Exponential L2–L∞ filtering for distributed delay systems with Markovian jumping parameters

Signal Processing 93 (2013) 206–216

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Exponential L2 2L1 filtering for distributed delay systems with Markovian jumping parameters$ Baoyong Zhang a,n, Yongmin Li b a b

School of Automation, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, PR China School of Science, Huzhou Teachers College, Huzhou 313000, Zhejiang, PR China

a r t i c l e in f o

abstract

Article history: Received 14 May 2012 Received in revised form 19 July 2012 Accepted 21 July 2012 Available online 27 July 2012

This paper is concerned with the problem of exponential L2 2L1 filter design for linear systems simultaneously with distributed delays, Markovian jumping parameters and norm-bounded parametric uncertainties. The purpose is to design full-order modedependent filters such that the filtering error system is not only mean-square robustly exponentially stable with a specified decay rate but also satisfies an L2 2L1 performance requirement. First, sufficient conditions for the stability and performance analysis of the filtering error system are derived based on a novel version of mode-dependent Lyapunov–Krasovskii functional. Then, delay-dependent and decay-rate-dependent conditions for the existence of desired filters are obtained in terms of linear matrix inequalities (LMIs). The filter coefficients can be computed by using feasible solutions of the presented LMIs. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed design method. & 2012 Elsevier B.V. All rights reserved.

Keywords: Decay-rate-dependent conditions Distributed delays Exponential stability L2 2L1 filtering Markovian jump systems

1. Introduction As an important class of hybrid systems, Markovian jump systems (MJSs) have received considerable attention during the past years. A great number of methods and results have been developed for stability analysis and control synthesis of MJSs; see, for example, [1,8,21,28,30,33,34,40] and the references therein. Time delays are always unavoidable when using MJSs to model practical systems. For example, it is known that packet loss and transmission delay are two important issues arising in networked control systems (NCSs) [20]. When the packet-loss is modeled as a

$ This work was supported in part by the National Natural Science Foundation of China under Grants 61104117 and 61174076, the China Postdoctoral Science Foundation under Grant 20100481089, and by the Fundamental Research Funds for the Central Universities under Grant NUST 2011ZDJH06. n Corresponding author. Tel.: þ 86 15365168859. E-mail addresses: [email protected] (B. Zhang), [email protected] (Y. Li).

0165-1684/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.sigpro.2012.07.027

Markovian process [44], the overall NCS is an MJS with time delays. More importantly, the existence of time delays always leads to instability of a control system [23,47]. For these reasons, it is significant to study MJSs with time delays. A notion of mean-square stochastic stability for MJSs with constant delays was introduced in [2], where a stability criterion was derived based on the eigenvalues of the system coefficient matrices. A linear matrix inequality (LMI) approach was developed in [4] to study the stochastic stability problem for MJSs with constant delays. For MJSs with time-varying delays, the exponential stability analysis problem was investigated in [29]. It is noted that the results given in [2,4,29] are independent from the size of delays. Such delay-independent results are generally more conservative than the delay-dependent ones [47]. Thus, recent research efforts have been focused on the development of delay-dependent conditions for analysis and synthesis problems of MJSs with time delays. For example, the problems of stability analysis, H1 control and H1 filtering were addressed in [3,5,10,18,22,27,31,37,38,41,46,50], respectively. The stabilization via dynamic output-feedback

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

controllers was studied in [6]. The passification problem was investigated in [48,51], where state-feedback and outputfeedback controllers were designed, respectively. It is worth mentioning that the distributed delays are not considered in the aforementioned works. Distributed delays always exist in the case when the number of summands in a system equation is increased and the differences between neighboring argument values are decreased [45]. The distributed delay systems have received much attention in the past years; see, for example, [11,12,36,45] and the references therein. The MJSs with distributed delays have been also studied. For example, the stochastic stability analysis for MJSs with both discrete and distributed delays was considered in [19,43], where delay-dependent stability criteria were given in terms of LMIs. Based on the notion of stochastic stability, the problems of H1 guaranteed cost control and H1 filtering for MJSs with mode-dependent mixed delays were addressed in [52,53], respectively. The exponential H1 filtering problem for stochastic MJSs with time-varying mixed delays was investigated in [32], where delaydependent conditions ensuring the exponential stability of the filtering error system were established. It is noted that the conditions proposed in [32] are independent from the decay rate. On the other hand, the L2 2L1 filtering (also called energy-to-peak filtering in the literature) is an important issue in the area of state estimation. The objective of this problem is to design an estimator such that the error system is stable and satisfies an energy-to-peak performance constraint. The L2 2L1 filtering problem was firstly considered in [16] for linear time-invariant systems. Since then, the problem has been investigated for a variety of dynamic systems, such as linear parameter-varying systems [39], switched systems [54], stochastic systems [7,14,49] and nonlinear time-delay systems [13,25]. In [26], the L2 2L1 filtering problem for discrete-time MJSs was studied, where reduced-order filters were designed by using the LMI approach. When time delays are considered in the discrete-time MJSs, the L2 2L1 filtering problem was addressed in [42]. The authors in [9] investigated the L2 2L1 filtering problem for a class of continuous-time MJSs with time-varying state delays and partly unknown transition probabilities. Very recently, the decentralized L2 2L1 filtering problem for interconnected MJSs with constant state delays was solved in [24]. However, to the best of the authors’ knowledge, the L2 2L1 filtering problem for uncertain MJSs with distributed delays has not been addressed in the literature, which is still open and needs to be solved. When studying the L2 2L1 filtering problem for uncertain MJSs with distributed delays, it is meaningful to consider the exponential stability with a specified decay rate. This needs to investigate the exponential estimate problem, which aims to develop decay-rate-dependent conditions for the exponential stability of the considered system. We note that the exponential estimate problem for MJSs with discrete delays has been addressed in [15,35], but the methods developed in these two works cannot be applied to solve the exponential L2 2L1 filtering problem for uncertain MJSs with distributed delays. Although the

207

exponential H1 filtering problem for MJSs with distributed delays has been studied in [32], the method developed therein fails to establish decay-rate-dependent conditions ensuring the exponential stability. Due to these reasons, it is necessary to develop an effective approach to solve the exponential L2 2L1 filtering problem for uncertain distributed delay MJSs under the consideration of exponential estimate. In this paper, we investigate the problem of exponential L2 2L1 filtering for linear systems that simultaneously contain distributed delays, Markovian jumping parameters and norm-bounded parametric uncertainties. Our purpose is to derive distributed-delay-dependent and decay-rate-dependent conditions for the existence of a mode-dependent filter guaranteeing that the filtering error system is exponentially stable and satisfies an L2 2L1 performance constraint. In order to reduce the conservatism of the obtained results, a novel modedependent Lyapunov–Krasovskii functional is employed and the integral-partitioning approach is applied. The obtained results are given in terms of LMIs, and the desired filter parameters can be computed by solving the presented LMIs. A numerical example is provided finally to illustrate the effectiveness of the proposed method. The novelty of this paper is threefold: (i) the exponential estimate is considered for the first time in the design of robust L2 2L1 filters for uncertain MJSs with distributed delays; (ii) a general version of mode-dependent Lyapunov–Krasovskii functional is employed and the integral-partitioning approach is applied, which could considerably reduce the conservatism of the obtained results; and (iii) the filter design result is obtained in terms of strict LMIs, which are easy to solve. Notations. Throughout this paper, for real symmetric matrices X and Y, the notation X ZY (respectively, X 4Y) means that the matrix XY is positive semi-definite (respectively, positive definite). Rn denotes the n dimensional Euclidean space and Rmn denotes the set of all m  n real matrices. I denotes an identity matrix of appropriate dimension. In denotes a n-dimensional identity matrix. On and On,m denote n  n-dimensional zero matrix and n  m-dimensional zero matrix, respectively. The superscript ‘T’ represents the transpose. The notation ‘n’ is used as an ellipsis for terms that are induced by symmetry. 9  9 denotes the Euclidean norm for vectors and J  J denotes the spectral norm for matrices. diagð. . .Þ stands for a block-diagonal matrix. L2 ½0,1Þ represents the space of square-integrable vector functions over ½0,1Þ. ðO,F ,PÞ is a probability space, where O is the sample space, F is the s-algebra of subsets of the sample space and P is the probability measure on F . Efg stands for the mathematical expectation operator with respect to the given probability measure P. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

2. System description and preliminaries On a given probability space ðO,F ,PÞ, we consider the following uncertain linear system with distributed delays

208

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

Using the filter in (5), when r t ¼ i 2 S, we obtain the filtering error system as follows:

and Markovian jumping parameters: 8 _ ¼ ½Aðr t Þ þ DAðt,r t ÞxðtÞ þ Bðr t ÞoðtÞ xðtÞ > > Rt > > > þ ½Ad ðr t Þ þ DAd ðt,r t Þ td xðaÞ da > > > > < yðtÞ ¼ ½Cðr t Þ þ DCðt,r t ÞxðtÞ þ Dðr t ÞoðtÞ Rt þ ½C d ðr t Þ þ DC d ðt,r t Þ td xðaÞ da > > > > > > zðtÞ ¼ Eðr t ÞxðtÞ > > > : xðtÞ ¼ jðtÞ, 8t 2 ½2d,0

ð1Þ

ð6Þ where "

where xðtÞ 2 Rn is the state; yðtÞ 2 Rq is the measured output; zðtÞ 2 Rp is the signal to be estimated; oðtÞ 2 Rl is the deterministic disturbance belonging to L2 ½0,1Þ; d 4 0 is a positive scalar representing the distributed delay; jðtÞ is a vector-valued initial continuous function defined on the interval ½2d,0; fr t g is a continuous-time Markovian process with right continuous trajectories, which takes values in a finite set S9f1,2, . . . ,sg with transition probability matrix P9½pij ss given by ( Prfr t þ h ¼ j9r t ¼ ig ¼

pij h þ oðhÞ, iaj 1 þ pii hþ oðhÞ, i ¼ j

ð2Þ

where h4 0, limh-0 ðoðhÞ=hÞ ¼ 0, pij Z0, for jai, is the transition rate from mode i at time t to mode j at time P t þ h, and pii ¼  sj ¼ 1,jai pij . For each r t 2 S, the matrices Aðr t Þ, Ad ðr t Þ, Bðr t Þ, Cðr t Þ, C d ðr t Þ, Dðr t Þ and Eðr t Þ are known, real and constant, while DAðt,r t Þ, DAd ðt,r t Þ, DCðt,r t Þ and DC d ðt,r t Þ are unknown time-varying matrices representing the parametric uncertainties, and they are assumed to take the form "

#

"

#

M 1 ðr t Þ DAðt,rt Þ DAd ðt,rt Þ Fðt,r t Þ½N1 ðr t Þ N 2 ðr t Þ ¼ M 2 ðr t Þ DCðt,rt Þ DC d ðt,r t Þ

ð3Þ

where M 1 ðr t Þ, M 2 ðr t Þ, N1 ðr t Þ and N2 ðr t Þ are known real matrices, and Fðt,r t Þ : R-Rgk , for all r t 2 S, are uncertain time-varying matrices satisfying Fðt,r t ÞT Fðt,r t Þ rI,

8r t 2 S

ð4Þ

The uncertainties DAðt,r t Þ, DAd ðt,r t Þ, DCðt,r t Þ and DC d ðt,r t Þ are said to be admissible if both the conditions in (3) and (4) are satisfied [46]. In the sequel, for each possible r t ¼ i 2 S, matrices Mðr t Þ and Mðt,r t Þ will be denoted by Mi and Mi(t), respectively; for example, Aðr t Þ ¼ Ai , Fðt,r t Þ ¼ F i ðtÞ, and so on [46]. This paper aims to design a mode-dependent linear filter taking the form (

x_ f ðtÞ ¼ Af ðr t Þxf ðtÞ þ Bf ðr t ÞyðtÞ zf ðtÞ ¼ C f ðr t Þxf ðtÞ

8 Rt _ ~ ~ ~ ~ ~ ~ > > < x ðtÞ ¼ ½A i þ M i F i ðtÞN i xðtÞ þ B i oðtÞ þ ½A di þ M i F i ðtÞN 2i  td xðaÞ da eðtÞ ¼ C~ i xðtÞ > > : xðtÞ ¼ j ~ ðtÞ, 8t 2 ½2d,0

ð5Þ

where xf ðtÞ 2 Rn is the filter state; zf ðtÞ 2 Rp is the filter output; Af ðr t Þ, Bf ðr t Þ and C f ðr t Þ are the filter coefficients to be determined. For each possible r t ¼ i 2 S, the matrices Af ðr t Þ, Bf ðr t Þ and C f ðr t Þ will be denoted by Afi , Bfi and C fi , respectively. The initial condition of the filter system is xf ð0Þ ¼ xf 0 . It is also assumed that xf ðtÞ ¼ xf 0 for any t 2 ½2d,0.

xðtÞ ¼ " A~ i ¼

xðtÞ xf ðtÞ Ai

# , 0

Bfi C i Afi " # M 1i ~i¼ , M Bfi M 2i

"

j~ ðtÞ ¼ # ,

jðtÞ

#

, eðtÞ ¼ zðtÞzf ðtÞ xf 0 " # " # Adi Bi A~ di ¼ , B~ i ¼ Bfi C di Bfi Di C~ i ¼ ½Ei C fi 

N~ i ¼ ½N 1i 0,

Throughout this paper, we adopt the following definitions. Definition 1 (Shu et al. [35]). When oðtÞ  0, the system in (6) is said to be mean-square robustly exponentially ~ ðtÞ defined on stable, if for all finite initial function j ½2d,0, all initial mode r 0 2 S and all admissible parametric uncertainties, there exist constants d 40 and k 4 0 such that the following inequality holds: 2

2

~ 9d Ef9xðtÞ9 g r kedt 9j

ð7Þ

~ ðaÞ9. In this case, the scalars d ~ 9d ¼ sup2d r a r 0 9j where 9j and k are called the decay rate and decay coefficient, respectively. Definition 2. For given scalars d 40 and g 40, the system in (6) is said to be mean-square robustly exponentially stable with a decay rate d and an L2 2L1 performance level g, if (i) the system with oðtÞ  0 is mean-square robustly exponentially stable with the decay rate being d in the sense of Definition 1, and (ii) for any non-zero oðtÞ 2 L2 ½0,1Þ, all admissible parametric uncertainties and under zero initial conditions, the following inequality holds: Z 1 2 2 9oðtÞ9 dt ð8Þ sup Ef9eðtÞ9 g r g2 t

0

The objective of this paper is to derive delay-dependent and decay-rate-dependent conditions for the existence of filter (5) such that the filtering error system in (6) is mean-square robustly exponentially stable with a specified decay rate d and a prescribed L2 2L1 performance level g. To achieve our objective, we need the following lemmas. Lemma 1 (Jensen’s inequality, Gu [17], Shu and Lam [36]). For any constant matrix M 2 Rmm with M 40, scalars b 4a, vector function v : ½a,b-Rm , such that the integrals in the following are well-defined, then "Z #T "Z # Z b

ðbaÞ a

b

vðaÞT MvðaÞ da Z

b

vðaÞ da

a

vðaÞ da

M

a

Lemma 2 (Feng and Lam [11]). For any constant matrix M 2 Rmm with M 4 0, scalars a o b r0, vector function

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

v : ½a,b-Rm , such that the integrals in the following are well-defined, then 2Z bZ t a2 b vðaÞT MvðaÞ da db 2 a tþb "Z Z #T "Z Z # b

t

b

vðaÞ da db

Z a

t

vðaÞ da db

M

tþb

a

medd=m ddm 2

r4 ¼

,

X T FðtÞY þY T FðtÞT X r E1 X T X þ EY T Y 3. Main results 3.1. Exponential stability analysis In this subsection, we study the problem of exponential stability analysis for the filtering error system in (6) with oðtÞ ¼ 0. The main result of this part is given as follows. Theorem 1. Given an integer m Z 1 and two scalars d 4 0 and d 4 0, system (6) with oðtÞ ¼ 0 is mean-square robustly exponentially stable with decay rate d, if there exist matrices P i 40, Q i 40, Q 40, Ri 4 0, R 40, Z i 4 0, Z 40, and scalars Ei 40, i 2 S, such that the following conditions hold for all i 2 S:

m

3

zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{m N i ¼ ½N~ i N 2i    N2i Ok,n 

Proof. Motivated by [12], we define the following augmented vector: 2 Rt 3 td=m xðaÞ da 6 7 7 6 ^ 6 7 6 R ZðtÞ ¼ 6 tðm2Þd=m xðaÞ da 7 ð13Þ 7 6 tðm1Þd=m 7 4R 5 tðm1Þd=m xðaÞ da td Then, we employ a mode-dependent Lyapunov–Krasovskii functional candidate for system (6) with oðtÞ ¼ 0 as follows: 4 X

Vðxt ,r t ,tÞ ¼

V i ðxt Þ

ð14Þ

i¼1

where xt ¼ xðt þ yÞ, y 2 ½2d,0, and V 1 ðxt Þ ¼ edt xðtÞT Pðr t ÞxðtÞ

V 2 ðxt Þ ¼

Z

t

edða þ d=mÞ ZðaÞT Q ðr t ÞZðaÞ da

td=m

Z

Z

0

t

edða þ d=mÞ ZðaÞT Q ZðaÞ da db

þ

pij Q j Q o0

2mþ dd þ ddedd=m 2medd=m

md md I 1 ¼ ½I2n O2n,ðm þ 1Þn , I 2 ¼ ½Omn,2n Imn Omn,n , I 3 ¼ ½Omn,3n Imn , I 4 ¼ ½In On,ðm þ 2Þn , I 5 ¼ ½On,2n In On,mn , I 6 ¼ ½In On  zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{ Ai ¼ ½A~ i A~ di    A~ di O2n,n ,

tþb

Lemma 3 (Cao and Lam [4]). Let X and Y be real constant matrices of appropriate dimensions and F(t) be a real matrix function satisfying FðtÞT FðtÞ r I. Then, for any scalar E 4 0, the following inequality holds:

s X

r3 ¼

209

tþb

d=m

ð9Þ

j¼1 s X

pij Rj R o 0

ð10Þ

V 3 ðxt Þ ¼

Z

Z

0

Z

edðabÞ xðaÞT Rðr t ÞxðaÞ da db

tþb

d=m

j¼1

t

Z

0

0

Z

t

edðayÞ xðaÞT RxðaÞ da db dy

þ s X

pij Z j Z o0

ð11Þ

j¼1

2 6 6 4

Ci þ E

T iN i N i

Z i I 6 Ai ~ T Pi I 1 M

where

0

Ci ¼ I T1 @

s X

n

n

Z i ~ T I T Zi M i 6

i

3 7 5

n 7 o0

Ei I

V 4 ðxt Þ ¼ ð12Þ

pij Pj þ dPi AI 1 þ I T1 Pi Ai þ ATi Pi I 1

  d þ edd=m I T2 Q i þ Q I 2 I T3 Q i I 3 þ I T4 ðr1 Ri þ r2 RÞI 4 m  m m T  m   I T5 Ri I 5 2 I 4  I 5 Z i I 4  I 5 d d d

r1 ¼

edd=m 1

m þ ddedd=m medd=m md

2

d

Z

0

Z

0

y

,

Z

_ aÞ da db dy _ aÞT Zðr t Þxð edðabÞ xð

tþb

Z

0

t

0

Z

0

Z

t

þ s

d=m

1

Z i ¼ r3 Z i þ r4 Z ,

Z

d=m

j¼1

r2 ¼

tþb

y

d=m

y

_ aÞ da db dy ds _ aÞT Z xð edðayÞ xð

tþb

Let L be the weak infinitesimal generator of the random process fxt ,r t g. Then, it is obtained that, for each r t ¼ i 2 S, 0 1 s X T@ dt pij Pj þ dPi AxðtÞ þ 2edt xðtÞT Pi x_ ðtÞ LV 1 ðxt Þ ¼ e xðtÞ j¼1

0

dt

T

¼ e WðtÞ

s X I T1 @ j¼1

1

pij Pj þ dPi AI 1 WðtÞ

~ i F i ðtÞN i WðtÞ þ 2edt WðtÞT I T1 P i ½Ai þ M 2 0 1 s X T4 T@ dt pij Pj þ dPi AI 1 þI T1 Pi Ai ¼ e WðtÞ I 1 j¼1

~ i F i ðtÞN i þ N T F i ðtÞT M ~ T Pi I 1 þ ATi P i I 1 þI T1 P i M i i

i

WðtÞ ð15Þ

210

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

By using Lemma 2, we have Z 0 Z t _ aÞT Z i xð _ aÞ da dy  xð

where Z

WðtÞ ¼ ½xðtÞT ZðtÞT

td

xðaÞT daT

tðm þ 1Þd=m

d=m

r

We get LV 2 ðxt Þ as follows: LV 2 ðxt Þ ¼

Z

t

e

dða þ d=mÞ

ZðaÞ

td=m

0 T@

s X

1

pij Q j Q AZðaÞ da

j¼1



 d þ edðt þ d=mÞ ZðtÞT Q i þ Q ZðtÞ m  T   d d Q i Z t edt Z t m m

ð16Þ

It follows from (9) and (16) that   d LV 2 ðxt Þ r edðt þ d=mÞ ZðtÞT Q i þ Q ZðtÞ m  T   d d Q i Z t edt Z t m m     d T ¼ edt WðtÞ edd=m I T2 Q i þ Q I 2 I T3 Q i I 3 WðtÞ m ð17Þ Now, LV 3 ðxt Þ is obtained as LV 3 ðxt Þ ¼

Z

Z

0

t

dðabÞ

e

s X

Z

0

pij Rj R AxðaÞ da db

t

edb db

td=m

xðaÞ Ri xðaÞ da

þ edt xðtÞT RxðtÞ

Z

0

d=m

Z

0

edy db dy

ð18Þ

y

Then, it follows from (10), (18) and (19) that LV 3 ðxt Þ r edt xðtÞT ðr1 Ri þ r2 RÞxðtÞ Z Z t m t xðaÞT daRi xðaÞ da edt d td=m td=m h i m ¼ edt WðtÞT I T4 ðr1 Ri þ r2 RÞI 4  I T5 Ri I 5 WðtÞ d

ð20Þ

It is also obtained that Z 0 Z 0Z t _ aÞT edðabÞ xð LV 4 ðxt Þ ¼ d=m y tþb 0 1 s X _ aÞ da db dy þ edt xðtÞ _ T Z i xðtÞ _ @ pij Z j Z Axð j¼1 0

Z

0

d=m

tþy

_ aÞT da dyZ i xð

Z

0 d=m

Z

t

_ aÞ da dy xð

tþy

" #T Z t 2m2 d xðtÞ ¼ 2 xðaÞ da m td=m d " # Z t d xðtÞ Z i xðaÞ da m td=m  m T  m  ¼ 2WðtÞT I 4  I 5 Z i I 4  I 5 WðtÞ d d

ð22Þ

This, together with (11) and (21), implies that  m T LV 4 ðxt Þ r edt x_ ðtÞT I T6 Z i I 6 x_ ðtÞ2edt WðtÞT I 4  I 5 Z i d  m   I 4  I 5 WðtÞ d ~ i F i ðtÞN i T I T Z i I 6 ¼ edt WðtÞT ½Ai þ M 6  m T dt ~ ½Ai þ M i F i ðtÞN i WðtÞ2e WðtÞT I 4  I 5 Z i d  m   I 4  I 5 WðtÞ ð23Þ d It follows from (15), (17), (20) and (23) that

T

ð19Þ



t

b ¼ C þI T P M ~ i F i ðtÞN i þ N T F i ðtÞT M ~ Pi I 1 C i i i 1 i i

By Lemma 1, it can be verified that Z t Z t Z m t  xðaÞT Ri xðaÞ da r xðaÞT daRi xðaÞ da d td=m td=m td=m

Z

d=m

Z

ð24Þ

where

T

e

0

b þðA þ M ~ i F i ðtÞN i ÞT I T Z i I 6 LVðxt ,i,tÞ r edt WðtÞT ½C i i 6 ~ i F i ðtÞN i ÞWðtÞ ðAi þ M

d=m

Z

d

2

Z

1

j¼1

þ edt xðtÞT Ri xðtÞ dt

xðaÞ

tþb

d=m

0 T@

tþy

2m2

edb db dyedt

Z

y

_ _ T Z xðtÞ da dy þ edt xðtÞ

Z

0

tþy

d=m

Z

0 d=m

Z 0Z s

t

0

_ aÞT Z i xð _ aÞ xð

edy db dy ds

y

ð21Þ

By using Lemma 3, we obtain that " # b C n i ~ i F i ðtÞN i Þ Z i Z i I 6 ðAi þ M " # " T # ~i Ci n I 1 Pi M ¼ þ ~ i F i ðtÞ½N i 0 Z i I 6 Ai Z i Zi I 6 M " # N Ti ~ T I T Zi  ~ T Pi I 1 M þ F i ðtÞT ½M i i 6 0 " # Ci þ Ei N Ti N i n r Z i I 6 Ai Z i " T # ~i I 1 Pi M ~T T ~T þ E1 i ~ i ½M i P i I 1 M i I 6 Z i  Zi I 6M

ð25Þ

ð26Þ

Applying the Schur complement equivalence to (12) yields that there always exists a sufficiently small scalar c 4 0 such that " # " T # ~i I 1 Pi M Ci þ Ei N Ti N i þcI n þ E1 i ~i Z i I 6 Ai Z i Zi I 6 M ~ T I T Zi o 0 ~ T Pi I 1 M ½M i i 6 This, together with (26), implies that " # b þcI C n i ~ i F i ðtÞN i Þ Z i o0 Z i I 6 ðAi þ M

ð27Þ

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

Applying the Schur complement equivalence to this inequality gives that b þ ðA þ M ~ i F i ðtÞN i ÞT I T Z i I 6 ðAi þ M ~ i F i ðtÞN i Þ o cI C i i 6

ð28Þ

2

2

LVðxt ,i,tÞ r cedt 9WðtÞ9 r cedt 9xðtÞ9

ð29Þ

By applying Dynkin’s formula, we have from (29) that EfVðxt ,r t ,tÞg r Vðx0 ,r 0 ,0Þ

ð30Þ

Recalling the Lyapunov–Krasovskii functional in (14), we obtain that 2

Vðx0 ,r 0 ,0Þ r maxfJPi Jg9xð0Þ9 þmaxfJQ i Jgedd=m i2S

0

dd=m

Z

þ JQ Je

Z

i2S

Z

0

þ maxfJZ i Jg

Z

0

0

Z

y

Z

e 9ZðaÞ9 da db

0

b 0

i2S

d=m

dðabÞ

2

0

þ JZ J

d=m

s

y

0

y

n5 ¼

Z

 d=m

9xðaÞ9 da þ n4

2

i2S

Z

0

2

_ aÞ9 da 9xð

ð31Þ

d=m

 d JQ J edd=m m i2S ! 2 d d maxfJRi Jg þ JRJ edd=m m i2S 2m2 ! 2 3 d d max fJZ Jg þ JZ J edd=m i 2m2 i2S 6m3

n2 ¼ maxfJQ i Jg þ

Recalling (13) and using Lemma 1, we have 2 m Z tðj1Þd=m X 2 9ZðtÞ9 ¼ xðaÞ da tjd=m j¼1 Z m tðj1Þd=m d X 2 r 9xðaÞ9 da m j ¼ 1 tjd=m Z d t 2 ¼ 9xðaÞ9 da m td

ð38Þ

where lmin ðP i Þ denotes the minimum eigenvalue of the matrix Pi. Then, it follows from (30), (37) and (38) that 2 ~ 92d , Ef9xðtÞ9 g r kedt 9j where k ¼ n5 =mini2S flmin ðPi Þg. Therefore, we conclude by Definition 1 that system (6) with oðtÞ ¼ 0 is mean-square robustly exponentially stable with decay rate d. &

i2S

n4 ¼

m3

EfVðxt ,r t ,tÞg Z minflmin ðP i Þgedt Ef9xðtÞ9 g

n1 ¼ maxfJPi Jg

n3 ¼

ð37Þ

Note from (14) that

b

where 

ð36Þ

n1 m3 þ n2 ðm þ1Þd3 þ n3 dm2 þ n4 dm23 m2

0

b

d=m 2

mm1 þðm þ 1Þdm2 m

where 2

Z

ð35Þ

where

edðayÞ 9xðaÞ9 da db dy

2

0

~ 9d 9x_ ðtÞ9 r m3 9j

~ 92d Vðx0 ,r 0 ,0Þ r n5 9j

edðabÞ 9xðaÞ9 da db

_ aÞ9 da db dy ds edðayÞ 9xð Z 0 2 2 r n1 9xð0Þ9 þ n2 9ZðaÞ9 da þ n3 Z

i2S

It follows from (34) that, when d=mr t r0,

Then, it follows from (31), (33) and (36) that

2

_ aÞ9 da db dy 9xð Z 0 Z 0Z 0Z 0

e

i2S

m2 ¼ maxfJA~ di J þ JM~ i JJN2i Jg

2 2 _ ~ 92d r 9x_ ðtÞ9 r m23 9j 9xðtÞ9

2

da

b

d=m

Z

d=m

0

b

d=m

þ maxfJRi Jg þ JRJ

Z

0

m1 ¼ maxfJA~ i J þJM~ i JJN~ i Jg

Using this inequality, we can obtain that, when d=m r t r0,

2

eda 9ZðaÞ9 da

d=m

ð34Þ

td

m3 ¼

i2S



It is also noted that, when oðtÞ ¼ 0, Z t 9xðaÞ9 da 9x_ ðtÞ9 r m1 9xðtÞ9 þ m2 where

Recalling (24), we have that

Z

211

ð32Þ

Therefore, for any t satisfying d=m rt r 0, we have Z 2 d 0 ðm þ 1Þd 2 2 ~ 92d 9ZðtÞ9 r 9xðaÞ9 da r 9j ð33Þ m ðm þ 1Þd=m m2

Remark 1. The idea of integral-partitioning approach, which was proposed in [12], is used when constructing the Lyapunov–Krasovskii functional in (14). This approach enables us to partition the distributed-delay term into m integrals and then construct an augmented Lyapunov– Krasovskii functional corresponding to the partitioned integrals. Obviously, the integral-partitioning approach introduces more flexibility in the obtained conditions. It is also shown in [12] that the Lyapunov–Krasovskii functional constructed using the integral-partitioning approach is much general and could lead to less conservative results. _ aÞT Z xð _ aÞ Remark 2. It is known that the integral of xð plays a key role in the development of delay-dependent stability conditions for time-delay systems [47]. In most of the works related to MJSs with time delays, such as [10,19,46], the positive-definite matrix Z is assumed to be independent from the system modes. This assumption is so strong that it might produce considerable conservatism. In order to relax the assumption, a Lyapunov–Krasovskii _ aÞT Zðr t Þxð _ aÞ was functional involving a double integral of xð employed in [50] for the stochastic stability analysis of MJSs with discrete delays, where Zðr t Þ is a mode-dependent positive-definite matrix. It has been shown in [50] that the use of mode-dependent matrix Zðr t Þ is helpful to reduce

212

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

the conservatism of the delay-dependent stability criteria for MJSs with discrete delays. This idea is utilized to construct the Lyapunov–Krasovskii functional in (14), _ aÞT Zðr t Þxð _ aÞ. which involves a triple integral of xð 3.2. L2 2L1 performance analysis In this subsection, we are going to analyze the L2 2L1 performance for the filtering error system in (6). The main result of this subsection is given as follows. Theorem 2. Given an integer mZ 1, and three scalars d 40, d 4 0 and g 40, the system in (6) is mean-square robustly exponentially stable with a decay rate d and an L2 2L1 performance level g, if there exist matrices Pi 40, Q i 40, Q 40, Ri 4 0, R 4 0, Z i 4 0, Z 4 0, and scalars Ei 4 0, i 2 S, such that (9), (10), (11) and the following conditions hold for all i 2 S: " # n P i o0 ð39Þ C~ i g2 I

2

Ci

n

6 ~T ¼6 4 B i Pi I 1 Z i I 6 Ai

I Z i I 6 B~ i

3

2 T ~ 3 I PM 7 6 1 i i7 n 7 0 þ 4 5F i ðtÞ½N i 0 0 5 ~i ZiI 6M Z i n

2

3 N Ti 6 7 ~ T Pi I 1 0 M ~ T I T Zi þ 4 0 5F i ðtÞT ½M i i 6 2 6 r6 4

0

Ci þ Ei N Ti N i

n

T B~ i P i I 1

I

n

3 7

n 7 5

Z i I 6 Ai Z i I 6 B~ i Z i 2 T ~ 32 T ~ 3T I 1 Pi M i I 1 Pi M i 76 7 1 6 0 0 þ Ei 4 54 5 o0 ~i ~i Zi I 6M Zi I 6M

ð42Þ

By applying the Schur complement equivalence, it follows from (42) that " # ~ C n 1i o0 ~ ~

C 2i

C 3i

This, together with (41), implies that 2 6 6 6 6 6 4

n

n

n

T B~ i P i I 1

I

n

Z i I 6 B~ i

n 7 7

Z i T ~ M i I T6 Z i

Z i I 6 Ai ~ T Pi I 1 M

0

i

2

3

Ci þ Ei N Ti N i

7

7o0 n 7 5 Ei I

ð40Þ

The notations involved in (40) are the same as those defined in Theorem1. Proof. It follows from (40) that the condition in (12) holds for all i 2 S. Therefore, by Theorem 1, system (6) with oðtÞ ¼ 0 is mean-square robustly exponentially stable with decay rate d. In the following, we shall prove that the condition in (8) is satisfied under zero initial conditions and for all nonzero disturbance oðtÞ 2 L2 ½0,1. To this end, we employ the Lyapunov–Krasovskii functional in (14). By following a similar line as in the proof of Theorem 1, we obtain that, when r t ¼ i, #" " #T " # ~ WðtÞ WðtÞ C n 2 1i LVðxt ,i,tÞedt 9oðtÞ9 r edt ~ ~ oðtÞ oðtÞ C C 2i

3i

ð41Þ

LVðxt ,r t ,tÞedt 9oðtÞ9 r0

ð43Þ

Then, under zero initial conditions, we have that Z t Z t 2 2 eda 9oðaÞ9 da r edt 9oðaÞ9 da EfVðxt ,r t ,tÞg r 0 0 Z 1 2 r edt 9oðaÞ9 da

ð44Þ

0

Note that EfVðxt ,r t ,tÞg Z edt EfxðtÞT Pðr t ÞxðtÞg

ð45Þ

It follows from (44) and (45) that, for r t ¼ i 2 S, Z 1 2 EfxðtÞT P i xðtÞg r 9oðaÞ9 da

ð46Þ

0

Applying the Schur complement equivalence to (39) yields that P i þ

1 ~T ~ C i C i o0

g2

This implies that, for any t Z 0, 2

T

9eðtÞ9 ¼ xðtÞT C~ i C~ i xðtÞ r g2 xðtÞT Pi xðtÞ

ð47Þ

This, together with (46), ensures that, for any t Z0, Z 1 2 2 Ef9eðtÞ9 g r g2 9oðaÞ9 da ð48Þ 0

where b þ ðA þ M ~ ¼C ~ i F i ðtÞN i ÞT I T Z i I 6 ðAi þ M ~ i F i ðtÞN i Þ C 1i i i 6 T

T

~ ¼ B~ P I 1 þ B~ I T Z I 6 ðA þ M ~ i F i ðtÞN i Þ C 2i i i i i 6 i T

~ ¼ I þ B~ I T Z I 6 B~ C 3i i i 6 i b is defined in (25). By Lemma 3 and the condition and C i in (40), it can be verified that 2 3 b C n n i 6 7 T 6 I n 7 B~ i Pi I 1 4 5 ~ i F i ðtÞN i Þ Z i I 6 B~ i Z i Z i I 6 ðAi þ M

Therefore, the condition in (8) holds.

&

3.3. Exponential L2 2L1 filter design The analysis results obtained in the previous subsection will be applied in this subsection to derive sufficient conditions for the existence of the desired L2 2L1 filters for system (1). Theorem 3. Consider the delayed Markovian jump system in (1). Given an integer m Z1, and three scalars d 4 0, d 4 0 and g 40, there exists a mode-dependent filter in the form of (5) such that the filtering error system in (6) is mean-square

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

robustly exponentially stable with a decay rate d and an L2 2L1 performance level g, if there exist matrices X i 4 0, Y 4 0, Q i 4 0, Q 40, Ri 40, R 4 0, Z i 40, Z 4 0, A^ fi , B^ fi , C^ fi , and scalars Ei 40, i 2 S, such that (9), (10), (11) and the following conditions hold for all i 2 S: 2 3 n n X i 6 Y Y n 7 ð49Þ 4 5 o0 ^ Ei Ei C fi g2 I 2

Ui 6 6 Gi 6 6 Li 4 Yi where

Ui ¼

n

n

I

n

Z i Bi 0

Z i M T1i Z i

0

I T1 @

s X

7 7 o0 n 7 5 Ei I n 7

ð50Þ

1

p



T

Gi ¼ ½BTi X i þDTi B^ fi BTi YI 1 , Li ¼ Z i ½Ai Ai I 1 þ Z i Adi H1 T

Yi ¼ ½MT1i X i þMT2i B^ fi MT1i YI 1 , F1i ¼ X i Ai þ B^ fi C i

A^ fi

#



Xi

Y



Y Y # X i Adi þ B^ fi C di F3i ¼ YAdi "

, YAi zfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflffl{m F4i ¼ ½N1i N 1i N2i    N2i Ok,n  zfflfflfflfflffl}|fflfflfflfflffl{m On , H2 ¼ ½In In On,ðm þ 1Þn  H1 ¼ ½On,2n In    In

F2i ¼

YAi

The other notations involved above are the same as those defined in Theorem1. When the conditions in (9), (10), (11), (49) and (50) are feasible, there exist nonsingular matrices Ui and V satisfying U i V ¼ IX i Y 1 . Then, the desired filter is obtained with the coefficient matrices given by 8 1 1 ^ ^ V A ¼ U 1 > i ðA fi X i Ai B fi C i ÞY > < fi ^ fi B Bfi ¼ U 1 ð51Þ i > > : C ¼ C^ Y 1 V 1 fi

T

Ui ¼ OT3 ðCi þ Ei N Ti N i ÞO3 , Gi ¼ B~ i P i I 1 O3 , Li ¼ Z i I 6 Ai O3 T

 d þ HT1 FT3i I 1 þ edd=m I T2 Q i þ Q I 2 I T3 Q i I 3 m

 m T T þ H2 r1 Ri þ r2 R H2  I 5 Ri I 5 d  m T  m  2 H2  I 5 Z i H2  I 5 þ Ei FT4i F4i d d

"

It can be verified that   Xi Y OT2 Pi O2 ¼ 40 Y Y

~ P i I 1 O3 Yi ¼ M i

T A T ij F1j þ dF1i þ F2i þ F2i I 1 þ I 1 F3i H1

j¼1

It is easy to see that the matrices O1i and O2 are nonsingular. Let " # Xi Ui P i 9O1i O1 ¼ 2 U Ti V T Y 1 ðX i YÞY 1 V 1

This implies that Pi 40. Let O3 9diagðO2 ,Iðm þ 1Þn Þ. Consider the conditions in Theorem 2 and the relationship in (51). Then, it is shown that

3

n

213

fi

Proof. The condition in (49) guarantees that   Xi Y 40 Y Y This inequality implies that IX i Y 1 is nonsingular. Choose a nonsingular matrix V and define U i 9ðIX i Y 1 Þ V 1 . It is obvious that Ui is nonsingular, and the matrices Ui and V satisfy U i V ¼ IX i Y 1 . Now, we introduce the following matrices: " #   Xi Y I I O1i 9 T , O2 9 Ui 0 0 VY

By noting these facts, it can be verified that (50) implies the condition in (40). Moreover, observing that C~ i O2 ¼ ½Ei Ei C^ , we have from (49) that the condition fi

in (39) holds. Consequently, the proof can be completed by applying Theorem 2. & Remark 3. Theorem 3 provides sufficient conditions for the existence of desired L2 2L1 filters for uncertain MJSs with distributed delays. It is noted that the conditions in Theorem 3 are LMIs that can be easily solved. When the LMIs in (9), (10), (11), (49) and (50) are feasible, the filter coefficients can be obtained according to (51). Remark 4. The conditions in Theorem 3 are dependent not only on the distributed delay d but also on the decay rate d. It can be verified that the scalars r1 , r2 , r3 and r4 are monotonic increasing with respect to d when m, d and g are fixed. This in turn implies that the matrices on the left-hand side of (50) are monotonic increasing with respect to d. Thus, the maximum value of the decay rate d ensuring the feasibility of the LMI conditions in Theorem 3 can be obtained when m, d and g are fixed. On the other hand, when m, d and d are fixed, it is possible to compute the minimum value of the performance index g that ensures the feasibility of the LMI conditions in Theorem 3. Remark 5. This paper is mainly concerned with MJSs with distributed delays, and thus the discrete delays are not considered. However, by applying the approach developed in this paper together with the methods in [10,50], the results obtained in this paper can be easily generalized to the case of MJSs with both discrete and distributed delays. 4. Numerical example In this section, we provide a numerical example to illustrate the effectiveness of the proposed design method. Consider system (1) with two modes (i.e. s¼2) and the system parameters are given by 2 3 2 3 3 1 0 5 1 0:2 6 7 6 7 1 5, A2 ¼ 4 0:2 7 0:6 5 A1 ¼ 4 0:3 2:5 0:1 0:3 3:8 0:2 2 4

214

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

2

3 2 3 0:2 0:1 0:6 0 0:3 0:6 6 7 6 7 0:5 0 5 Ad1 ¼ 4 0:5 1 0:8 5, Ad2 ¼ 4 0:1 0 1 1:5 0:6 1 0:8 2 3 2 3 2 3 2 3 0:1 1 0:6 0:1 6 7 6 7 6 7 6 7 B1 ¼ 4 0 5, B2 ¼ 4 0:5 5, M 11 ¼ 4 0 5, M 12 ¼ 4 0:1 5 1 0 0:2 0 N 11 ¼ ½0:2 0 0:1,

N 12 ¼ ½0:1 0:1 0

N 21 ¼ ½0:1 0:2 0,

N 22 ¼ ½0 0:1 0:2

C 1 ¼ ½0:8 0:3 0,

C 2 ¼ ½0:5 0:2 0:3

C d1 ¼ ½0:2 0:3 0:6, E1 ¼ ½0:5 0:1 1, D1 ¼ 0:2,

D2 ¼ 0:5,

C d2 ¼ ½0 0:6 0:2

E2 ¼ ½0 1 0:6 M 21 ¼ 0:2,

M 22 ¼ 0:1

When the time delay d and the performance index

g are fixed, the maximum value of the decay rate d can be computed by solving the LMI conditions in Theorem 3. Assume that d ¼1.5, g ¼ 2:5 and p22 ¼ 1:3. Then, the maximum allowed decay rate d is given in Table 1 via different methods. It is seen from the table that the maximum allowed decay rate d obtained when m¼5 is larger than those obtained when m ¼1, m ¼2 and m ¼3. On the other hand, when the time delay d and the decay rate d are fixed, we can compute the minimum value of g that ensures the feasibility of the LMIs in Theorem 3. Assume that d ¼2, d ¼ 0:48 and p22 ¼ 1:3. Table 2 shows the minimum allowed value of g obtained by using different methods. In this table, ‘–’ means that there does not exist a scalar g 4 0 such that the LMI conditions in Theorem 3 are feasible. For instance, when m ¼2 and p11 ¼ 1, the conditions in Theorem 3 with g being a decision variable are not feasible, and thus we cannot get the allowed value of g. It is found from Table 2 that the method with m¼5 gives smaller values of g. It is shown from Tables 1 and 2 that, for this example, the method with larger m may lead to less conservative results. Therefore, the integral-partitioning approach

Table 1 Maximum allowed d via different methods (d ¼1.5, g ¼ 2:5, p22 ¼ 1:3). Methods p11 ¼ 0:2 p11 ¼ 0:5 p11 ¼ 1 Number of decision variables m¼ 1 m¼ 2 m¼ 3 m¼ 5

0.8038 1.3414 1.4941 1.5044

0.8023 1.3094 1.4504 1.4654

0.8003 1.2702 1.3932 1.4227

104 149 221 446

introduces more flexibility in the LMI conditions of Theorem 3. This indicates that it is much effective to apply the integral-partitioning approach to solve the exponential L2 2L1 filtering problem for uncertain MJSs with distributed delays. In addition, the number of decision variables involved in the LMI conditions of Theorem 3 is provided in Tables 1 and 2. It is seen that the number of decision variables increases as the parameter m increases. This means that, when m is larger, the complexity of the conditions in Theorem 3 is higher. Thus, there is a trade-off between the conservatism and complexity of the proposed results. Now, we are in a position to compute the parameters of a desired filter based on Theorem 3. To this end, we choose   0:2 0:2 d ¼ 2, d ¼ 0:48, g ¼ 0:45, P ¼ 1:3 1:3 For this example, it is found that the conditions in Theorem 3 are not feasible for any m r4. However, the LMI conditions in (9), (10), (11), (49) and (50) are feasible when m ¼5. In this case, we know by Theorem 3 that there exists a mode-dependent filter in the form of (5) such that the filtering error system in (6) is mean-square robustly exponentially stable with decay rate 0.48 and L2 2L1 performance level 0.45. When m¼5, by solving the LMIs in (9), (10), (11), (49) and (50), we obtain 2 3 10:2355 12:7697 3:3577 6 12:7697 40:5922 1:3052 7 X1 ¼ 4 5 3:3577

1:3052

3 3:3779 0:8508 1:7356 6 7 X 2 ¼ 4 0:8508 15:3867 0:2889 5 1:7356 0:2889 5:6039 2 3 1:7573 1:3911 1:4049 6 7 Y ¼ 4 1:3911 9:4670 0:6948 5 1:4049 0:6948 4:7366 2 3 8:7406 13:0453 3:9205 6 1:4918 7 A^ f 1 ¼ 4 0:9941 13:9225 5 1:4030 7:5048 22:3817 2 3 7:1132 8:5273 3:6243 6 1:6815 7 A^ f 2 ¼ 4 6:5900 55:5545 5 7:5650 2:9099 15:7721 2 3 2 3 10:9271 2:3950 6 7 6 7 B^ f 1 ¼ 4 32:8993 5, B^ f 2 ¼ 4 3:8807 5 19:7001

Table 2 Minimum allowed g via different methods (d ¼ 2, d ¼ 0:48, p22 ¼ 1:3). Methods

p11 ¼ 0:2 p11 ¼ 0:5 p11 ¼ 1 Number of decision variables

m¼ 1 m¼ 2 m¼ 3 m¼ 5

– 1.4940 0.5893 0.4472

– 2.6464 0.5950 0.4482

– – 0.5925 0.4433

104 149 221 446

11:4222

2

2:7930

C^ f 1 ¼ 0:3601

0:6046

0:5804

C^ f 2 ¼ 0:1700

0:5787

0:1465

To compute the filtering system coefficients, we choose the nonsingular matrix V as 2 3 0:5 0:08 0:02 6 7 V ¼ 4 0:16 0:01 0:1 5 0:08 0:02 0:03

B. Zhang, Y. Li / Signal Processing 93 (2013) 206–216

Then, by solving the equation U i V ¼ IX i Y 1 , we obtain 2 3 22:8730 16:6736 56:1865 6 7 U 1 ¼ 4 44:9512 33:7238 122:8738 5 2:5264 11:8826 6:8847 2 3 3:9660 3:3561 9:3074 6 7 U 2 ¼ 4 8:0602 7:9949 24:6171 5 0:5108 1:1909 4:2177 According to (51), the parameters of the desired L2 2L1 filter are obtained as 2 3 0:1765 9:4433 7:3609 6 7 Af 1 ¼ 4 1:4220 3:1204 3:3897 5 0:6701 2:7895 4:0670 2 3 28:9300 66:0317 94:2589 6 7 Af 2 ¼ 4 63:5711 237:3863 282:6516 5 13:2977 3

55:6100 67:7429 2 3 0:1465 6 0:6995 7 6 11:1834 7 Bf 1 ¼ 4 5, B f 2 ¼ 4 5 0:8637 3:8377 2

2:1539

C f 1 ¼ ½1:0180 2:4166 3:5904 C f 2 ¼ ½0:7300 0:7748 2:4173

5. Conclusions In this paper, the problem of exponential L2 2L1 filter design for uncertain MJSs with constant distributed delays has been studied. LMI-based conditions for the solvability of the addressed problem have been obtained based on a novel Lyapunov–Krasovskii functional together with the integral-partitioning approach. The presented results are dependent not only on the distributed delays but also on the decay rate. A numerical example has been provided to show the effectiveness of the proposed design method. The approach developed in this paper is also efficient for the problems of state-feedback and outputfeedback control of MJSs with constant distributed delays. Moreover, it is meaningful to apply our approach to study the exponential control and filtering problems for MJSs with mode-dependent and/or time-varying distributed delays. All these issues are important and challenging.

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