3295, 1:3418, D and the reduced-order filtering level of the system ðSÞ is g ¼ 0:394. In the following, we will show the approximation performance of the obtained reduced-order H1 filtering. With the initial state of x~ ð0Þ ¼ ½0 0 0 0T , Fig. 1b is outputs trajectories of the original system and the first-order system subject to disturbance input uðtÞ ¼ et=8 sinð2tÞ under the systems jump mode Fig. 1a, while Fig. 2b is subject to disturbance input uðtÞ ¼ sinð2tÞ under the system jumps mode Fig. 2a. We can see the obtained reduced-order model approximates the original model very well.
6 7 6 0:1885 0:7920 0:2425 0:9164 7 6 7 P2 ¼ 6 7, 6 0:3942 0:2425 1:3660 1:3099 7 4 5 1:3395 0:9164 1:3099 3:0306 2 3 6:0983 1:6232 2:1582 2:7232 6 7 6 1:6232 1:7214 1:1819 1:4939 7 6 7 X1 ¼ 6 7, 6 2:1582 1:1819 2:6026 2:2374 7 4 5 3:2017 3 3:8733 6 7 6 2:7098 3:4661 1:2913 2:8038 7 6 7 X2 ¼ 6 7, 6 1:7263 1:2913 1:8970 1:9733 7 4 5 3:8733 2:8038 1:9733 3:7426 3 2 0:6166 0:3461 0:1885 0:5396 6 7 6 0:3461 2:8414 0:6090 1:2000 7 6 7 Q ¼6 7, 6 0:1885 0:6090 2:7983 1:8427 7 4 5 0:5396 1:2000 1:8427 3:4330 2 3 3:4386 1:3381 1:1190 1:6088 6 7 6 1:3381 1:1642 0:8811 1:0902 7 6 7 Z¼6 7. 6 1:1190 0:8811 1:2256 1:1417 7 4 5 2
2:7232 5:2215
1:6088
1:4939 2:7098
1:0902
2:2374 1:7263
1:1417
5. Conclusions This paper has presented an approach for the reducedorder filtering design for Markovian jump systems with mode-dependent time-varying delays. The designed filtering can guarantee the error gain satisfying a prescribed scalar. Sufficient conditions for the existence of the reduced-order H1 filtering have been proposed in terms of LMIs with equality constraints. The HinfRFP algorithm is used to handle the non-convex constraints. A numerical example has been given to demonstrate the validity of the theoretical results. The results obtained in this paper may be extended to Markovian jump systems with uncertainties.
1:5381
systems jump mode
Then the parameters of the first-order model can be computed as A 1 ¼ 4:6461, A d1 ¼ 3:5469, B 1 ¼ 0:7479, 1 ¼ 0:1506; A 2 ¼ 2:4290, C 1 ¼ 0:8624, C d1 ¼ 0:4812, D
2.5 2 1.5 1 0.5 0
5
10
15
20
25
30
35
t 0.5
outputs
original filtering reduced order filtering
0
−0.5 0
5
10
15
20
25
t Fig. 1. Systems jump mode and outputs under uðtÞ ¼ et=8 sinð2tÞ.
30
35
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systems jump mode
2.5 2 1.5 1 0.5 0
5
10
15
20
25
30
35
t 1.5 original filtering reduced order filtering
outputs
1 0.5 0 −0.5 −1 0
5
10
15
20
25
30
35
t Fig. 2. Systems jump mode and outputs under uðtÞ ¼ sinð2tÞ.
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