filtering for continuous-time Markovian jump ... - Semantic Scholar
Signal Processing 93 (2013) 2392–2407
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A new design of HN filtering for continuous-time Markovian jump systems with time-varying delay and partially accessible mode information$ Yanling Wei a, Jianbin Qiu a,n, Hamid Reza Karimi b, Mao Wang a a b
Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, PR China Department of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad, Norway
a r t i c l e in f o
abstract
Article history: Received 20 November 2012 Received in revised form 30 January 2013 Accepted 20 February 2013 Available online 5 March 2013
In this paper, the delay-dependent H1 filtering problem for a class of continuous-time Markovian jump linear systems with time-varying delay and partially accessible mode information is investigated by an indirect approach. The generality lies in that the systems under consideration are subject to a Markov stochastic process with exactly known and partially unknown transition rates. By utilizing the model transformation idea, an input–output approach is employed to transform the time-delayed filtering error system into a feedback interconnection formulation. Invoking the results from the scaled small gain theorem, an improved version of bounded real lemma is obtained based on a Markovian Lyapunov–Krasovskii functional. The underlying full-order and reduced-order H1 filtering synthesis problems are formulated by a linearization technique. Via solving a set of linear matrix inequalities, the desired filters can therefore be constructed. The results developed in this paper are less conservative than existing ones in the literature, which are illustrated by two simulation examples. & 2013 Elsevier B.V. All rights reserved.
Keywords: Markovian jump system H1 filtering Partially accessible mode information Scaled small gain theorem Time-varying delay
1. Introduction State estimation has long been a significant and active research issue due to its theoretical and practical significance in engineering applications and signal processing. Many results on estimation and filtering design for different kinds of dynamic systems have been obtained [1–10]. Specifically, H1 filtering, whose main advantages are insensitive to the exact knowledge of the external noise signals and more robust to parameter uncertainties in systems [1,11], has attracted considerable attention. Therefore, recently, a number of studies on H1 filtering problem for discrete-time or continuoustime systems with delays and uncertainties have been reported in the open literature, e.g., [1,10–15]. On another research front line, as an important class of stochastic hybrid systems, Markovian jump linear systems (MJLSs) have been extensively studied due to their powerful modeling ability of Markov process in many fields, such as robot manipulator systems, aircraft control systems, medical and physical systems and so on [13,15,16]. Differing from the nondeterministic switching in switched systems, the modes evolution in MJLSs is determined by a Markov chain [17,18].
$ This work was supported in part by the National Natural Science Foundation of China (61004038), in part by the National 973 Project of China (2009CB320600), in part by the Program for New Century Excellent Talents in University, in part by the Fundamental Research Funds for the Central Universities (HIT.BRETIII.201214), in part by the Alexander von Humboldt Foundation of Germany. n Corresponding author. Tel.: þ 86 451 86402350. E-mail addresses: [email protected] (Y. Wei), [email protected], [email protected] (J. Qiu), [email protected] (H. Reza Karimi), [email protected] (M. Wang).
0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.02.014
Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
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In traditional analysis and design, it is usually assumed that the mode information in the Markov chain is definitely known. In fact, it is questionable and generally expensive to obtain all the transition mode information even for a simple system [19–21]. Recently, some attention has been drawn to the more general cases with partially accessible mode information. It is noted that for the discrete-time case, the transition probabilities are all non-negative, and the sum of transition probabilities is 1 for each row in a transition probability matrix (TPM) [22], which facilitates the mathematical derivations in the discrete-time context. On the other hand, for the continuous-time case with partially accessible mode information, the difficulty is that the unknown elements may locate in diagonal of the transition rate matrix (TRM), which are nonpositive. Therefore, it is more difficult and challenging to investigate the corresponding filtering analysis and design problems for continuous-time MJLSs with partially accessible mode information. On the other hand, time-delays are frequently encountered in MJLSs. Recently, there have appeared some results on robust control and filtering design for MJLSs with time-delays and it has been shown that the delay-dependent conditions reflect the reality better in general [12,23–26]. In addition, it has also been shown that model transformation, which may incur additional dynamics to degrade the performance of original system, is a main source of design conservatism [27–30]. Recently, a new model transformation based on the input–output approach was proposed for time-delay systems [31–34]. Unlike the previous transformations, a new input–output model is introduced for the original system by employing a twoterm approximation method. Then, the stability analysis and synthesis of systems with time-varying delay are transformed into a scaled small gain (SSG) problem [31,35–39]. However, to the authors’ best knowledge, there exist few related studies on delay-dependent H1 filtering for continuous-time MJLSs with time-varying delay and partially accessible mode information based on the SSG techniques. Especially, it is expected that the conservatism of filtering design conditions shall be further reduced by utilizing the input–output approach, which inspires us for this study. According to the issues mentioned above, we will propose a new delay-dependent H1 filtering design method for continuous-time MJLSs in the presence of time-varying delay and partially accessible mode information based on SSG theorem. By introducing the lower bounds of the unknown diagonal elements, together with the convexification of unknown transition rates, the difficulty that the unknown elements appear in diagonal of TRM is overcome without introducing additional conservatism. A two-term approximation approach is used to transform the filtering error system into a feedback interconnection form, which contains a forward subsystem with two constant delays and a feedback one with norm-bounded uncertainties. Then, based on a Markovian Lyapunov–Krasovskii functional combined with SSG theorem, new delay-dependent H1 performance analysis conditions are derived. By applying a linearization procedure, both the full-order and reduced-order filtering design results can be constructed in terms of linear matrix inequalities. Two examples will be given to illustrate the advantages of the proposed results over the existing ones. Notations: The notations used throughout the paper are standard. T 1 JT 2 represents the series connection of mapping T1 and T2. Rn and Rmn denote, respectively, the n-dimensional Euclidean space, and the set of all m n real matrices; P 4 0 means that P is real symmetric and positive definite; SymfAg is the shorthand notation for A þ AT ; I and 0 represent the identity matrix and a zero matrix, respectively; ðO,F ,PÞ denotes a complete probability space, in which O is the sample space, F is the s algebra of subsets of the sample space, and P is the probability measure on F ; E½ stands for the mathematical expectation; kk denotes the Euclidean norm of a vector or its induced norm of a matrix; signals that are square integrable over ½0,1Þ is denoted by L2 ½0,1Þ with the norm J J2 . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation and preliminaries A continuous-time Markovian jump linear system (MJLS) with an interval time-varying delay in the state, defined in a complete probability space ðO,F ,PÞ, is governed by state-space equation of the form: _ ¼ AðrðtÞÞxðtÞ þ Ad ðrðtÞÞxðtdðtÞÞ þBðrðtÞÞwðtÞ ðSÞ : xðtÞ yðtÞ ¼ CðrðtÞÞxðtÞ þ C d ðrðtÞÞxðtdðtÞÞ þ D1 ðrðtÞÞwðtÞ zðtÞ ¼ LðrðtÞÞxðtÞ þ Ld ðrðtÞÞxðtdðtÞÞ þ D2 ðrðtÞÞwðtÞ xðtÞ ¼ ft ,
t 2 ½d2 ,0,
ð1Þ
where xðtÞ 2 Rn is the state vector; wðtÞ 2 Rm denotes the disturbance input vector, which belongs to L2 ½0,1Þ; yðtÞ 2 Rp is the measured output; zðtÞ 2 Rq is the signal to be estimated, and d(t) is a time-varying delay satisfying _ r m o1, where d and d are known constant scalars representing the lower and upper 0 r d1 r dðtÞ rd2 o 1 and dðtÞ 1 2 delay bounds, respectively. In (1), ft is a vector-valued initial continuous function defined on the interval ½d2 ,0; the process frðtÞ,t Z 0g is a continuous-time homogeneous Markov chain with right continuous trajectories and taking values in a finite set I :¼ f1, . . . ,Ng with transition rate matrix (TRM) L :¼ ½lij NN given by ( lij hþ oðhÞ, iaj Prfrðt þhÞ ¼ j9rðtÞ ¼ ig ¼ 1 þ lii h þoðhÞ, i ¼ j where h 40, limh-0 ðoðhÞ=hÞ ¼ 0, and lij Z0, for jai, is the transition rate (TR) from mode i at time t to mode j at time t þ h, P and lii ¼ N j ¼ 1,jai lij . In the sequel, for each possible rðtÞ ¼ i, i 2 I , the system matrices of the ith mode are denoted by ðAi ,Adi ,Bi ,C i ,C di ,D1i ,Li ,Ldi ,D2i Þ, which are real and known matrices with appropriate dimensions.
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The TRs of the stochastic process in this paper are considered to be partially available. For instance, in the case of a system (S) with four operation modes, the TRM may be as 2 3 l l^ 12 l^ 13 l14 6 11 7 6 l^ 7 ^ 6 21 l 22 l23 l24 7 6 7 6 l31 l^ 32 l33 l^ 34 7 4 5 l l^ l^ l^ 41
42
43
44
[ where the elements labeled with the hat ‘‘^’’ represent the unknown TRs. For notational clarity, 8i 2 I , we describe I ¼ I ðiÞ k ðiÞ ðiÞ ^ is unknowng. Also, throughout this paper, we denote as follows: I :¼ fj : l is knowng, and I :¼ fj : l I ðiÞ ij ij uk k uk P ðiÞ ^ lðiÞ l . For tractability reasons, we further restrict the unknown diagonal element l^ ii as lðiÞ k :¼ d r l ii , where ld j2I ðiÞ ij k
provides a lower bound for the unknown element l^ ii [21]. The objective of this paper is to design a filter of the following general structure to estimate z(t), ^ Þ : x^_ ðtÞ ¼ Afi xðtÞ ^ þ Bfi yðtÞ ðS ^ þ Dfi yðtÞ, z^ ðtÞ ¼ C fi xðtÞ
ð2Þ
n^
q
^ n^ n
^ np
qn^
qp
^ 2 R ðn^ r nÞ is the filter state; z^ ðtÞ 2 R is the estimation of z(t); Afi 2 R , Bfi 2 R , C fi 2 R and Dfi 2 R , where xðtÞ 8i 2 I , are filter gains to be determined. It is noted that n^ ¼ n for the full-order filtering and n^ o n for the reduced-order filtering. Augmenting the model in (1) to include the states of the filter in (2), we obtain the following filtering error system: ðS Þ : x_ ðtÞ ¼ A i xðtÞ þ A di ExðtdðtÞÞ þB i wðtÞ zðtÞ ¼ L i xðtÞ þ L di ExðtdðtÞÞ þ D i wðtÞ T
xðtÞ ¼ ½ft 0T , T
t 2 ½d2 ,0, T
ð3Þ
T
where xðtÞ :¼ ½x ðtÞ x^ ðtÞ , zðtÞ :¼ zðtÞz^ ðtÞ and " # " # " # Ai Adi Bi 0 A i :¼ , A di :¼ , B i :¼ , Bfi C di Bfi D1i Bfi C i Afi L i :¼ ½Li Dfi C i C fi , D i :¼ D2i Dfi D1i ,
L di :¼ Ldi Dfi C di ,
E :¼ ½In 0nn^ :
ð4Þ
More precisely, we introduce the following definitions for the filtering error system (S ) in (3), which are essential for the later development. Definition 1 (Zhang et al. [25]). The filtering error system (S ) in (3) is said to be stochastically stable if under wðtÞ ¼ 0 and ^ any initial condition xð0Þ 2 Rðn þ nÞ , rð0Þ 2 I , Z t Jxða,xð0Þ,rð0ÞÞJ2 da o1: lim E t-1
0
Definition 2 (Zhang et al. [25]). Given a scalar g 4 0, the filtering error system (S ) in (3) is said to be stochastically stable with an H1 disturbance attenuation performance index g if it is stochastically stable with wðtÞ ¼ 0, and under zero initial condition, Z 1 Z 1 E z T ðtÞzðtÞ dt o g2 wT ðtÞwðtÞ dt, 80awðtÞ 2 L2 ½0,1Þ: 0
0
For the filtering design based on (2), it is required that the original system (1) should be stochastically stable. The purpose of this paper is to design a delay-dependent H1 filter such that the filtering error system (S ) in (3) is stochastically stable with a prescribed H1 performance index g by an indirect approach. The key procedure with this approach is to introduce a new model transformation by employing a two-term approximation for MJLSs with timevarying delay, which converts system (S ) into an interconnection structure. A particular direction of this transformation is that the H1 filtering design problem is reformulated into the scaled small gain (SSG) problem of the new input–output model. More details on this approach can be found in [28,31,36]. Here, we just retrospect some preparative notions on the SSG theorem. Consider an interconnected system consisting of two subsystems, L1 : xðtÞ ¼ GZðtÞ,
L2 : ZðtÞ ¼ DxðtÞ,
ð5Þ
which is shown in Fig. 1, where the forward subsystem L1 is known and time-invariant with operator G mapping Z to x, and the feedback one L2 is unknown and time-varying with operator D 2 D :¼ fD : JDJ1 r 1g. As a direct ramification of
Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
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Fig. 1. Interconnected system consisting of two subsystems.
the SSG theorem [31,33,35,36], a sufficient condition for the robust stability of the interconnected system formed by L1 and L2 in (5) is provided as follows. Lemma 1. Consider (5) and assume that L1 is internally stable. The closed-loop system formed by L1 and L2 is robustly stable Z for all D 2 D if JT x JGJT 1 Z Rxx : T Z ,T x Z J1 o1 holds for some matrices fT x ,T Z g 2 T with T :¼ ffT Z ,T x g 2 R 1 nonsingular; JT Z JDJT x J1 r1g. 3. Main results In this section, we concentrate on transforming system (S ) in (3) into an interconnection of two subsystems as in (5) and analyzing the scaled small gain (SSG) of the forward subsystem. A new delay-dependent H1 performance criterion will firstly be proposed for system (S ) with complete known mode information, and then a bounded real lemma (BRL) and corresponding filtering synthesis for the underlying systems with partially accessible mode information will be developed. 3.1. Model transformation Considering the filtering error system (S ) in (3), the term xðtdðtÞÞ is approximated by 12 ½xðtd1 Þ þxðtd2 Þ, which brings to the approximation error, 1 2
Z d ðtÞ ¼ xðtdðtÞÞ ½xðtd1 Þ þxðtd2 Þ ¼ :¼
1 2 1 2
Z Z
dðtÞ
_ þ aÞ da xðt
d2
1 2
Z
d1
_ þ aÞ da xðt
dðtÞ
d1
rðaÞxd ðt þ aÞ da,
ð6Þ
d2
_ and where xd ðtÞ :¼ xðtÞ, ( 1 when a r dðtÞ, rðaÞ :¼ 1 when a 4 dðtÞ: Normalizing Z d ðtÞ in (6) by multiplying 2=d with d ¼ d2 d1 , (6) can be rewritten as, Z 1 d1 Zd ðtÞ :¼ Dd ðtÞxd ðtÞ ¼ rðaÞxd ðt þ aÞ da, d d2
ð7Þ
where Zd ðtÞ :¼ ð2=dÞðZ d ÞðtÞ, and operator Dd : xd /Zd , which includes the uncertainties pulled out from xðtdðtÞÞ in the original time-delay system (S ). Substituting (7) into system (S ) in (3), the filtering error system (S ) is converted into a feedback system with the forward subsystem ðLd1 Þ and the feedback one ðLd2 Þ, described by 8 1 d > > > x_ ðtÞ ¼ A i xðtÞ þ A di E½xðtd1 Þ þ xðtd2 Þ þ A di Zd ðtÞ þ B i wðtÞ > > 2 2 > > < 1 d ðLd1 Þ : zðtÞ ¼ L i xðtÞ þ L di E½xðtd1 Þ þxðtd2 Þ þ L di Zd ðtÞ þ D i wðtÞ > 2 2 > > > > 1 d > > : xd ðtÞ ¼ Ai xðtÞ þ Adi ½xðtd1 Þ þ xðtd2 Þ þ Adi Zd ðtÞ þBi wðtÞ 2 2 ðLd2 Þ : Zd ðtÞ ¼ Dd ðtÞxd ðtÞ,
T
xðtÞ ¼ ½ft 0T ,
t 2 ½d2 ,0:
ð8Þ
In light of the transformation (6), the original system (S ) is transformed into a feedback interconnection form, which contains a forward subsystem (Ld1 ) with two known constant delays and a feedback one (Ld2 ) with delay uncertainties. The stability problem of system (S ) can therefore be interpreted as a robust performance analysis problem for the nominal system (Ld1 ) in the face of norm-bounded uncertainties Dd ðtÞ. Of course, the stability of the transformed system in (8)
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implies the stability of the original time-delay system in (3). Specifically, the following result can be concluded which, in the meantime, provides a possible choice of the scaling matrices fT Z ,T x g 2 T . Lemma 2. For any given invertible matrix X, the operator Dd in (8) enjoys the property JX JDd JX 1 J1 r 1. Proof. By virtue of Jensen inequality [28], considering zero initial condition and exchanging the order of integration, it follows that Z t 1 ZTd ðaÞX T X Zd ðaÞ da ¼ 2 E 0 d 8 " #T j 2 I ðiÞ uk > > = if i 2 I ðiÞ , k > > > ;
ðiÞ
QðiÞ þ QðiÞ R2 lk ðQ 1j þ Q 3j Þ r0, k1 k3 ðiÞ
QðiÞ R1 lk Q 2j r 0, k2
j 2 I ðiÞ uk
ðiÞ QðiÞ R2 lk Q 3j r 0, k3
I ðiÞ uk
j2
ðiÞ
ðiÞ
ð31Þ
9 > j 2 I ðiÞ uk > > =
ðiÞ
þ QðiÞ R2 þ ld ðQ 1i þQ 3i Þðld þ lk ÞðQ 1j þ Q 3j Þ r 0, QðiÞ k1 k3 ðiÞ
ðiÞ
ðiÞ
j 2 I ðiÞ uk
ðiÞ
ðiÞ
ðiÞ
j 2 I ðiÞ uk
QðiÞ R1 þ ld Q 2i ðld þ lk ÞQ 2j r 0, k2 QðiÞ R2 þ ld Q 3i ðld þ lk ÞQ 3j r 0, k3
> > > ;
if i 2 I ðiÞ , uk
ð32Þ
where 2
3
T 1 2 P i A di þ E Z 1
1 2 P i A di
d 2 P i A di
U2
14 ð1mÞQ 1i þ 12 Z 2
4d ð1mÞQ 1i
6 n 4
n
14 ð1mÞQ 1i Q 3i 12 Z 2
n
n
U1
6 6 n
U :¼ 6 6
:¼ QðiÞ k1
X
n
lij Q 1j , QðiÞ :¼ k2
j2I ðiÞ k
X
4d ð1mÞQ 1i 2 d4
lij Q 2j , QðiÞ :¼ k3
j2I ðiÞ k
X
2
ð1mÞQ 1i d2 Z 2
7 7 7 7, 7 5
lij Q 3j ,
ð33Þ
j2I ðiÞ k
with
U 1 :¼
8 < U1 þ P ðiÞ lðiÞ P j ,
j 2 I ðiÞ , k k uk ðiÞ ðiÞ : U1 þ P þ l Pi ðlðiÞ þ lðiÞ ÞP j , d d k k
:¼ P ðiÞ k
X j2I ðiÞ k
if i 2 I ðiÞ , k j2
I ðiÞ , uk
if i 2 I ðiÞ , uk
lij P j , ðiÞ
and S, F12 , F13 , F23 , F33 , U1 , and U2 are defined in Proposition1; ld is a given lower bound for the unknown diagonal element l^ ii . Proof. Based on Proposition 1, it is shown that the filtering error system in (8) subject to completely known TRs is stochastically stable with a prescribed H1 performance g if (9)–(12) hold. Since the diagonal elements in the TRM may contain unknown ones, we shall separate the proof of Theorem 1 into two cases, i 2 I ðiÞ and i 2 I ðiÞ . k uk ðiÞ (i) i 2 I k . ðiÞ ðiÞ In this case, i 2 I ðiÞ implies that lii is known, then it is straightforward that lk r0. Here, we only need to consider lk o0, k ðiÞ since if lk ¼ 0, condition (9) can be readily viewed as a direct result. P Notice that the term N j ¼ 1 lij P j in (9) can be treated as N X
lij Pj ¼ P ðiÞ þ k
j¼1
ðiÞ l^ ij P j ¼ P ðiÞ lk k
j2I ðiÞ uk
:¼ where P ðiÞ k becomes N X
X
P
lij Pj , j2I ðiÞ k
lij Pj ¼
j¼1
X l^ ij ðiÞ
j2I ðiÞ uk
lk
X l^ ij ðiÞ
j2I ðiÞ uk
lk
Pj ,
ð34Þ
P ðiÞ ðiÞ and the elements l^ ij , j 2 I ðiÞ , are unknown. Since 0 r l^ ij =lk r1 and ðl^ ij =lk Þ ¼ 1, (34) uk j2I ðiÞ uk
ðiÞ
½P ðiÞ lk Pj : k
ð35Þ
Thus, for 0 r l^ ij r lk , the left-hand side of inequality (9) can be rewritten as 2 3 U S F12 F13 X l^ ij 6 X l^ ij 7 g2 I F23 5 :¼ F¼ F, 4 n ðiÞ ðiÞ lk lk j2I ðiÞ j2I ðiÞ ðiÞ
uk
n
n
F33
ð36Þ
uk
ðiÞ U 1 ¼ U1 þP ðiÞ lk P j , k
and U1 , S, F12 , F13 , F23 , and F33 are defined in (13). Then, (9) holds if where U is defined in (33) with and only if F o0 in (36), which implies that, in the presence of unknown elements l^ ij , i 2 I ðiÞ , j 2 I ðiÞ , inequality (30) is k uk equivalent to (9). (ii) i 2 I ðiÞ . uk ðiÞ ðiÞ Following a similar argument as in (i), we only consider l^ ii olk here, since if l^ ii ¼ lk , then the ith row of the TRM is completely known.
Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
Equivalently, for this case, the term N X
X
lij Pj ¼ P ðiÞ þ l^ ii Pi þ k
j¼1
PN
j¼1
2401
lij Pj in (9) can be expressed as
l^ ij P j ¼ P ðiÞ k
j2I ðiÞ ,jai uk
X
þ l^ ii P i þ ðl^ ii lk Þ ðiÞ
l^ ij
ðiÞ l^ ii lk j2I ðiÞ ,jai
Pj ,
ð37Þ
uk
P ðiÞ k
P
:¼ j2I ðiÞ lij P j . k P ðiÞ ðiÞ Likewise, since 0 r l^ ij =ðl^ ii lk Þ r1 and j2I ðiÞ ,jai l^ ij =ðl^ ii lk Þ ¼ 1, we have
where
uk
N X
X
l^ ij
j2I ðiÞ ,jai uk
l^ ii lk
lij Pj ¼
j¼1
½P ðiÞ þ l^ ii P i ðl^ ii þ lk ÞP j : k ðiÞ ðiÞ
ð38Þ
Correspondingly, we can rewrite the left-hand side of the inequality in (9) as 2 3 U S F12 F13 ^l X ij 6 7 g2 I F23 5 F¼ 4 n ^ lðiÞ l ðiÞ ii k j2I ,jai n
uk
:¼
X
l^
j2I ðiÞ ,jai uk
l^ ii lk
ij ðiÞ
n
F33
F,
ð39Þ
ðiÞ þ l^ ii Pi ðl^ ii þ lk ÞPj , and U1 , S, F12 , F13 , F23 , and F33 are defined in (13). where U is defined in (33) with U 1 ¼ U1 þP ðiÞ k Deduced from (39), F o 0 is equivalent to 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ F :¼ 4 n , jai: ð40Þ uk
n
n
F33
ðiÞ For tractability, by introducing a lower bound ld for the unknown element l^ ii , we have ðiÞ ^ lðiÞ d r l ii olk ,
ð41Þ
ðiÞ ðiÞ which implies that l^ ii may take any value in ½ld ,lk þ E for some sufficiently small E o0. Then l^ ii can be directly written as a convex combination ðiÞ l^ ii ¼ klðiÞ k þ kE þ ð1kÞld ,
ð42Þ
where 0 r k r 1. Since l^ ii in (42) depends on k linearly, and (40) therefore needs only to be satisfied for k ¼ 0 and k ¼ 1, that is, (40) holds if and only if the following inequalities in (43)–(44) simultaneously hold: 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ , jai, ð43Þ 4 n uk n
n
F33 ðiÞ
where U is defined in (33) with U 1 ¼ U1 þ P ðiÞ lk P i þ EðPi P j Þ, and k 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ , jai, 4 n uk n
n
ðiÞ
ðiÞ
where U is defined in (33) with U 1 ¼ U1 þ P ðiÞ þ ld ðP i P j Þlk P j . Since E is small enough, (43) holds if and only if k 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ , jai, 4 n uk n
n
ð44Þ
F33
ð45Þ
F33 ðiÞ
where U is defined in (33) with U 1 ¼ U1 þ P ðiÞ lk Pi , which is implied by (44) when j ¼i, j 2 I ðiÞ . Hence, the inequality (9) k uk can be replaced by (30) in the context 8j 2 I ðiÞ . uk In addition, following a similar procedure presented above, the conditions in (31)–(32) can also be obtained based on (10)–(12), respectively. In summary, with the presence of unknown elements in the TRM, one can readily conclude that the filtering error system in (8) is stochastically stable with a prescribed H1 performance index g if (30)–(32) hold. & Remark 3. In order to render the unknown diagonal elements numerically tractable, in Theorem 1, the lower bounds for the unknown diagonal elements are introduced. Together with the property that the sum of each row is zero in a TRM,
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Y. Wei et al. / Signal Processing 93 (2013) 2392–2407 ðiÞ
ðiÞ
a convex combination in (42) can thus be obtained. Thanks to the convex combination of ld and lk , sufficient conditions for the H1 performance analysis have been derived in Theorem 1. Yet, the obtained conditions are no loss of generality, ðiÞ since the lower bound ld , of l^ ii is allowed to be arbitrarily small. In the following, we will give the filtering design result in the presence of partially accessible mode information, which relies heavily on the delay-dependent BRL presented in Theorem 1.
3.3. Delay-dependent H1 filtering design In this subsection, we consider both the full-order and reduced-order H1 filtering designs. It is shown that the parametrized representation of the filter gains can be obtained in terms of the feasible solutions to a set of linear matrix inequalities (LMIs). Theorem 2. The filtering error system in (8) with partially accessible mode information is stochastically stable with a ^ ^ HP ið2Þ ðn þ nÞðn þ nÞ , guaranteed H1 performance g, if there exist positive-definite symmetric matrices P i ¼ ½Pið1Þ n Pið2Þ 2 R ^ ^ ^ ^ fS,Q 1i ,Q 2i ,Q 3i ,R1 ,R2 ,Z 1 ,Z 2 g 2 Rnn , and matrices A fi 2 Rnn , B fi 2 Rnp , C fi 2 Rqn , and Dfi 2 Rqp , for each mode i 2 I , such that the conditions (31)–(32) and the following LMIs hold: 2 3 ^ 12 F ^ 13 ^ 11 F F 6 ^ 23 7 ^ :¼ 6 n 7 o 0, j 2 I ðiÞ , g2 I F F ð46Þ 4 5 uk ^ n n F 33 where 2 ^ 11 F
C1
6 n 6 6 6 :¼ 6 n 6 6 n 4 n
n
C3 þ Z 1 C5 C6
n
n
n
n
C2 C4
14 ð1 ÞQ 1i þ 12 Z 2 14 ð1 ÞQ 1i Q 3i 12 Z 2
4d ð1 4d ð1
m
mÞQ 1i mÞQ 1i
m
2
2
d4 ð1mÞQ 1i d2 Z 2 S
n
T
3
dC3 dC5
C3 C5
7 7 7 7 7, 7 7 5
T
^ 12 :¼ ½BT P ið1Þ þ DT B HT BT HP ið2Þ þ DT B 0m3n T , F i 1i fi i 1i fi T H :¼ ½In^ 0nðn ^ ^ , nÞ T
T
T
T
^ 13 :¼ ½d1 A ET Z 1 d2 A ET Z 2 A ET S L , F i i i i ^ 23 :¼ ½d1 BT Z 1 d2 BT Z 2 BT S D2i D D1i , F fi i i i ^ 33 :¼ diagfZ 1 ,Z 2 ,S,In g, F A i :¼ ½Ai 0nn
1 2
Adi
1 2
Adi
d 2Adi ,
1 2
Ldi 12 Dfi C di
Ldi 12 Dfi C di 2d Ldi 2dDfi C di , 9 ðiÞ > C1 :¼ G1 þP ð1iÞ lk P jð1Þ , j 2 I ðiÞ > k uk > = ðiÞ ð2iÞ ðiÞ C2 :¼ G2 þHP k lk HP jð2Þ , j 2 I uk if i 2 I ðiÞ , k > > > ðiÞ ð2iÞ ðiÞ ; C4 :¼ SymfA fi g þ P k lk Pjð2Þ , j 2 I uk
L i :¼ ½Li Dfi C i C fi
1 2
ðiÞ ðiÞ ðiÞ C1 :¼ G1 þP ð1iÞ þ ld Pið1Þ ðld þ lk ÞP jð1Þ , j 2 I ðiÞ k uk ðiÞ ðiÞ ðiÞ C2 :¼ G2 þHP ð2iÞ þ ld HPið2Þ ðld þ lk ÞHP jð2Þ , j 2 I ðiÞ k uk ðiÞ ðiÞ ðiÞ C4 :¼ Sym fA fi g þ P ð2iÞ þ ld P ið2Þ ðld þ lk ÞPjð2Þ , j 2 I ðiÞ k uk
9 > > > = > > > ;
if i 2 I ðiÞ , uk
C3 :¼ 12 Pið1Þ Adi þ 12HB fi C di , C5 :¼ 12 PTið2Þ HT Adi þ 12B fi C di , C6 :¼ 14 ð1mÞQ 1i Q 2i Z 1 12Z 2 , G1 :¼ SymfPið1Þ Ai þ HB fi C i g þQ 1i þ Q 2i þ Q 3i þ d1 R1 þd2 R2 Z 1 , T
G2 :¼ HA fi þ ATi HP ið2Þ þC Ti B fi , P ð1iÞ :¼ k
X j2I ðiÞ k
lij P jð1Þ , P ð2iÞ :¼ k
X j2I ðiÞ k
lij P jð2Þ :
ð47Þ
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Moreover, if the above conditions have a set of feasible solutions ðP ið1Þ ,Pið2Þ ,S,Q 1i ,Q 2i ,Q 3i ,R1 , R2 ,Z 1 ,Z 2 ,A fi ,B fi ,C fi ,Dfi Þ, then an ^ admissible n-order filter in the form of (2) can be obtained as Afi ¼ P 1 ið2Þ A fi ,
Bfi ¼ P1 ið2Þ B fi ,
C fi ¼ C fi ,
Dfi ¼ Dfi :
ð48Þ
Proof. It follows from Theorem 1 that if we can show (30)–(32), then the claimed result follows. For simplicity in filter synthesis procedure, we first partition the Lyapunov matrices Pi in Theorem 1 as " # P ið1Þ HPð2Þ Pi ¼ , ð49Þ n P ð3Þ ^
^
^
^
nn T , P ð2Þ 2 Rnn , and P ð3Þ 2 Rnn . Then, similar to [14], performing a congruent transforwhere H :¼ ½In^ 0nðn ^ ^ , P ið1Þ 2 R nÞ mation to Pi by diagfIn , Pð2Þ P 1 g yields, ð3Þ 2 3 2 3 T P ið1Þ HP ð2Þ P ið1Þ HP ð2Þ P 1 P ð3Þ ð2Þ 4 5 :¼ 4 5: ð50Þ T n Pð2Þ P1 n P ð2Þ ð3Þ P ð2Þ
Thus, without loss of generality, we can directly specify the Lyapunov matrices as 2 3 Pið1Þ HPið2Þ 5: Pi ¼ 4 n Pið2Þ
ð51Þ
It is noted that in this way the matrix variables P ið2Þ are set as Markovian and can be absorbed directly by the filter gain variables Afi and Bfi by introducing A fi ¼ Pið2Þ Afi ,
B fi ¼ P ið2Þ Bfi ,
i 2 I:
ð52Þ
On the other hand, Pi 40 implies that Pið2Þ is nonsingular. Then, the filter gains can be constructed by (48). This completes the proof. & Remark 4. Theorem 2 provides a new delay-dependent condition on H1 filtering synthesis problem for continuous-time MJLSs in (1) with time-varying delay and partially accessible mode information. Following the similar arguments as in [14], the condition in Theorem 2 can be readily extended to the cases in which the systems contain parametric uncertainties. It is also worth mentioning that the condition given in Theorem 2 depends on the derivative of time-varying _ r m o1. However, it can be readily generalized for the case of delay-derivative-independent by setting delay with dðtÞ Q 1i ¼ 0, i 2 I in Theorem 2 for the underlying systems. 4. Simulation studies In this section, two simulation examples are provided to demonstrate the effectiveness and less conservatism of the proposed approach. Example 1. Consider a continuous-time Markovian jump linear time-delay system in the form of (1), borrowed from [25] with some modifications. The system is with two modes and the following parameters: 2 3 3 1 0 0:2 0:1 0:6 1 2 3 6 A1 Ad1 B1 2 0:5 1 0:8 0 7 6 0:3 2:5 7 6 7 6C 7 6 0 1 2:5 1 7 4 1 C d1 D11 5 ¼ 6 0:1 0:3 3:8 7, 6 7 L1 Ld1 D21 0:3 0 0:2 0:3 0:6 0:2 5 4 0:8 0:5 0:1 1 0 0:1 0:2 0:1 2 3 2:5 0:5 0:1 0 0:3 0:6 0:6 2 3 6 A2 Ad2 B2 0:1 0:5 0 0:5 7 6 0:1 3:5 0:3 7 7 6 6C 7 7: 0:1 1 2 0:6 1 0:8 0 4 2 C d2 D12 5 ¼ 6 6 7 6 7 L2 Ld2 D22 0:3 0 0:6 0:2 0:5 5 4 0:5 0:2 0 1 1:6 0:2 0:1 0 0:1
The objective is to design a filter of the form (2) such that the resulting filtering error system (3) is stochastically stable with a guaranteed H1 performance. To this end, suppose transition rates (TRs) l11 ¼ 1:2, l22 ¼ 0:3, and choose the lower and upper delay bounds, respectively, as d1 ¼ 0 and d2 ¼0.7 with m ¼ 0:2. It is noted that the result given in [12] is not applicable to the filtering design for the above system with time-varying delay and it has also been found there is no feasible solution based on the method proposed in [25]. Nevertheless, by applying Theorem 2 proposed in this paper, we indeed obtain the feasible solutions of gmin ¼ 0:2709 for the full-order filter, gmin ¼ 0:3252 for the 2-order filter, and
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gmin ¼ 0:3946 for the 1-order filter, respectively. A more detailed comparison of the obtained performances based on Theorem 2 for different cases is summarized in Table 1, where m^ denotes that m is unknown, which indicates the delayderivative-independent filtering design results, as mentioned in Remark 4. The above example has clearly demonstrated that the results proposed in this paper are more general and less conservative than the existing ones proposed in [12,25]. To illustrate the effectiveness of the results proposed in this paper more comprehensively, in the following we present another example. Example 2. Consider a continuous-time Markovian jump linear time-delay system in (1) with four modes, and the system parameters are given as follows: 2 3 3:5 0:8 0:9 1:3 1 2 3 A1 Ad1 B1 6 7 6C 7 6 0:6 3:3 0:7 2:1 0 7 4 1 C d1 D11 5 ¼ 6 7, 4 0:8 0:3 0:2 0:3 0:2 5 L1 Ld1 D21 0:5 0:1 0 0:1 0:1 2 3 2 1 0 1 0:7 2 3 A2 Ad2 B2 6 2 1 0 0:1 7 6C 7 6 0 7 4 2 C d2 D12 5 ¼ 6 7, 4 0:9 2:1 1:5 0 0:6 5 L2 Ld2 D22 0 1 0:3 0:5 0:1 2 3 3 1 0:9 0 0:9 2 3 A3 Ad3 B3 6 7 6C 7 6 0:3 2:5 1 1:1 2:3 7 4 3 C d3 D13 5 ¼ 6 7, 4 0:6 0:7 1:2 0:7 0:3 5 L3 Ld3 D23 0:3 0:1 0:3 0:1 0:3 2 3 2 0 1 0 0:8 2 3 A4 Ad4 B4 6 7 6C 7 6 0 0:9 1 1 0:2 7 4 4 C d4 D14 5 ¼ 6 7: 4 0:8 2:2 1:2 0:1 0:5 5 L4 Ld4 D24 1 0 0:2 0:1 0:2 Three different cases for the transition rate matrix (TRM) are given in Table 2, where the elements labeled with the hat ‘‘ ^ ’’ ð2Þ represent the unknown TRs. In Case 2, we restrict the unknown diagonal element l^ 22 with a lower bound ld ¼ 1:5, and ð1Þ ð2Þ ð3Þ ð4Þ also assign ld ¼ 1:3, ld ¼ 1:5, ld ¼ 2:5, and ld ¼ 1:2 a priori for Case 3, respectively. By applying Theorem 2 with the lower and upper delay bounds d1 ¼0.1, and d2 ¼0.65, respectively, a detailed comparison of the obtained minimum H1 performance indices gmin for both full-order and reduced-order filters with different delay-derivatives and three TRM cases is shown in Table 3, where m^ denotes that m is unknown, which indicates the delay-derivative-independent filtering design results, as mentioned in Remark 4. The results given in Table 3 clearly show that the filtering design method proposed in this paper is effective. _ r0:9. By applying Theorem 2, Specifically, in the following, we consider the time-varying delay 0:1 rdðtÞ r 0:5, with dðtÞ the feasible solutions of gmin ¼ 0:5870 for the full-order filter and gmin ¼ 0:8509 for the reduced-order filter are obtained, Table 1 Minimum H1 performances for different cases based on Theorem 2 with d1 ¼ 0. Filter-order
m ¼ 0:2
m ¼ 0:5
m ¼ 0:9
m^
d2 ¼ 0:7 Full-order 2-order 1-order
0.2709 0.3252 0.3946
0.3390 0.3927 0.4766
0.4355 0.5155 0.6119
0.4373 0.5193 0.6356
d2 ¼ 0:9 Full-order 2-order 1-order
0.3229 0.3684 0.4590
0.4379 0.4979 0.5894
0.8915 1.2484 1.7210
1.3834 2.9093 5.0887
Table 2 Three different TRMs. Case 1: completely known 2
1:3 6 0:3 6 6 4 0:1 0:4
0:2
0:8
1:3
0:5
0:9 0:2
2:5 0:6
Case 2: partially known 3
0:3 0:5 7 7 7 1:5 5 1:2
2
1:3 6 ^ 6 l 21 6 6 4 0:1 0:4
0:2 l^
l^ 13
l^ 32
2:5
0:2
0:6
22
0:5
l^ 14
Case 3: completely unknown 3
7 0:5 7 7 7 ^ l 34 5 1:2
2
l^ 6 11 6 l^ 6 21 6^ 6 l 31 4 l^ 41
l^ 12 l^ 22 l^ 32 l^ 42
l^ 13 l^ 23 l^ 33 l^ 43
3
l^ 14 7 l^ 24 7 7 ^l 7 7 34 5 l^ 44
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Table 3 Minimum H1 performances for different TRMs with d1 ¼0.1 and d2 ¼0.65. TRMs
Filter-order
m ¼ 0:2
m ¼ 0:5
m ¼ 0:9
m^
Case 1
Full-order Reduced-order Full-order Reduced-order Full-order Reduced-order
0.5458 0.7354 0.7121 0.9106 0.7507 0.9258
0.6186 0.8557 0.7724 1.0939 0.8356 1.1395
0.6606 0.9362 0.8282 1.1861 0.9136 1.2231
0.6611 0.9368 0.8292 1.1865 0.9161 1.2236
Case 2 Case 3
0.5
system mode
0.3 0.2
4 3 2 1
0.1
0
2 Time(s)
4
0 Time−varying delays d(t)
−0.1 Time delay
Filtering error
Case 1 Case 2 Case 3
Modes evolution r(t)
0.4
−0.2 −0.3 −0.4
0.5 0.4 0.3 0.2 0.1 0
2 Time
−0.5 0
0.5
1
1.5
2
4
2.5
3
3.5
4
Time(s) Fig. 2. Responses of zðtÞ for three TRM cases (full-order).
respectively, under Case 2 shown in Table 2. The filter gains are given as follows: 2 3 " # 8:3334 7:5832 13:9418 Af 1 Bf 1 6 7 ¼ 4 4:9212 35:0598 34:5833 5, C f 1 Df 1 0:5429 0:0237 0:0340 2 3 " # 18:0870 19:5453 9:2342 Af 2 Bf 2 6 0:3476 0:6985 0:4796 7 ¼4 5, C f 2 Df 2 0:1918 1:1420 0:0311 2 3 " # 3:5271 3:4687 3:9715 Af 3 Bf 3 6 0:2084 15:4139 10:7591 7 ¼4 5, C f 3 Df 3 0:1722 0:8969 0:8439 2 3 " # 33:4322 78:8817 27:5667 Af 4 Bf 4 6 7 ¼ 4 178:9554 394:3563 138:1362 5, C f 4 Df 4 1:0083 0:1646 0:0383 for the full-order case, and " # Af 1 Bf 1 14:3477 10:2293 ¼ , C f 1 Df 1 0:8068 0:2902 " # Af 2 Bf 2 2:7626 0:7768 ¼ , C f 2 Df 2 0:8577 0:3148 " # Af 3 Bf 3 7:0410 2:6561 ¼ , C f 3 Df 3 0:9081 0:4145 " # Af 4 Bf 4 4:4234 0:2000 ¼ , C f 4 Df 4 0:9782 0:0022 for the reduced-order case. The feasible solutions for the other two TRM cases shown in Table 2 are omitted for brevity. With the above obtained filters and for three different TRM cases shown in Table 2, the time responses of the full-order and reduced-order estimation errors under one of possible mode evolutions are shown in Figs. 2 and 3, respectively, where the initial condition is selected as xð0Þ ¼ ½0:5 0:3 0 0T , time-varying delay dðtÞ ¼ 0:3 þ 0:2 sinð4:5tÞ, and the disturbance
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0.5
system mode
0.3 0.2
4 3 2 1
0.1
0
2 Time(s)
0
4
Time−varying delays d(t)
−0.1 Time delay
Filtering error
Case 1 Case 2 Case 3
Modes evolution r(t)
0.4
−0.2 −0.3 −0.4
0.5 0.4 0.3 0.2 0.1 0
2 Time
−0.5 0
0.5
1
1.5
2
4
2.5
3
3.5
4
Time(s) Fig. 3. Responses of zðtÞ for three TRM cases (reduced-order).
input wðtÞ ¼ 10e3t sinð0:015tÞ. The simulation results further indicate that, whether the TRs are known or not, the designed filters are feasible and effective. 5. Conclusions The delay-dependent H1 filtering design problem for a class of continuous-time Markovian jump linear systems (MJLSs) with time-varying delay and partially accessible mode information has been investigated in this paper. The partially known transition rates have been dealt with by a convex combination method. A new input–output model has been presented by employing a novel approximation for delayed state. On the basis of scaled small gain theorem and constructing a Markovian Lyapunov–Krasovskii functional, an improved bounded real lemma has been derived. Meanwhile, the filtering design results are established by a linearization technique for both full-order and reduced-order cases. Finally, two illustrative examples are presented to demonstrate the effectiveness and less conservatism of the proposed approach over the existing results. It is noted that the extensions of the proposed method to the controller design for continuous-time MJLSs with multiple time-varying delays and defective mode information, which simultaneously involves the exactly known, partially unknown and uncertain transition rates, deserve further investigation. Applications of the proposed theoretical results to some real-world complex systems such as the networked control systems (NCSs) [40], vertical take-off landing (VTOL) helicopter systems [41], short-term interest rate (STIR) in financial economics systems [42] etc., are also part of our future works.
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A new design of HN filtering for continuous-time Markovian jump systems with time-varying delay and partially accessible mode information$ Yanling Wei a, Jianbin Qiu a,n, Hamid Reza Karimi b, Mao Wang a a b
Research Institute of Intelligent Control and Systems, Harbin Institute of Technology, Harbin 150001, PR China Department of Engineering, Faculty of Engineering and Science, University of Agder, N-4898 Grimstad, Norway
a r t i c l e in f o
abstract
Article history: Received 20 November 2012 Received in revised form 30 January 2013 Accepted 20 February 2013 Available online 5 March 2013
In this paper, the delay-dependent H1 filtering problem for a class of continuous-time Markovian jump linear systems with time-varying delay and partially accessible mode information is investigated by an indirect approach. The generality lies in that the systems under consideration are subject to a Markov stochastic process with exactly known and partially unknown transition rates. By utilizing the model transformation idea, an input–output approach is employed to transform the time-delayed filtering error system into a feedback interconnection formulation. Invoking the results from the scaled small gain theorem, an improved version of bounded real lemma is obtained based on a Markovian Lyapunov–Krasovskii functional. The underlying full-order and reduced-order H1 filtering synthesis problems are formulated by a linearization technique. Via solving a set of linear matrix inequalities, the desired filters can therefore be constructed. The results developed in this paper are less conservative than existing ones in the literature, which are illustrated by two simulation examples. & 2013 Elsevier B.V. All rights reserved.
Keywords: Markovian jump system H1 filtering Partially accessible mode information Scaled small gain theorem Time-varying delay
1. Introduction State estimation has long been a significant and active research issue due to its theoretical and practical significance in engineering applications and signal processing. Many results on estimation and filtering design for different kinds of dynamic systems have been obtained [1–10]. Specifically, H1 filtering, whose main advantages are insensitive to the exact knowledge of the external noise signals and more robust to parameter uncertainties in systems [1,11], has attracted considerable attention. Therefore, recently, a number of studies on H1 filtering problem for discrete-time or continuoustime systems with delays and uncertainties have been reported in the open literature, e.g., [1,10–15]. On another research front line, as an important class of stochastic hybrid systems, Markovian jump linear systems (MJLSs) have been extensively studied due to their powerful modeling ability of Markov process in many fields, such as robot manipulator systems, aircraft control systems, medical and physical systems and so on [13,15,16]. Differing from the nondeterministic switching in switched systems, the modes evolution in MJLSs is determined by a Markov chain [17,18].
$ This work was supported in part by the National Natural Science Foundation of China (61004038), in part by the National 973 Project of China (2009CB320600), in part by the Program for New Century Excellent Talents in University, in part by the Fundamental Research Funds for the Central Universities (HIT.BRETIII.201214), in part by the Alexander von Humboldt Foundation of Germany. n Corresponding author. Tel.: þ 86 451 86402350. E-mail addresses: [email protected] (Y. Wei), [email protected], [email protected] (J. Qiu), [email protected] (H. Reza Karimi), [email protected] (M. Wang).
0165-1684/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2013.02.014
Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
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In traditional analysis and design, it is usually assumed that the mode information in the Markov chain is definitely known. In fact, it is questionable and generally expensive to obtain all the transition mode information even for a simple system [19–21]. Recently, some attention has been drawn to the more general cases with partially accessible mode information. It is noted that for the discrete-time case, the transition probabilities are all non-negative, and the sum of transition probabilities is 1 for each row in a transition probability matrix (TPM) [22], which facilitates the mathematical derivations in the discrete-time context. On the other hand, for the continuous-time case with partially accessible mode information, the difficulty is that the unknown elements may locate in diagonal of the transition rate matrix (TRM), which are nonpositive. Therefore, it is more difficult and challenging to investigate the corresponding filtering analysis and design problems for continuous-time MJLSs with partially accessible mode information. On the other hand, time-delays are frequently encountered in MJLSs. Recently, there have appeared some results on robust control and filtering design for MJLSs with time-delays and it has been shown that the delay-dependent conditions reflect the reality better in general [12,23–26]. In addition, it has also been shown that model transformation, which may incur additional dynamics to degrade the performance of original system, is a main source of design conservatism [27–30]. Recently, a new model transformation based on the input–output approach was proposed for time-delay systems [31–34]. Unlike the previous transformations, a new input–output model is introduced for the original system by employing a twoterm approximation method. Then, the stability analysis and synthesis of systems with time-varying delay are transformed into a scaled small gain (SSG) problem [31,35–39]. However, to the authors’ best knowledge, there exist few related studies on delay-dependent H1 filtering for continuous-time MJLSs with time-varying delay and partially accessible mode information based on the SSG techniques. Especially, it is expected that the conservatism of filtering design conditions shall be further reduced by utilizing the input–output approach, which inspires us for this study. According to the issues mentioned above, we will propose a new delay-dependent H1 filtering design method for continuous-time MJLSs in the presence of time-varying delay and partially accessible mode information based on SSG theorem. By introducing the lower bounds of the unknown diagonal elements, together with the convexification of unknown transition rates, the difficulty that the unknown elements appear in diagonal of TRM is overcome without introducing additional conservatism. A two-term approximation approach is used to transform the filtering error system into a feedback interconnection form, which contains a forward subsystem with two constant delays and a feedback one with norm-bounded uncertainties. Then, based on a Markovian Lyapunov–Krasovskii functional combined with SSG theorem, new delay-dependent H1 performance analysis conditions are derived. By applying a linearization procedure, both the full-order and reduced-order filtering design results can be constructed in terms of linear matrix inequalities. Two examples will be given to illustrate the advantages of the proposed results over the existing ones. Notations: The notations used throughout the paper are standard. T 1 JT 2 represents the series connection of mapping T1 and T2. Rn and Rmn denote, respectively, the n-dimensional Euclidean space, and the set of all m n real matrices; P 4 0 means that P is real symmetric and positive definite; SymfAg is the shorthand notation for A þ AT ; I and 0 represent the identity matrix and a zero matrix, respectively; ðO,F ,PÞ denotes a complete probability space, in which O is the sample space, F is the s algebra of subsets of the sample space, and P is the probability measure on F ; E½ stands for the mathematical expectation; kk denotes the Euclidean norm of a vector or its induced norm of a matrix; signals that are square integrable over ½0,1Þ is denoted by L2 ½0,1Þ with the norm J J2 . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. 2. Problem formulation and preliminaries A continuous-time Markovian jump linear system (MJLS) with an interval time-varying delay in the state, defined in a complete probability space ðO,F ,PÞ, is governed by state-space equation of the form: _ ¼ AðrðtÞÞxðtÞ þ Ad ðrðtÞÞxðtdðtÞÞ þBðrðtÞÞwðtÞ ðSÞ : xðtÞ yðtÞ ¼ CðrðtÞÞxðtÞ þ C d ðrðtÞÞxðtdðtÞÞ þ D1 ðrðtÞÞwðtÞ zðtÞ ¼ LðrðtÞÞxðtÞ þ Ld ðrðtÞÞxðtdðtÞÞ þ D2 ðrðtÞÞwðtÞ xðtÞ ¼ ft ,
t 2 ½d2 ,0,
ð1Þ
where xðtÞ 2 Rn is the state vector; wðtÞ 2 Rm denotes the disturbance input vector, which belongs to L2 ½0,1Þ; yðtÞ 2 Rp is the measured output; zðtÞ 2 Rq is the signal to be estimated, and d(t) is a time-varying delay satisfying _ r m o1, where d and d are known constant scalars representing the lower and upper 0 r d1 r dðtÞ rd2 o 1 and dðtÞ 1 2 delay bounds, respectively. In (1), ft is a vector-valued initial continuous function defined on the interval ½d2 ,0; the process frðtÞ,t Z 0g is a continuous-time homogeneous Markov chain with right continuous trajectories and taking values in a finite set I :¼ f1, . . . ,Ng with transition rate matrix (TRM) L :¼ ½lij NN given by ( lij hþ oðhÞ, iaj Prfrðt þhÞ ¼ j9rðtÞ ¼ ig ¼ 1 þ lii h þoðhÞ, i ¼ j where h 40, limh-0 ðoðhÞ=hÞ ¼ 0, and lij Z0, for jai, is the transition rate (TR) from mode i at time t to mode j at time t þ h, P and lii ¼ N j ¼ 1,jai lij . In the sequel, for each possible rðtÞ ¼ i, i 2 I , the system matrices of the ith mode are denoted by ðAi ,Adi ,Bi ,C i ,C di ,D1i ,Li ,Ldi ,D2i Þ, which are real and known matrices with appropriate dimensions.
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The TRs of the stochastic process in this paper are considered to be partially available. For instance, in the case of a system (S) with four operation modes, the TRM may be as 2 3 l l^ 12 l^ 13 l14 6 11 7 6 l^ 7 ^ 6 21 l 22 l23 l24 7 6 7 6 l31 l^ 32 l33 l^ 34 7 4 5 l l^ l^ l^ 41
42
43
44
[ where the elements labeled with the hat ‘‘^’’ represent the unknown TRs. For notational clarity, 8i 2 I , we describe I ¼ I ðiÞ k ðiÞ ðiÞ ^ is unknowng. Also, throughout this paper, we denote as follows: I :¼ fj : l is knowng, and I :¼ fj : l I ðiÞ ij ij uk k uk P ðiÞ ^ lðiÞ l . For tractability reasons, we further restrict the unknown diagonal element l^ ii as lðiÞ k :¼ d r l ii , where ld j2I ðiÞ ij k
provides a lower bound for the unknown element l^ ii [21]. The objective of this paper is to design a filter of the following general structure to estimate z(t), ^ Þ : x^_ ðtÞ ¼ Afi xðtÞ ^ þ Bfi yðtÞ ðS ^ þ Dfi yðtÞ, z^ ðtÞ ¼ C fi xðtÞ
ð2Þ
n^
q
^ n^ n
^ np
qn^
qp
^ 2 R ðn^ r nÞ is the filter state; z^ ðtÞ 2 R is the estimation of z(t); Afi 2 R , Bfi 2 R , C fi 2 R and Dfi 2 R , where xðtÞ 8i 2 I , are filter gains to be determined. It is noted that n^ ¼ n for the full-order filtering and n^ o n for the reduced-order filtering. Augmenting the model in (1) to include the states of the filter in (2), we obtain the following filtering error system: ðS Þ : x_ ðtÞ ¼ A i xðtÞ þ A di ExðtdðtÞÞ þB i wðtÞ zðtÞ ¼ L i xðtÞ þ L di ExðtdðtÞÞ þ D i wðtÞ T
xðtÞ ¼ ½ft 0T , T
t 2 ½d2 ,0, T
ð3Þ
T
where xðtÞ :¼ ½x ðtÞ x^ ðtÞ , zðtÞ :¼ zðtÞz^ ðtÞ and " # " # " # Ai Adi Bi 0 A i :¼ , A di :¼ , B i :¼ , Bfi C di Bfi D1i Bfi C i Afi L i :¼ ½Li Dfi C i C fi , D i :¼ D2i Dfi D1i ,
L di :¼ Ldi Dfi C di ,
E :¼ ½In 0nn^ :
ð4Þ
More precisely, we introduce the following definitions for the filtering error system (S ) in (3), which are essential for the later development. Definition 1 (Zhang et al. [25]). The filtering error system (S ) in (3) is said to be stochastically stable if under wðtÞ ¼ 0 and ^ any initial condition xð0Þ 2 Rðn þ nÞ , rð0Þ 2 I , Z t Jxða,xð0Þ,rð0ÞÞJ2 da o1: lim E t-1
0
Definition 2 (Zhang et al. [25]). Given a scalar g 4 0, the filtering error system (S ) in (3) is said to be stochastically stable with an H1 disturbance attenuation performance index g if it is stochastically stable with wðtÞ ¼ 0, and under zero initial condition, Z 1 Z 1 E z T ðtÞzðtÞ dt o g2 wT ðtÞwðtÞ dt, 80awðtÞ 2 L2 ½0,1Þ: 0
0
For the filtering design based on (2), it is required that the original system (1) should be stochastically stable. The purpose of this paper is to design a delay-dependent H1 filter such that the filtering error system (S ) in (3) is stochastically stable with a prescribed H1 performance index g by an indirect approach. The key procedure with this approach is to introduce a new model transformation by employing a two-term approximation for MJLSs with timevarying delay, which converts system (S ) into an interconnection structure. A particular direction of this transformation is that the H1 filtering design problem is reformulated into the scaled small gain (SSG) problem of the new input–output model. More details on this approach can be found in [28,31,36]. Here, we just retrospect some preparative notions on the SSG theorem. Consider an interconnected system consisting of two subsystems, L1 : xðtÞ ¼ GZðtÞ,
L2 : ZðtÞ ¼ DxðtÞ,
ð5Þ
which is shown in Fig. 1, where the forward subsystem L1 is known and time-invariant with operator G mapping Z to x, and the feedback one L2 is unknown and time-varying with operator D 2 D :¼ fD : JDJ1 r 1g. As a direct ramification of
Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
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Fig. 1. Interconnected system consisting of two subsystems.
the SSG theorem [31,33,35,36], a sufficient condition for the robust stability of the interconnected system formed by L1 and L2 in (5) is provided as follows. Lemma 1. Consider (5) and assume that L1 is internally stable. The closed-loop system formed by L1 and L2 is robustly stable Z for all D 2 D if JT x JGJT 1 Z Rxx : T Z ,T x Z J1 o1 holds for some matrices fT x ,T Z g 2 T with T :¼ ffT Z ,T x g 2 R 1 nonsingular; JT Z JDJT x J1 r1g. 3. Main results In this section, we concentrate on transforming system (S ) in (3) into an interconnection of two subsystems as in (5) and analyzing the scaled small gain (SSG) of the forward subsystem. A new delay-dependent H1 performance criterion will firstly be proposed for system (S ) with complete known mode information, and then a bounded real lemma (BRL) and corresponding filtering synthesis for the underlying systems with partially accessible mode information will be developed. 3.1. Model transformation Considering the filtering error system (S ) in (3), the term xðtdðtÞÞ is approximated by 12 ½xðtd1 Þ þxðtd2 Þ, which brings to the approximation error, 1 2
Z d ðtÞ ¼ xðtdðtÞÞ ½xðtd1 Þ þxðtd2 Þ ¼ :¼
1 2 1 2
Z Z
dðtÞ
_ þ aÞ da xðt
d2
1 2
Z
d1
_ þ aÞ da xðt
dðtÞ
d1
rðaÞxd ðt þ aÞ da,
ð6Þ
d2
_ and where xd ðtÞ :¼ xðtÞ, ( 1 when a r dðtÞ, rðaÞ :¼ 1 when a 4 dðtÞ: Normalizing Z d ðtÞ in (6) by multiplying 2=d with d ¼ d2 d1 , (6) can be rewritten as, Z 1 d1 Zd ðtÞ :¼ Dd ðtÞxd ðtÞ ¼ rðaÞxd ðt þ aÞ da, d d2
ð7Þ
where Zd ðtÞ :¼ ð2=dÞðZ d ÞðtÞ, and operator Dd : xd /Zd , which includes the uncertainties pulled out from xðtdðtÞÞ in the original time-delay system (S ). Substituting (7) into system (S ) in (3), the filtering error system (S ) is converted into a feedback system with the forward subsystem ðLd1 Þ and the feedback one ðLd2 Þ, described by 8 1 d > > > x_ ðtÞ ¼ A i xðtÞ þ A di E½xðtd1 Þ þ xðtd2 Þ þ A di Zd ðtÞ þ B i wðtÞ > > 2 2 > > < 1 d ðLd1 Þ : zðtÞ ¼ L i xðtÞ þ L di E½xðtd1 Þ þxðtd2 Þ þ L di Zd ðtÞ þ D i wðtÞ > 2 2 > > > > 1 d > > : xd ðtÞ ¼ Ai xðtÞ þ Adi ½xðtd1 Þ þ xðtd2 Þ þ Adi Zd ðtÞ þBi wðtÞ 2 2 ðLd2 Þ : Zd ðtÞ ¼ Dd ðtÞxd ðtÞ,
T
xðtÞ ¼ ½ft 0T ,
t 2 ½d2 ,0:
ð8Þ
In light of the transformation (6), the original system (S ) is transformed into a feedback interconnection form, which contains a forward subsystem (Ld1 ) with two known constant delays and a feedback one (Ld2 ) with delay uncertainties. The stability problem of system (S ) can therefore be interpreted as a robust performance analysis problem for the nominal system (Ld1 ) in the face of norm-bounded uncertainties Dd ðtÞ. Of course, the stability of the transformed system in (8)
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Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
implies the stability of the original time-delay system in (3). Specifically, the following result can be concluded which, in the meantime, provides a possible choice of the scaling matrices fT Z ,T x g 2 T . Lemma 2. For any given invertible matrix X, the operator Dd in (8) enjoys the property JX JDd JX 1 J1 r 1. Proof. By virtue of Jensen inequality [28], considering zero initial condition and exchanging the order of integration, it follows that Z t 1 ZTd ðaÞX T X Zd ðaÞ da ¼ 2 E 0 d 8 " #T j 2 I ðiÞ uk > > = if i 2 I ðiÞ , k > > > ;
ðiÞ
QðiÞ þ QðiÞ R2 lk ðQ 1j þ Q 3j Þ r0, k1 k3 ðiÞ
QðiÞ R1 lk Q 2j r 0, k2
j 2 I ðiÞ uk
ðiÞ QðiÞ R2 lk Q 3j r 0, k3
I ðiÞ uk
j2
ðiÞ
ðiÞ
ð31Þ
9 > j 2 I ðiÞ uk > > =
ðiÞ
þ QðiÞ R2 þ ld ðQ 1i þQ 3i Þðld þ lk ÞðQ 1j þ Q 3j Þ r 0, QðiÞ k1 k3 ðiÞ
ðiÞ
ðiÞ
j 2 I ðiÞ uk
ðiÞ
ðiÞ
ðiÞ
j 2 I ðiÞ uk
QðiÞ R1 þ ld Q 2i ðld þ lk ÞQ 2j r 0, k2 QðiÞ R2 þ ld Q 3i ðld þ lk ÞQ 3j r 0, k3
> > > ;
if i 2 I ðiÞ , uk
ð32Þ
where 2
3
T 1 2 P i A di þ E Z 1
1 2 P i A di
d 2 P i A di
U2
14 ð1mÞQ 1i þ 12 Z 2
4d ð1mÞQ 1i
6 n 4
n
14 ð1mÞQ 1i Q 3i 12 Z 2
n
n
U1
6 6 n
U :¼ 6 6
:¼ QðiÞ k1
X
n
lij Q 1j , QðiÞ :¼ k2
j2I ðiÞ k
X
4d ð1mÞQ 1i 2 d4
lij Q 2j , QðiÞ :¼ k3
j2I ðiÞ k
X
2
ð1mÞQ 1i d2 Z 2
7 7 7 7, 7 5
lij Q 3j ,
ð33Þ
j2I ðiÞ k
with
U 1 :¼
8 < U1 þ P ðiÞ lðiÞ P j ,
j 2 I ðiÞ , k k uk ðiÞ ðiÞ : U1 þ P þ l Pi ðlðiÞ þ lðiÞ ÞP j , d d k k
:¼ P ðiÞ k
X j2I ðiÞ k
if i 2 I ðiÞ , k j2
I ðiÞ , uk
if i 2 I ðiÞ , uk
lij P j , ðiÞ
and S, F12 , F13 , F23 , F33 , U1 , and U2 are defined in Proposition1; ld is a given lower bound for the unknown diagonal element l^ ii . Proof. Based on Proposition 1, it is shown that the filtering error system in (8) subject to completely known TRs is stochastically stable with a prescribed H1 performance g if (9)–(12) hold. Since the diagonal elements in the TRM may contain unknown ones, we shall separate the proof of Theorem 1 into two cases, i 2 I ðiÞ and i 2 I ðiÞ . k uk ðiÞ (i) i 2 I k . ðiÞ ðiÞ In this case, i 2 I ðiÞ implies that lii is known, then it is straightforward that lk r0. Here, we only need to consider lk o0, k ðiÞ since if lk ¼ 0, condition (9) can be readily viewed as a direct result. P Notice that the term N j ¼ 1 lij P j in (9) can be treated as N X
lij Pj ¼ P ðiÞ þ k
j¼1
ðiÞ l^ ij P j ¼ P ðiÞ lk k
j2I ðiÞ uk
:¼ where P ðiÞ k becomes N X
X
P
lij Pj , j2I ðiÞ k
lij Pj ¼
j¼1
X l^ ij ðiÞ
j2I ðiÞ uk
lk
X l^ ij ðiÞ
j2I ðiÞ uk
lk
Pj ,
ð34Þ
P ðiÞ ðiÞ and the elements l^ ij , j 2 I ðiÞ , are unknown. Since 0 r l^ ij =lk r1 and ðl^ ij =lk Þ ¼ 1, (34) uk j2I ðiÞ uk
ðiÞ
½P ðiÞ lk Pj : k
ð35Þ
Thus, for 0 r l^ ij r lk , the left-hand side of inequality (9) can be rewritten as 2 3 U S F12 F13 X l^ ij 6 X l^ ij 7 g2 I F23 5 :¼ F¼ F, 4 n ðiÞ ðiÞ lk lk j2I ðiÞ j2I ðiÞ ðiÞ
uk
n
n
F33
ð36Þ
uk
ðiÞ U 1 ¼ U1 þP ðiÞ lk P j , k
and U1 , S, F12 , F13 , F23 , and F33 are defined in (13). Then, (9) holds if where U is defined in (33) with and only if F o0 in (36), which implies that, in the presence of unknown elements l^ ij , i 2 I ðiÞ , j 2 I ðiÞ , inequality (30) is k uk equivalent to (9). (ii) i 2 I ðiÞ . uk ðiÞ ðiÞ Following a similar argument as in (i), we only consider l^ ii olk here, since if l^ ii ¼ lk , then the ith row of the TRM is completely known.
Y. Wei et al. / Signal Processing 93 (2013) 2392–2407
Equivalently, for this case, the term N X
X
lij Pj ¼ P ðiÞ þ l^ ii Pi þ k
j¼1
PN
j¼1
2401
lij Pj in (9) can be expressed as
l^ ij P j ¼ P ðiÞ k
j2I ðiÞ ,jai uk
X
þ l^ ii P i þ ðl^ ii lk Þ ðiÞ
l^ ij
ðiÞ l^ ii lk j2I ðiÞ ,jai
Pj ,
ð37Þ
uk
P ðiÞ k
P
:¼ j2I ðiÞ lij P j . k P ðiÞ ðiÞ Likewise, since 0 r l^ ij =ðl^ ii lk Þ r1 and j2I ðiÞ ,jai l^ ij =ðl^ ii lk Þ ¼ 1, we have
where
uk
N X
X
l^ ij
j2I ðiÞ ,jai uk
l^ ii lk
lij Pj ¼
j¼1
½P ðiÞ þ l^ ii P i ðl^ ii þ lk ÞP j : k ðiÞ ðiÞ
ð38Þ
Correspondingly, we can rewrite the left-hand side of the inequality in (9) as 2 3 U S F12 F13 ^l X ij 6 7 g2 I F23 5 F¼ 4 n ^ lðiÞ l ðiÞ ii k j2I ,jai n
uk
:¼
X
l^
j2I ðiÞ ,jai uk
l^ ii lk
ij ðiÞ
n
F33
F,
ð39Þ
ðiÞ þ l^ ii Pi ðl^ ii þ lk ÞPj , and U1 , S, F12 , F13 , F23 , and F33 are defined in (13). where U is defined in (33) with U 1 ¼ U1 þP ðiÞ k Deduced from (39), F o 0 is equivalent to 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ F :¼ 4 n , jai: ð40Þ uk
n
n
F33
ðiÞ For tractability, by introducing a lower bound ld for the unknown element l^ ii , we have ðiÞ ^ lðiÞ d r l ii olk ,
ð41Þ
ðiÞ ðiÞ which implies that l^ ii may take any value in ½ld ,lk þ E for some sufficiently small E o0. Then l^ ii can be directly written as a convex combination ðiÞ l^ ii ¼ klðiÞ k þ kE þ ð1kÞld ,
ð42Þ
where 0 r k r 1. Since l^ ii in (42) depends on k linearly, and (40) therefore needs only to be satisfied for k ¼ 0 and k ¼ 1, that is, (40) holds if and only if the following inequalities in (43)–(44) simultaneously hold: 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ , jai, ð43Þ 4 n uk n
n
F33 ðiÞ
where U is defined in (33) with U 1 ¼ U1 þ P ðiÞ lk P i þ EðPi P j Þ, and k 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ , jai, 4 n uk n
n
ðiÞ
ðiÞ
where U is defined in (33) with U 1 ¼ U1 þ P ðiÞ þ ld ðP i P j Þlk P j . Since E is small enough, (43) holds if and only if k 2 3 U S F12 F13 6 7 g2 I F23 5 o 0, j 2 I ðiÞ , jai, 4 n uk n
n
ð44Þ
F33
ð45Þ
F33 ðiÞ
where U is defined in (33) with U 1 ¼ U1 þ P ðiÞ lk Pi , which is implied by (44) when j ¼i, j 2 I ðiÞ . Hence, the inequality (9) k uk can be replaced by (30) in the context 8j 2 I ðiÞ . uk In addition, following a similar procedure presented above, the conditions in (31)–(32) can also be obtained based on (10)–(12), respectively. In summary, with the presence of unknown elements in the TRM, one can readily conclude that the filtering error system in (8) is stochastically stable with a prescribed H1 performance index g if (30)–(32) hold. & Remark 3. In order to render the unknown diagonal elements numerically tractable, in Theorem 1, the lower bounds for the unknown diagonal elements are introduced. Together with the property that the sum of each row is zero in a TRM,
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Y. Wei et al. / Signal Processing 93 (2013) 2392–2407 ðiÞ
ðiÞ
a convex combination in (42) can thus be obtained. Thanks to the convex combination of ld and lk , sufficient conditions for the H1 performance analysis have been derived in Theorem 1. Yet, the obtained conditions are no loss of generality, ðiÞ since the lower bound ld , of l^ ii is allowed to be arbitrarily small. In the following, we will give the filtering design result in the presence of partially accessible mode information, which relies heavily on the delay-dependent BRL presented in Theorem 1.
3.3. Delay-dependent H1 filtering design In this subsection, we consider both the full-order and reduced-order H1 filtering designs. It is shown that the parametrized representation of the filter gains can be obtained in terms of the feasible solutions to a set of linear matrix inequalities (LMIs). Theorem 2. The filtering error system in (8) with partially accessible mode information is stochastically stable with a ^ ^ HP ið2Þ ðn þ nÞðn þ nÞ , guaranteed H1 performance g, if there exist positive-definite symmetric matrices P i ¼ ½Pið1Þ n Pið2Þ 2 R ^ ^ ^ ^ fS,Q 1i ,Q 2i ,Q 3i ,R1 ,R2 ,Z 1 ,Z 2 g 2 Rnn , and matrices A fi 2 Rnn , B fi 2 Rnp , C fi 2 Rqn , and Dfi 2 Rqp , for each mode i 2 I , such that the conditions (31)–(32) and the following LMIs hold: 2 3 ^ 12 F ^ 13 ^ 11 F F 6 ^ 23 7 ^ :¼ 6 n 7 o 0, j 2 I ðiÞ , g2 I F F ð46Þ 4 5 uk ^ n n F 33 where 2 ^ 11 F
C1
6 n 6 6 6 :¼ 6 n 6 6 n 4 n
n
C3 þ Z 1 C5 C6
n
n
n
n
C2 C4
14 ð1 ÞQ 1i þ 12 Z 2 14 ð1 ÞQ 1i Q 3i 12 Z 2
4d ð1 4d ð1
m
mÞQ 1i mÞQ 1i
m
2
2
d4 ð1mÞQ 1i d2 Z 2 S
n
T
3
dC3 dC5
C3 C5
7 7 7 7 7, 7 7 5
T
^ 12 :¼ ½BT P ið1Þ þ DT B HT BT HP ið2Þ þ DT B 0m3n T , F i 1i fi i 1i fi T H :¼ ½In^ 0nðn ^ ^ , nÞ T
T
T
T
^ 13 :¼ ½d1 A ET Z 1 d2 A ET Z 2 A ET S L , F i i i i ^ 23 :¼ ½d1 BT Z 1 d2 BT Z 2 BT S D2i D D1i , F fi i i i ^ 33 :¼ diagfZ 1 ,Z 2 ,S,In g, F A i :¼ ½Ai 0nn
1 2
Adi
1 2
Adi
d 2Adi ,
1 2
Ldi 12 Dfi C di
Ldi 12 Dfi C di 2d Ldi 2dDfi C di , 9 ðiÞ > C1 :¼ G1 þP ð1iÞ lk P jð1Þ , j 2 I ðiÞ > k uk > = ðiÞ ð2iÞ ðiÞ C2 :¼ G2 þHP k lk HP jð2Þ , j 2 I uk if i 2 I ðiÞ , k > > > ðiÞ ð2iÞ ðiÞ ; C4 :¼ SymfA fi g þ P k lk Pjð2Þ , j 2 I uk
L i :¼ ½Li Dfi C i C fi
1 2
ðiÞ ðiÞ ðiÞ C1 :¼ G1 þP ð1iÞ þ ld Pið1Þ ðld þ lk ÞP jð1Þ , j 2 I ðiÞ k uk ðiÞ ðiÞ ðiÞ C2 :¼ G2 þHP ð2iÞ þ ld HPið2Þ ðld þ lk ÞHP jð2Þ , j 2 I ðiÞ k uk ðiÞ ðiÞ ðiÞ C4 :¼ Sym fA fi g þ P ð2iÞ þ ld P ið2Þ ðld þ lk ÞPjð2Þ , j 2 I ðiÞ k uk
9 > > > = > > > ;
if i 2 I ðiÞ , uk
C3 :¼ 12 Pið1Þ Adi þ 12HB fi C di , C5 :¼ 12 PTið2Þ HT Adi þ 12B fi C di , C6 :¼ 14 ð1mÞQ 1i Q 2i Z 1 12Z 2 , G1 :¼ SymfPið1Þ Ai þ HB fi C i g þQ 1i þ Q 2i þ Q 3i þ d1 R1 þd2 R2 Z 1 , T
G2 :¼ HA fi þ ATi HP ið2Þ þC Ti B fi , P ð1iÞ :¼ k
X j2I ðiÞ k
lij P jð1Þ , P ð2iÞ :¼ k
X j2I ðiÞ k
lij P jð2Þ :
ð47Þ
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Moreover, if the above conditions have a set of feasible solutions ðP ið1Þ ,Pið2Þ ,S,Q 1i ,Q 2i ,Q 3i ,R1 , R2 ,Z 1 ,Z 2 ,A fi ,B fi ,C fi ,Dfi Þ, then an ^ admissible n-order filter in the form of (2) can be obtained as Afi ¼ P 1 ið2Þ A fi ,
Bfi ¼ P1 ið2Þ B fi ,
C fi ¼ C fi ,
Dfi ¼ Dfi :
ð48Þ
Proof. It follows from Theorem 1 that if we can show (30)–(32), then the claimed result follows. For simplicity in filter synthesis procedure, we first partition the Lyapunov matrices Pi in Theorem 1 as " # P ið1Þ HPð2Þ Pi ¼ , ð49Þ n P ð3Þ ^
^
^
^
nn T , P ð2Þ 2 Rnn , and P ð3Þ 2 Rnn . Then, similar to [14], performing a congruent transforwhere H :¼ ½In^ 0nðn ^ ^ , P ið1Þ 2 R nÞ mation to Pi by diagfIn , Pð2Þ P 1 g yields, ð3Þ 2 3 2 3 T P ið1Þ HP ð2Þ P ið1Þ HP ð2Þ P 1 P ð3Þ ð2Þ 4 5 :¼ 4 5: ð50Þ T n Pð2Þ P1 n P ð2Þ ð3Þ P ð2Þ
Thus, without loss of generality, we can directly specify the Lyapunov matrices as 2 3 Pið1Þ HPið2Þ 5: Pi ¼ 4 n Pið2Þ
ð51Þ
It is noted that in this way the matrix variables P ið2Þ are set as Markovian and can be absorbed directly by the filter gain variables Afi and Bfi by introducing A fi ¼ Pið2Þ Afi ,
B fi ¼ P ið2Þ Bfi ,
i 2 I:
ð52Þ
On the other hand, Pi 40 implies that Pið2Þ is nonsingular. Then, the filter gains can be constructed by (48). This completes the proof. & Remark 4. Theorem 2 provides a new delay-dependent condition on H1 filtering synthesis problem for continuous-time MJLSs in (1) with time-varying delay and partially accessible mode information. Following the similar arguments as in [14], the condition in Theorem 2 can be readily extended to the cases in which the systems contain parametric uncertainties. It is also worth mentioning that the condition given in Theorem 2 depends on the derivative of time-varying _ r m o1. However, it can be readily generalized for the case of delay-derivative-independent by setting delay with dðtÞ Q 1i ¼ 0, i 2 I in Theorem 2 for the underlying systems. 4. Simulation studies In this section, two simulation examples are provided to demonstrate the effectiveness and less conservatism of the proposed approach. Example 1. Consider a continuous-time Markovian jump linear time-delay system in the form of (1), borrowed from [25] with some modifications. The system is with two modes and the following parameters: 2 3 3 1 0 0:2 0:1 0:6 1 2 3 6 A1 Ad1 B1 2 0:5 1 0:8 0 7 6 0:3 2:5 7 6 7 6C 7 6 0 1 2:5 1 7 4 1 C d1 D11 5 ¼ 6 0:1 0:3 3:8 7, 6 7 L1 Ld1 D21 0:3 0 0:2 0:3 0:6 0:2 5 4 0:8 0:5 0:1 1 0 0:1 0:2 0:1 2 3 2:5 0:5 0:1 0 0:3 0:6 0:6 2 3 6 A2 Ad2 B2 0:1 0:5 0 0:5 7 6 0:1 3:5 0:3 7 7 6 6C 7 7: 0:1 1 2 0:6 1 0:8 0 4 2 C d2 D12 5 ¼ 6 6 7 6 7 L2 Ld2 D22 0:3 0 0:6 0:2 0:5 5 4 0:5 0:2 0 1 1:6 0:2 0:1 0 0:1
The objective is to design a filter of the form (2) such that the resulting filtering error system (3) is stochastically stable with a guaranteed H1 performance. To this end, suppose transition rates (TRs) l11 ¼ 1:2, l22 ¼ 0:3, and choose the lower and upper delay bounds, respectively, as d1 ¼ 0 and d2 ¼0.7 with m ¼ 0:2. It is noted that the result given in [12] is not applicable to the filtering design for the above system with time-varying delay and it has also been found there is no feasible solution based on the method proposed in [25]. Nevertheless, by applying Theorem 2 proposed in this paper, we indeed obtain the feasible solutions of gmin ¼ 0:2709 for the full-order filter, gmin ¼ 0:3252 for the 2-order filter, and
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gmin ¼ 0:3946 for the 1-order filter, respectively. A more detailed comparison of the obtained performances based on Theorem 2 for different cases is summarized in Table 1, where m^ denotes that m is unknown, which indicates the delayderivative-independent filtering design results, as mentioned in Remark 4. The above example has clearly demonstrated that the results proposed in this paper are more general and less conservative than the existing ones proposed in [12,25]. To illustrate the effectiveness of the results proposed in this paper more comprehensively, in the following we present another example. Example 2. Consider a continuous-time Markovian jump linear time-delay system in (1) with four modes, and the system parameters are given as follows: 2 3 3:5 0:8 0:9 1:3 1 2 3 A1 Ad1 B1 6 7 6C 7 6 0:6 3:3 0:7 2:1 0 7 4 1 C d1 D11 5 ¼ 6 7, 4 0:8 0:3 0:2 0:3 0:2 5 L1 Ld1 D21 0:5 0:1 0 0:1 0:1 2 3 2 1 0 1 0:7 2 3 A2 Ad2 B2 6 2 1 0 0:1 7 6C 7 6 0 7 4 2 C d2 D12 5 ¼ 6 7, 4 0:9 2:1 1:5 0 0:6 5 L2 Ld2 D22 0 1 0:3 0:5 0:1 2 3 3 1 0:9 0 0:9 2 3 A3 Ad3 B3 6 7 6C 7 6 0:3 2:5 1 1:1 2:3 7 4 3 C d3 D13 5 ¼ 6 7, 4 0:6 0:7 1:2 0:7 0:3 5 L3 Ld3 D23 0:3 0:1 0:3 0:1 0:3 2 3 2 0 1 0 0:8 2 3 A4 Ad4 B4 6 7 6C 7 6 0 0:9 1 1 0:2 7 4 4 C d4 D14 5 ¼ 6 7: 4 0:8 2:2 1:2 0:1 0:5 5 L4 Ld4 D24 1 0 0:2 0:1 0:2 Three different cases for the transition rate matrix (TRM) are given in Table 2, where the elements labeled with the hat ‘‘ ^ ’’ ð2Þ represent the unknown TRs. In Case 2, we restrict the unknown diagonal element l^ 22 with a lower bound ld ¼ 1:5, and ð1Þ ð2Þ ð3Þ ð4Þ also assign ld ¼ 1:3, ld ¼ 1:5, ld ¼ 2:5, and ld ¼ 1:2 a priori for Case 3, respectively. By applying Theorem 2 with the lower and upper delay bounds d1 ¼0.1, and d2 ¼0.65, respectively, a detailed comparison of the obtained minimum H1 performance indices gmin for both full-order and reduced-order filters with different delay-derivatives and three TRM cases is shown in Table 3, where m^ denotes that m is unknown, which indicates the delay-derivative-independent filtering design results, as mentioned in Remark 4. The results given in Table 3 clearly show that the filtering design method proposed in this paper is effective. _ r0:9. By applying Theorem 2, Specifically, in the following, we consider the time-varying delay 0:1 rdðtÞ r 0:5, with dðtÞ the feasible solutions of gmin ¼ 0:5870 for the full-order filter and gmin ¼ 0:8509 for the reduced-order filter are obtained, Table 1 Minimum H1 performances for different cases based on Theorem 2 with d1 ¼ 0. Filter-order
m ¼ 0:2
m ¼ 0:5
m ¼ 0:9
m^
d2 ¼ 0:7 Full-order 2-order 1-order
0.2709 0.3252 0.3946
0.3390 0.3927 0.4766
0.4355 0.5155 0.6119
0.4373 0.5193 0.6356
d2 ¼ 0:9 Full-order 2-order 1-order
0.3229 0.3684 0.4590
0.4379 0.4979 0.5894
0.8915 1.2484 1.7210
1.3834 2.9093 5.0887
Table 2 Three different TRMs. Case 1: completely known 2
1:3 6 0:3 6 6 4 0:1 0:4
0:2
0:8
1:3
0:5
0:9 0:2
2:5 0:6
Case 2: partially known 3
0:3 0:5 7 7 7 1:5 5 1:2
2
1:3 6 ^ 6 l 21 6 6 4 0:1 0:4
0:2 l^
l^ 13
l^ 32
2:5
0:2
0:6
22
0:5
l^ 14
Case 3: completely unknown 3
7 0:5 7 7 7 ^ l 34 5 1:2
2
l^ 6 11 6 l^ 6 21 6^ 6 l 31 4 l^ 41
l^ 12 l^ 22 l^ 32 l^ 42
l^ 13 l^ 23 l^ 33 l^ 43
3
l^ 14 7 l^ 24 7 7 ^l 7 7 34 5 l^ 44
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Table 3 Minimum H1 performances for different TRMs with d1 ¼0.1 and d2 ¼0.65. TRMs
Filter-order
m ¼ 0:2
m ¼ 0:5
m ¼ 0:9
m^
Case 1
Full-order Reduced-order Full-order Reduced-order Full-order Reduced-order
0.5458 0.7354 0.7121 0.9106 0.7507 0.9258
0.6186 0.8557 0.7724 1.0939 0.8356 1.1395
0.6606 0.9362 0.8282 1.1861 0.9136 1.2231
0.6611 0.9368 0.8292 1.1865 0.9161 1.2236
Case 2 Case 3
0.5
system mode
0.3 0.2
4 3 2 1
0.1
0
2 Time(s)
4
0 Time−varying delays d(t)
−0.1 Time delay
Filtering error
Case 1 Case 2 Case 3
Modes evolution r(t)
0.4
−0.2 −0.3 −0.4
0.5 0.4 0.3 0.2 0.1 0
2 Time
−0.5 0
0.5
1
1.5
2
4
2.5
3
3.5
4
Time(s) Fig. 2. Responses of zðtÞ for three TRM cases (full-order).
respectively, under Case 2 shown in Table 2. The filter gains are given as follows: 2 3 " # 8:3334 7:5832 13:9418 Af 1 Bf 1 6 7 ¼ 4 4:9212 35:0598 34:5833 5, C f 1 Df 1 0:5429 0:0237 0:0340 2 3 " # 18:0870 19:5453 9:2342 Af 2 Bf 2 6 0:3476 0:6985 0:4796 7 ¼4 5, C f 2 Df 2 0:1918 1:1420 0:0311 2 3 " # 3:5271 3:4687 3:9715 Af 3 Bf 3 6 0:2084 15:4139 10:7591 7 ¼4 5, C f 3 Df 3 0:1722 0:8969 0:8439 2 3 " # 33:4322 78:8817 27:5667 Af 4 Bf 4 6 7 ¼ 4 178:9554 394:3563 138:1362 5, C f 4 Df 4 1:0083 0:1646 0:0383 for the full-order case, and " # Af 1 Bf 1 14:3477 10:2293 ¼ , C f 1 Df 1 0:8068 0:2902 " # Af 2 Bf 2 2:7626 0:7768 ¼ , C f 2 Df 2 0:8577 0:3148 " # Af 3 Bf 3 7:0410 2:6561 ¼ , C f 3 Df 3 0:9081 0:4145 " # Af 4 Bf 4 4:4234 0:2000 ¼ , C f 4 Df 4 0:9782 0:0022 for the reduced-order case. The feasible solutions for the other two TRM cases shown in Table 2 are omitted for brevity. With the above obtained filters and for three different TRM cases shown in Table 2, the time responses of the full-order and reduced-order estimation errors under one of possible mode evolutions are shown in Figs. 2 and 3, respectively, where the initial condition is selected as xð0Þ ¼ ½0:5 0:3 0 0T , time-varying delay dðtÞ ¼ 0:3 þ 0:2 sinð4:5tÞ, and the disturbance
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0.5
system mode
0.3 0.2
4 3 2 1
0.1
0
2 Time(s)
0
4
Time−varying delays d(t)
−0.1 Time delay
Filtering error
Case 1 Case 2 Case 3
Modes evolution r(t)
0.4
−0.2 −0.3 −0.4
0.5 0.4 0.3 0.2 0.1 0
2 Time
−0.5 0
0.5
1
1.5
2
4
2.5
3
3.5
4
Time(s) Fig. 3. Responses of zðtÞ for three TRM cases (reduced-order).
input wðtÞ ¼ 10e3t sinð0:015tÞ. The simulation results further indicate that, whether the TRs are known or not, the designed filters are feasible and effective. 5. Conclusions The delay-dependent H1 filtering design problem for a class of continuous-time Markovian jump linear systems (MJLSs) with time-varying delay and partially accessible mode information has been investigated in this paper. The partially known transition rates have been dealt with by a convex combination method. A new input–output model has been presented by employing a novel approximation for delayed state. On the basis of scaled small gain theorem and constructing a Markovian Lyapunov–Krasovskii functional, an improved bounded real lemma has been derived. Meanwhile, the filtering design results are established by a linearization technique for both full-order and reduced-order cases. Finally, two illustrative examples are presented to demonstrate the effectiveness and less conservatism of the proposed approach over the existing results. It is noted that the extensions of the proposed method to the controller design for continuous-time MJLSs with multiple time-varying delays and defective mode information, which simultaneously involves the exactly known, partially unknown and uncertain transition rates, deserve further investigation. Applications of the proposed theoretical results to some real-world complex systems such as the networked control systems (NCSs) [40], vertical take-off landing (VTOL) helicopter systems [41], short-term interest rate (STIR) in financial economics systems [42] etc., are also part of our future works.
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