Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback

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Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback$ Hao Shena, Zhen Wangb, Xia Huangc,n, Jing Wanga a

School of Electrical Engineering and Information, Anhui University of Technology, Ma'anshan 243002, China b College of Information Science and Engineering, Shandong University of Science and Technology, Qingdao 266590, China c College of Information and Electrical Engineering, Shandong University of Science and Technology, Qingdao 266590, China Received 1 September 2012; received in revised form 8 February 2013; accepted 18 February 2013

Abstract This paper deals with the problem of the dissipative control for a class of nonlinear Markovian jump systems through Takagi–Sugeno fuzzy model approach. The transition rates of Markovian process under consideration are assumed to be partly known. We aim to design retarded feedback controllers such that the resulting closed-loop system is stochastically stable and strictly ðQ; S; RÞθdissipative. By introducing a novel augmented Lyapunov functional and some free Markovian switching matrices, some sufficient conditions for the solvability of the above problem are given in terms of linear matrix inequalities. Finally, two numerical examples are given to demonstrate the effectiveness of our proposed approach. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction The Takagi–Sugeno (T–S) fuzzy model approach is an efficient and widely accepted technique to cope with the analysis and design of nonlinear systems [1–5]. The idea is to represent or ☆ This work was supported by the National Natural Science Foundation of China under Grants 61104007, 61004078, the Natural Science Foundation of Anhui Province under Grant 1308085QF119, the Key Foundation of Natural Science for Colleges and Universities in Anhui province under Grants KJ2012A049, KJ2013Z018, the Research Foundation for Young Scientists of Anhui University of Technology under Grant QZ201112. n Corresponding author. Tel.: +86 1396 983 2984. E-mail addresses: [email protected] (H. Shen), [email protected] (Z. Wang), [email protected], [email protected] (X. Huang).

0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.02.031 Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

approximate the nonlinear systems by T–S fuzzy model, which is described by smoothly blending a set of simple linear subsystems together through the membership functions [6–9]. As a result, based on this fuzzy model, some simple control designs have been developed to achieve control performance of many complex nonlinear systems, such as the Henon system, chaotic Lorenz system, and truck trailer system [10–13]. For this reason, there have been lots of results on the stability analysis and control synthesis for T–S fuzzy systems. To mention a few, by solving some convex optimization problems with linear matrix inequality (LMI) constraints, some effective H ∞ controller design approaches were presented for continuous-time T–S fuzzy systems [14–16] and discrete-time T–S fuzzy systems [17–20], respectively. In [21,22], several H ∞ filters were designed, which guarantee that the corresponding filtering error systems are stable with γdisturbance rejection. More recently, the problems of passivity analysis and passive control for T–S fuzzy delayed systems were studied in [23], where some delaydependent conditions for the solvability of the problems were presented by using LMI relaxation technique. It should be pointed out, however, that all these problems come down to the dissipativity analysis and dissipative control problems. As noted in [24–27], in the dissipativity theory, a useful framework for the study of nonlinear control systems is provided by using an input–output description regarding the system energy. However, it is not easy to construct an efficient controller to deal with the dissipative design problem of nonlinear control systems [28]. In this case, the T–S fuzzy model may be used, and a fuzzy controller is developed to achieve the dissipative design goal of the original nonlinear systems. Therefore, the fuzzy dissipative control is a problem worthy of study. On the other hand, in many practical systems, the abrupt phenomena cannot be ignored and lead to the changes of system parameters. The study of Markovian jump systems, therefore, has been the subject of extensive research activity [29–33]. A large portion of the literature has focused on the analysis and synthesis of linear Markovian jump systems. In context of nonlinear Markovian jump systems, by using the T–S fuzzy model approach, some results have been derived and widely applied in many practical systems such as circuit systems [34,35], a singlelink robot arm [36], electrical power systems [37], and backing up control of a computer simulated truck–trailer [38,39]. In these papers, the transition rates are assumed to be completely known. This assumption may limit the scope of the applications of these results because it may be prohibitively expensive, and indeed sometimes impossible, to measure all transition rates [40–45]. Taking partly known transition rates into account, some sufficient conditions for the solvability of the fuzzy stabilization problem for Markovian jump nonlinear systems were presented in [46], which were further improved in [47]. In addition, time delay is often encountered in control signal transmission channel [48,49,51]. When information on the size of the time delay is available, retarded controllers can achieve better performance than traditional controllers since not only the current but also the past control signals are used [52]. Moreover, on the basis of the T–S fuzzy model approach, the dissipative control problem for nonlinear Markovian jump systems has not been fully investigated so far. These motivate the present study. This paper is concerned with the dissipative control problem for a class of nonlinear Markovian jump systems via the T–S fuzzy model approach. We aim to design retarded controller such that the resulting closed-loop system is stochastically mean-square stable and strictly ðQ; R; SÞθdissipative. For this purpose, some sufficient conditions are established for the solvability of the problem based on LMI formulations combined with the Lyapunov– Krasovskii method. A retarded controller is constructed and two numerical examples are given to illustrate the effectiveness of the proposed method. It is worth pointing out that the dissipative control problem considered here includes the H ∞ control problem or passivity based control Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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problem as a special case. Furthermore, the restriction on the completely known knowledge of transition rates of Markov process is removed. In order to express the relationship among the transition rates, some useful slack matrices are given on the basis of investigating the properties that the sum of each row is zero in a transition rate matrix. In addition, a useful Lemma (see Lemma 2) and an argument Lyapunov–Krasovskii functional are introduced due to their potential capability of reducing the conservatism of the proposed method. Notation: Throughout this paper, for symmetric matrices X and Y, the notation X≥Y (respectively, X4Y) means that the matrix X−Y is positive semi-definite (respectively, positive definite); I is the identity matrix with appropriate dimension. The notation MT represents the transpose of the matrix M; Efg denotes the expectation operator with respect to some probability measure P; L2 ½0; ∞Þ is the space of square-integrable vector functions over ½0; ∞Þ; ∥  ∥ refers to the Euclidean vector norm; ∥  ∥2 stands for the usual L2 ½0; ∞Þ norm. Matrices, if not explicitly stated, are assumed to have compatible dimensions. The symbol n is used to denote a matrix which can be inferred by symmetry. For the integer l∈½1; k þ 2, el and e~ l represent the block entry matrices, for example, e2 ¼ ½0 I 0 ⋯ 0Tnðkþ2Þn and e~ 2 ¼ ½0 I 0 0 0 ⋯ 0Tnðkþ4Þn , respectively. SymfXg denotes the expression X þ X T .

2. System descriptions and definitions Consider a class of nonlinear Markovian jump systems, which can be described by the following fuzzy model ðΣÞ: Plant Rule i: IF θ1 ðtÞ is μi1 and θ2 ðtÞ is μi2 and … and θp ðtÞ is μip THEN x_ ðtÞ ¼ Ai ðsðtÞÞxðtÞ þ Bi ðsðtÞÞuðtÞ þ Di ðsðtÞÞωðtÞ; zðtÞ ¼ C i ðsðtÞÞxðtÞ þ C1i ðsðtÞÞuðtÞ þ C 2i ðsðtÞÞωðtÞ;

ð1Þ sð0Þ ¼ s0 ;

ð2Þ

where θj ðtÞ; μij ; i ¼ 1; …; r; j ¼ 1; …; p, are the premise variables and the fuzzy sets, respectively; r is the number of IF–THEN rules; xðtÞ∈Rn is the system state; uðtÞ∈Rm is the control input; ωðtÞ∈Rq is the noise signal which is assumed to be an arbitrary signal in L2 ½0; ∞Þ; zðtÞ∈Rp is the controlled output; fsðtÞg is a continuous-time Markovian process with right continuous trajectories and takes values in a finite set S ¼ f1; 2; …; N g with transition probability matrix Π≜fπ αβ g given by ( α≠β; π αβ Δ þ oðΔÞ; Prfsðt þ ΔÞ ¼ β∥sðtÞ ¼ αg ¼ 1 þ π αα Δ þ oðΔÞ; α ¼ β; where Δ40, limΔ-0 ðoðΔÞ=ΔÞ ¼ 0, and π αβ ≥0, for β≠α, is the transition rate from mode α at time t to mode β at time t þ Δ and π αα ¼ − ∑ π αβ :

ð3Þ

β∈S;β≠α

The set S comprises the operation modes of the system and for each possible value of sðtÞ ¼ α; α∈S. The transition rates described system in Eqs. (1) and (2) are considered to be partially available, that is, some elements in the transition rate matrix Π are unmeasurable. For instance, the transition rate matrix for system (1) and (2) with N modes may be Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

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expressed as 2

? 6π 6 21 6 Ψ ¼6 6 ? 6 ⋮ 4 ?

? π 22

? ?

⋯ ⋯

π 32 ⋮

? ⋮

⋯ ⋱

?

πN 3

⋯

3 π 1N ? 7 7 7 ? 7 7: ⋮ 7 5 π NN

For notational simplicity, ∀α∈S, we denote S αk ≜fβ : π αβ is knowng;

S αuk ≜fβ : π αβ is unknowng:

By fuzzy blending, the overall fuzzy model is inferred as follows: Further, if S αk a|, we set S αk ¼ {k α1 , kα2 ,y,kαm ,yN 1 } with k αm ¼ a. r

x_ ðtÞ ¼ ∑ hi ðθðtÞÞfAi;α xðtÞ þ Bi;α uðtÞ þ Di;α ωðtÞg;

ð4Þ

i¼1 r

zðtÞ ¼ ∑ hi ðθðtÞÞfCi;α xðtÞ þ C 1i;α uðtÞ þ C 2i;α ωðtÞg; i¼1

sð0Þ ¼ s0 ;

ð5Þ

where θðtÞ ¼ ½θ1 ðtÞ; …; θp ðtÞ, hi ðθðtÞÞ ¼ vi ðθðtÞÞ=∑ri ¼ 1 vi ðθðtÞÞ, vi ðθðtÞÞ ¼ ∏pj ¼ 1 μij ðθj ðtÞÞ, and μij ðθj ðtÞÞ is the grade of membership of θj ðtÞ in μij . Suppose vi ðθðtÞÞ≥0, i ¼ 1; 2; …; r; ∑ri ¼ 1 vi ðθðtÞÞ40 for all t. Therefore, hi ðθðtÞÞ≥0 for i ¼ 1; 2; …; r, and ∑ri ¼ 1 hi ðθðtÞÞ ¼ 1 for all t. In this paper, we consider the following fuzzy retarded feedback controller: Control Rule i: IF θ1 ðtÞ is μi1 and θ2 ðtÞ is μi2 and ⋯ and θp ðtÞ is μip THEN uðtÞ ¼ K 1i;α xðtÞ þ K 2i;α xðt−τÞ;

ð6Þ

where τ is a given constant; K 1i;α and K 2i;α ; i ¼ 1; 2; …; r, are controller gains to be designed, the controller can also be rewritten as r

uðtÞ ¼ ∑ hi ðθðtÞÞfK 1i;α xðtÞ þ K 2i;α xðt−τÞg: i¼1

Substituting this expression into Eqs. (4) and (5), we have the resulting closed-loop system ðΣ Þ: x_ ðtÞ ¼ Aα ðhÞxðtÞ þ Bα ðhÞxðt−τÞ þ Dα ðhÞωðtÞ;

ð7Þ

zðtÞ ¼ Cα ðhÞxðtÞ þ C1α ðhÞxðt−τÞ þ C2α ðhÞωðtÞ;

ð8Þ

where r

r

Aα ðhÞ ¼ ∑ ∑ hi ðθðtÞÞhj ðθðtÞÞfAi;α þ Bi;α K 1j;α g; i¼1j¼1 r r

Bα ðhÞ ¼ ∑ ∑ hi ðθðtÞÞhj ðθðtÞÞBi;α K 2j;α ; i¼1j¼1 r r

Cα ðhÞ ¼ ∑ ∑ hi ðθðtÞÞhj ðθðtÞÞfCi;α þ C1i;α K 1j;α g; i¼1j¼1 r r

C1α ðhÞ ¼ ∑ ∑ hi ðθðtÞÞhj ðθðtÞÞC 1i;α K 2j;α ; i¼1j¼1

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]] r

Dα ðhÞ ¼ ∑ hi ðθðtÞÞDi;α ; i¼1

5

r

C2α ðhÞ ¼ ∑ hi ðθðtÞÞC 2i;α : i¼1

Throughout the paper we shall use the following definitions. Definition 1 (Shen et al. [53]). The fuzzy Markovian jump system ðΣÞ with ωðtÞ ¼ 0 is said to be stochastically mean-square stable, if there exists a constant scalar Mðx0 ; s0 Þ40 such that the following condition holds for any initial condition ðx0 ; s0 Þ : Z ∞  T x ðtÞxðtÞ dtjx0 ; s0 ≤Mðx0 ; s0 Þ: ð9Þ E 0

Definition 2 (Hill and Moylan [24], Feng et al. [25]). Given a scalar θ40, real matrices Q ¼ QT and R ¼ RT and matrix S, the fuzzy Markovian jump system ðΣ Þ is said to be stochastically mean-square stable and strictly ðQ; R; SÞθdissipative, if the fuzzy Markovian jump system ðΣ Þ is stochastically mean-square stable when ωðtÞ ¼ 0 and for any ϝ≥0, under zero initial state, the following condition is satisfied: Ef〈z; Qz〉ϝ þ 2〈z; Sω〉ϝ þ 〈ω; Rω〉g≥θ〈ω; ω〉ϝ ; ð10Þ Rϝ T where the notation 〈z; Qz〉ϝ represents 0 z ðtÞQzðtÞ dt, and the other symbols are similarly defined. Remark 1. The left-hand side of inequality (10) is usually defined as the energy supply function of the system (1) and (2). As noted in [25], we also assume that Q≤0 and there exists a matrix Q1 such that −Q ¼ QT1 Q1 . The dissipative control problem under consideration here is more general than the H ∞ control and passivity-based control problems. In fact, if Q ¼ −I, S ¼ 0, and R ¼ θI þ γ 2 I, the expression in Eq. (10) becomes an H ∞ performance function; while if Q ¼ 0, S ¼ I, and R ¼ θI þ γI, then the expression in Eq. (10) degenerates into a passivity performance function. The fuzzy dissipative control problem to be dealt with is formulated as follows: for given matrices Q, S, R and scalars θ40, τ40, and the system (1)–(2), determine a fuzzy retarded state feedback controller in the form of Eq. (6) such that the resulting closed-loop system ðΣ Þ is stochastically mean-square stable and strictly ðQ; R; SÞθdissipative. Before proceeding further, we introduce the following lemmas, which play key roles in deriving our main results. Lemma 1 (Gu et al. [50]). For any constant matrix M∈Rnn , M ¼ M T 40, vector function ϖ : ½0; r-Rn such that the integrations in the following are well defined, then Z r  Z r T Z r ϖðsÞ dsÞ M ϖðsÞ ds ≤r ϖ T ðsÞMϖðsÞ ds: 0

0

0

Lemma 2. For any symmetric positive-definite matrices U ¼ ½U ab ∈Rðkþ2Þnðkþ2Þn for a≤b∈ ½1; k þ 2, and a given scalar k40, if there exists a vector function xðtÞ∈Rn such that the integrations concerned are well defined, then 2ξT1 ðtÞU ξ_ 1 ðtÞ ¼ 2ξT ðtÞΦT1 UΦ2 ξðtÞ;

ð11Þ

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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where

Z

f ðlÞ ¼

t

ðs−t þ τÞl xT ðsÞ ds;

t−τ T

ξ1 ðtÞ ¼ ½x ðtÞ f ð0Þ f ð1Þ ⋯ f ðkÞT ; ξðtÞ ¼ ½xT ðtÞ xT ðt−τÞ ωT ðtÞ f ð0Þ f ð1Þ ⋯ f ðkÞT ; Φ1 ¼ ½e1 0 0 e2 e3 ⋯ ekþ2 ; Φ2 ¼ ½Φ12 Φ22 e1 Dα ðhÞ −e3 ⋯ −kekþ2 0; k

Φ12 ¼ e1 Aα ðhÞ þ e2 þ ∑ τl elþ2 ; l¼1

Φ22 ¼ e1 Bα ðhÞ−e2 :

ξ1 ðtÞ and ξðtÞ, it is easy to see 3 0 0 0 0 ⋯ 0 0 0 I 0 ⋯ 07 7 7 0 0 0 I ⋯ 07 7ξðtÞ ¼ Φ1 ξðtÞ: 7 ⋮ ⋮ ⋮ ⋮ ⋱ ⋮5 0 0 0 0 0 ⋯ I

Proof. In light of 2 I 60 6 6 ξ1 ðtÞ ¼ 6 60 6 4⋮

ð12Þ

On the other hand, by some mathematical operations, we have Z t df ðlÞ l T ¼ τ x ðtÞ−l ðs−t þ τÞl−1 xT ðsÞ ds ¼ τl xT ðtÞ−lf ðl−1Þ: dt t−τ It is not difficult to obtain that 2 Aα ðhÞ Bα ðhÞ Dα ðhÞ 0 6 I −I 0 0 6 6 6 τI 0 0 −I ξ_ 1 ðtÞ ¼ 6 6 τ2 I 0 0 0 6 6 4 ⋮ ⋮ ⋮ ⋮ k τI 0 0 0 ¼ ½Φ12

0

⋯

0

⋯

0 −2I

⋯ ⋯

⋮ 0

⋱ ⋯

0

ð13Þ

3

07 7 7 07 7ξðtÞ 07 7 7 05 0

Φ22

e1 Dα ðhÞ −e3 ⋯ −kekþ2 0ξðtÞ: ð14Þ T T T _ This, together with Eq. (12), gives rise to 2ξ1 ðtÞU ξ 1 ðtÞ ¼ 2ξ ðtÞΦ1 UΦ2 ξðtÞ. The proof is now complete. □ Lemma 3 (Xu et al. [32]). Let A, B, and M40 be real matrices of appropriate dimensions. Then AB þ ðABÞT ≤AMAT þ BT M −1 B: 3. Dissipativity analysis In this section, we will present a delay-dependent sufficient condition for the fuzzy Markovian jump system ðΣ Þ, which ensures that the system ðΣ Þ is stochastically mean-square stable and strictly ðQ; R; SÞθdissipative. Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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Theorem 1. Given scalars θ40, τ40, an integer k40, and matrices Q ¼ QT , S, R ¼ RT , the system ðΣ Þ is stochastically mean-square stable and strictly ðQ; R; SÞθdissipative, if there exist matrices V α 40, V40, U ¼ ½U ab ∈Rðkþ2Þnðkþ2Þn 40 for a≤b∈½1; k þ 2, W l 40 for l∈½1; k, Z∈R2n2n 40, Pα , T cα ðhÞ ¼ ∑ri ¼ 1 hi ðθðtÞÞT ci;α for c ¼ 1; 2; 3, such that the following matrix inequalities hold for each α∈S: Ξ 0α ðhÞ ¼ Symf~e 1 PTα Aα ðhÞ þ ΦT1 UΦ2 −~e 3 SCα ðhÞg þ Φ3 þ τ2 Φ4 ZΦT4 −Φ5 ZΦT5 −CTα ðhÞQCα ðhÞ−~e 1 ∑ π αβ T 2α ðhÞ~e T4 −~e 4 ∑ π αβ T 2α ðhÞ~e T1 o0; β∈S k

ð15Þ

β∈S k

Ξ 1α ðhÞ ¼ ∑β∈Sk π αβ V β −V−∑β∈S k π αβ T 3α ðhÞ≤0; # Pβ −τT 1α ðhÞ −T 2α ðhÞ Ξ 2α ðhÞ ¼ ≤0; −T 2α ðhÞ V β −T 3α ðhÞ

ð16Þ

"

" Ξ 3α ðhÞ ¼

τT 1α ðhÞ−Pβ

T 2α ðhÞ

T 2α ðhÞ

T 3α ðhÞ−V β

β∈S αuk ; β≠α;

ð17Þ

β∈S αuk ; β ¼ α;

ð18Þ

# ≤0;

where Aα ðhÞ ¼ Aα ðhÞ~e T1 þ Bα ðhÞ~e T2 þ Dα ðhÞ~e T3 ; Cα ðhÞ ¼ Cα ðhÞ~e T1 þ C1α ðhÞ~e T2 þ C2α ðhÞ~e T3 ; Φ3 ¼ diagfΦ13 ; −V α ; −ðR−θIÞ; τ−1 Ξ 1α ðhÞ; −3W 1 ; …; −ð2k þ 1ÞW k g; k

Φ13 ¼ ∑ π αβ Pβ þ τV þ V α þ ∑ τ2lþ2 W l −τ ∑ π αβ T 1α ðhÞ; l¼1

β∈S k

β∈S k

Φ5 ¼ ½~e 4 e~ 1 −~e 2 :

Φ4 ¼ ½~e 1 ATα ðhÞ;

Proof. Define a new process fðxt ; αÞ; t≥0g by xt ¼ xðt þ sÞ; −τ≤s≤0. Choose a Markovian switched Lyapunov functional candidate as 4

Vðxt ; α; tÞ ¼ ∑ V l ðxt ; α; tÞ;

ð19Þ

l¼1

where Z V 1 ðxt ; α; tÞ ¼ xT ðtÞPα xðtÞ þ Z V 2 ðxt ; α; tÞ ¼

0

Z

t

xT ðsÞV α xðsÞ ds;

t−τ t

xT ðϑÞVxðϑÞ dϑ ds; Z t k ðs−t þ τÞ2lþ1 xT ðsÞW l xðsÞ ds; V 3 ðxt ; α; tÞ ¼ ξT1 ðtÞUξ1 ðtÞ þ τ ∑ l ¼ 1 t−τ Z 0Z t T V 4 ðxt ; α; tÞ ¼ τ ξ2 ðϑÞZξ2 ðϑÞ dϑ ds; ξ2 ðtÞ ¼ ½xT ðtÞ x_ T ðtÞT ; −τ

−τ

tþs

tþs

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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and ξ1 ðtÞ is shown in Eq. (11). Then, we define the weak infinitesimal operator L as in [54], it can be deduced that for each α∈S, Z t T LV 1 ðxt ; α; tÞ ¼ ∑ π αβ ðx ðtÞPβ xðtÞ þ xT ðsÞV β xðsÞ dsÞ β∈S

t−τ

þx ðtÞV α xðtÞ−x ðt−τÞV α xðt−τÞ þ 2xT ðtÞPα Aα ðhÞξðtÞ: T

T

Similarly, we have

Z

LV 2 ðxt ; α; tÞ ¼ τxT ðtÞVxðtÞ−

t

xT ðsÞVxðsÞ ds;

ð20Þ

ð21Þ

t−τ k

k

LV 3 ðxt ; α; tÞ≤2ξT1 ðtÞU ξ_ 1 ðtÞ þ xT ðtÞ ∑ τ2lþ2 W l xðtÞ− ∑ fð2l þ 1Þf ðlÞW l f T ðlÞg l¼1

l¼1

k

k

l¼1

l¼1

¼ 2ξT ðtÞΦT1 UΦ2 ξðtÞ þ xT ðtÞ ∑ τ2lþ2 W l xðtÞ− ∑ fð2l þ 1Þf ðlÞW l f T ðlÞg; and

Z LV 4 ðxt ; α; tÞ ¼ τ2 ξT2 ðtÞZξ2 ðtÞ−τ Z ≤τ2 ξT2 ðtÞZξ2 ðtÞ−

Note that

t

t−τ t t−τ

ξT2 ðsÞZξ2 ðsÞ ds

ξT2 ðsÞ dsZ

Z

t

ξ2 ðsÞ ds:

t

ð23Þ

t−τ

# # " xðtÞ xðtÞ ξ2 ðtÞ ¼ ¼ x_ ðtÞ Aα ðhÞxðtÞ þ Bα ðhÞxðt−τÞ þ Dα ðhÞωðtÞ " T # e~ 1 ¼ ξðtÞ ¼ ΦT4 ξðtÞ; Aα ðhÞ

Z

ð22Þ

"

Z ξ2 ðsÞ ds ¼

t−τ

t t−τ

"

" Rt # " T # # xðsÞ e~ 4 t−τ xðsÞ ds ¼ T T ξðtÞ: ds ¼ x_ ðsÞ e~ 1 −~e 2 xðtÞ−xðt−τÞ

ð24Þ

ð25Þ

Then, it follows from Eqs. (23) to (25) that LV 4 ðxt ; α; tÞ≤ξT ðtÞðτ2 Φ4 ZΦT4 −Φ5 ZΦT5 ÞξðtÞ:

ð26Þ

Due to the fact that the sum of all the elements in every row of Π is zero, we have " # Z t T 1α ðhÞ T 2α ðhÞ T − ∑ π αβ ξ3 ðt; sÞ ξ ðt; sÞ ds ¼ 0; T 2α ðhÞ T 3α ðhÞ 3 t−τ β∈S

ð27Þ

where ξ3 ðt; sÞ ¼ ½xT ðtÞ xT ðsÞT , which means that for each α∈S Z t 0 ¼ −τ ∑ π αβ xT ðtÞT 1α ðhÞxðtÞ− ∑ π αβ xT ðtÞT 2α ðhÞ xðsÞ ds β∈S

β∈S

t−τ

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Z − ∑ π αβ β∈S

t

Z

t

xT ðsÞ dsT 2α ðhÞxðtÞ− ∑ π αβ β∈S

t−τ

xT ðsÞT 3α ðhÞxðsÞ ds:

9

ð28Þ

t−τ

From Eqs. (20) to (28), it can be concluded that LVðxt ; α; tÞ ¼ LV 1 ðxt ; α; tÞ þ LV 2 ðxt ; α; tÞ þ LV 3 ðxt ; α; tÞ þ LV 4 ðxt ; α; tÞ ¼ τxT ðtÞVxðtÞ þ xT ðtÞV α xðtÞ−xT ðt−τÞV α xðt−τÞ þ 2xT ðtÞPα Aα ðhÞξðtÞ Z t Z t T T þ ∑ π αβ ðx ðtÞPβ xðtÞ þ x ðsÞV β xðsÞ dsÞ− xT ðsÞVxðsÞ ds β∈S

t−τ

t−d k

k

l¼1

l¼1

þ2ξT ðtÞΦT1 UΦ2 ξðtÞ þ xT ðtÞ ∑ τ2lþ2 W l xðtÞ− ∑ fð2l þ 1Þf ðlÞW l f T ðlÞg þτ2 Φ4 ZΦT4 −Φ5 ZΦT5 −τ ∑ π αβ xT ðtÞT 1α ðhÞxðtÞ Z − ∑ π αβ xT ðtÞT 2α ðhÞ β∈S

Z

− ∑ π αβ β∈S

β∈S

t

xðsÞ ds t−τ

t

Z

xT ðsÞ dsT 2α ðhÞxðtÞ− ∑ π αβ β∈S

t−τ

t−τ

xT ðsÞT 3α ðhÞxðsÞ ds:

ð29Þ

t−τ

It follows from Ξ 1α ðhÞ≤0 and Lemma 1 that 0 1T Z Z t Z t T @ A τ x ðsÞΞ 1α ðhÞxðsÞ ds≤ xðsÞ ds Ξ 1α ðhÞ t−τ

t

t

 xðsÞ ds :

t−τ

Substituting this expression into Eq. (29) yields Z LVðxt ; α; tÞ≤ξT ðtÞΞ 4α ðhÞξðtÞ þ Z −

∑

β∈S αuk ;β ¼ α

π αβ

t

t−τ

∑

β∈S αuk ;β≠α

π αβ

t

t−τ

ξT3 ðt; sÞΞ 2α ðhÞξ3 ðt; sÞ ds

ξT3 ðt; sÞΞ 3α ðhÞξ3 ðt; sÞ ds;

ð30Þ

where ~ 3 þ τ2 Φ4 ZΦT4 Ξ 4α ðhÞ ¼ Symf~e 1 PTα Aα ðhÞ þ ΦT1 UΦ2 g þ Φ −Φ5 ZΦT5 −~e 1 ∑ π αβ T 2α ðhÞ~e T4 −~e 4 ∑ π αβ T 2α ðhÞ~e T1 ; β∈S k

β∈S k

~ 3 ¼ diagfΦ13 ; −V α ; 0; τ−1 Ξ 1α ðhÞ; −3W 1 ; …; −ð2k þ 1ÞW k g: Φ Noting −Q40, we have Ξ 4α ðhÞoΞ 0α ðhÞo0 when ωðtÞ ¼ 0. Moreover, as aforesaid, one can see π αβ ≥0, for β≠α, and π αα ¼ −∑β∈S;β≠α π αβ , and consequently it can be deduced from Eqs. (15)–(18) and (30) that LVðxt ; α; tÞoδ∥xðtÞ∥2 for a sufficiently small δ40 and xðtÞ≠0. Following a similar line in the proof of Theorem 1 in [53], it can be shown that Z ∞  E xðt; xð0Þ; sð0ÞÞT xðt; xð0Þ; sð0ÞÞ dtjxð0Þ; sð0Þ o∞; 0

that is, system ðΣ Þ is stochastically mean-square stable. Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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10

In what follows, we will prove that system ðΣ Þ is strictly ðQ; R; SÞθdissipative. To this end, we introduce the following performance index: J ϝ ¼ Ef−〈z; Qz〉ϝ −2〈z; Sω〉ϝ −〈ω; Rω〉ϝ þ θ〈ω; ω〉ϝ g Z ϝ  ½−zT ðtÞQzðtÞ−2zT ðtÞSωðtÞ−ωT ðtÞRωðtÞ þ θωT ðtÞωðtÞ dt ; ¼E

ð31Þ

0

where ϝ40. It can be shown from Eqs. (15) to (18) that for each α∈S, LVðxt ; α; tÞ−zT ðtÞQzðtÞ−2zT ðtÞSωðtÞ−ωT ðtÞRωðtÞ þ θωT ðtÞωðtÞ Z t ξT3 ðt; sÞΞ 2α ðhÞξ3 ðt; sÞ ds ≤ξT ðtÞΞ 0α ðhÞξðtÞ þ ∑ π αβ t−τ β∈S αuk ;β≠α t ξT3 ðt; sÞΞ 3α ðhÞξ3 ðt; sÞ t−τ

Z −

∑

β∈S αuk ;β ¼ α

π αβ

ð32Þ

dso0:

Considering this and Eq. (31), we have Z ϝ  T T T T J ϝ ≤E ½−z ðtÞQzðtÞ−2z ðtÞSωðtÞ−ω ðtÞRωðtÞ þ θω ðtÞωðtÞ þ LVðxt ; α; tÞ dt o0; 0

ð33Þ for any non-zero ωðtÞ∈L2 ½0, ∞Þ. This implies Z ϝ  Z E ½zT ðtÞQzðtÞ þ 2zT ðtÞSωðtÞ þ ωT ðtÞRωðtÞ dt 4θ 0

ϝ

 ωðtÞT ωðtÞ dt ;

ð34Þ

0

and then the condition (10) is satisfied. Hence, the system ðΣ Þ is stochastically stable and strictly ðQ; R; SÞθdissipative according to Definition 2. The proof of this theorem is now complete. □ 4. Fuzzy retarded feedback controller design We are now ready to present a solution to the fuzzy retarded feedback controller design for the nonlinear Markovian jump systems ðΣÞ. Theorem 2. Consider the system ðΣÞ and let scalars θ40, τ40, an integer k40, matrices Q ¼ QT , S, R ¼ RT be given. Then there exists an admissible retarded feedback controller in the form of Eq. (6) such that the resulting closed-loop system is stochastically stable and strictly ðQ; R; SÞθdissipative, if there exist matrices X40; V~ α 40, V~ 40, U ¼ ½U ab ∈ ~ l 40 for l∈½1; k, Z~ ∈R2n2n 40, P~ α , Y 1j;α , Y 2j;α , T 1j;α , Rðkþ2Þnðkþ2Þn 40 for a≤b∈½1; k þ 2, W T 2j;α , T 3j;α such that the following LMIs hold for each α∈S and j∈½1; r : ( " 0 # Ωii;α ðϱÞ Ω1ii;α ðϱÞ 1; α∈S αk ; β∈S αk ; 1≤i≤r; ð35Þ o0; ϱ ¼ 2; α∈S αuk ; β∈S αk ; n Ω2ii;α "

Ω0ij;α ðϱÞ þ Ω0ji;α ðϱÞ Ω3ij;α ðϱÞ n

Ω4ij;α ðϱÞ

#

( o0;

ϱ¼

Ξ 1j;α ¼ ∑β∈S k π αβ V~ β −V~ −∑β∈S k π αβ T~ 3j;α ≤0;

1; α∈S αk ; β∈S αk ; 2; α∈S αuk ; β∈S αk ; 1≤j≤r;

1≤ioj≤r;

ð36Þ ð37Þ

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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2

−T~ 2j;α V~ β −T~ 3j;α

−τT~ 1j;α 6 −T~ Ξ 2j;α ¼ 4 2j;α X

X

"

τT~ 1j;α þ P~ β −2X Ξ 3j;α ¼ T~ 2j;α where

≤0; T~ 3j;α −V~ β

ϒ 11 ðϱÞ ϒ 12 6 n ϒ 22 6 6 0 6 n n Ωij;α ðϱÞ ¼ 6 6 n n 4 2 6 Ω1ii;α ðϱÞ ¼ 4

Θ1 ðϱÞ~e 1

1≤j≤r; β∈S αuk ; β≠α;

ϒ 13 ϒ 23

ϒ 13 ϒ 24

ϒ 33

0

n

ϒ 44

n

n

n

ΘT2

ΘT3

ΘT4ii

0 0 0 0 0 0 pffiffiffi T 2 pffiffiffi 2Θ 2 2Θ1 ðϱÞ~e 1 6 Ω3ij;α ðϱÞ ¼ 4 0 0 0 0

ð38Þ

#

T~ 2j;α

2

n

3

0 7 5≤0; ~ −P β

0

11

1≤j≤r; β∈S αuk ; β ¼ α;

ð39Þ

3 ϒ 15 ϒ 25 7 7 7 ϒ 35 7 7; 0 7 5 ϒ 55 ΘT4ii

3

7 0 0 5; 0 0 pffiffiffi T 2Θ3 ΘT4ij 0 0

ΘT4ji

ΘT4ij

0 0

0 0

0 0

ΘT4ji

3

7 0 5; 0

Ω2ii;α ¼ diagfΘ5 ðϱÞ; −P~ α ; −X; −P~ α ; −Xg; Ω4ij;α ðϱÞ ¼ diagfΘ5 ðϱÞ; −P~ α ; −X; −P~ α ; −P~ α ; −X; −Xg; with ( ϒ 11 ðϱÞ ¼

ϒ 11 −π αα ðP~ α −2XÞ; ϒ 11 ;

ϱ ¼ 1; ϱ ¼ 2;

( U~ ab ¼

k

U ab ;

a≤b;

T U ba ;

a4b;

kþ2

~ l −τ ∑ π αβ T~ 1j;α þ ∑ τl−2 SymfU~ 1l g þ τ2 Z~ 11 −Z~ 22 ; ϒ 11 ¼ V~ α þ τV~ þ ∑ τ2lþ2 W β∈S αk

l¼1

ϒ 12 ¼ −U~ 12 þ Z~ 22 ;

ϒ 22 ¼ −V~ α −Z~ 22 ;

ϒ 13 ¼ −XC Ti;α S−Y T1j;α C T1i;α S;

ϒ 23 ¼ −Y T2j;α C T1i;α S;

ϒ 33 ¼ −ðR−θIÞ−SymfSC 2i;α g; " k

ϒ 14 ¼ ∑

l¼2

l¼2

T τl−2 U~ 2l −U~ 13 −

∑

β∈S αk

T ϒ 24 ¼ ½Z~ 12 −U~ 22 −U~ 23 ⋯ −U~ 2ðkþ1Þ −U~ 2ðkþ2Þ ; T π αβ T~ 2j;α −Z~ 12

T U~ 3l −2U~ 14

⋯

T U~ ðkþ1Þl −kU~ 1ðkþ2Þ

T U~ ðkþ2Þl

# ;

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

12

2 6 6 6 6 ϒ 44 ¼ 6 6 6 4

−1 ~ ϒ ð1;1Þ 44 þ τ Ξ 1j;α −Z 11

ϒ ð1;2Þ 44

⋯

ϒ ð1;kÞ 44

n

~ ϒ ð2;2Þ 44 −3W 1

n

n

⋯ ⋱

n

n

n

ϒ ð2;kÞ 44 ⋮ ðk;kÞ ~ k−1 ϒ 44 −ð2k−1ÞW

n

n

n

n

−U~ 3ðkþ2Þ −2U~ 4ðkþ2Þ

3

7 7 7 7 7; ⋮ 7 −k U~ ðkþ2Þðkþ2Þ ; m 7 5 ~k −ð2k þ 1ÞW

~ ~ ϒ ða;bÞ 44 ¼ −bU ðaþ1Þðbþ2Þ −aU ðaþ2Þðbþ1Þ ; ϒ 15 ¼ ½ðC i;α X þ C1i;α Y 1j;α ÞT τðAi;α X þ Bi;α Y 1j;α ÞT ; ϒ 25 ¼ ½ðC 1i;α Y 2j;α ÞT τðBi;α Y 2j;α ÞT ;

ϒ 35 ¼ ½C T2i;α τDTi;α ;

ϒ 55 ¼ diagf−ð−QÞ−1 ; Z~ 22 −2Xg; i 8h pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi > < π αka1 X ⋯ π α;kam−1 X π α;kamþ1 X ⋯ π αN 1 X ; i h Θ1 ðϱÞ ¼ pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffia > : π αk1 X ⋯ π α;kam X ⋯ π αN 1 X ;

ϱ ¼ 1; ϱ ¼ 2;

k

Θ3 ¼ e~ T1 ðU~ 11 þ τ2 Z~ 12 Þ þ ∑ e~ Tðlþ4Þ U~ 1ðlþ2Þ ;

Θ2 ¼ X e~ T1 ;

l¼0

Θ4ij ¼ ðAi;α X þ Bi;α Y 1j;α Þ~e T1 þ Bi;α Y 2j;α e~ T2 þ Di;α e~ T3 ; i 8h > < −P~ ka1 ⋯ −P~ kam−1 −P~ kamþ1 ⋯ −P~ N 1 ; i h Θ5 ðϱÞ ¼ > : −P~ ka1 ⋯ −P~ kam ⋯ −P~ N 1 ;

ϱ ¼ 1; ϱ ¼ 2:

In this case, the desired retarded state feedback controller gains can be given as K 1j;α ¼ Y 1j;α X −1 ;

K 2j;α ¼ Y 2j;α X −1 ; 1≤j≤r; 1≤α≤N :

ð40Þ

Proof. Defining new matrix variables for any integer a; b∈½1; k þ 2; l∈½1; k, −1 Pα ¼ P~ α ;

Z 11 ¼ X −1 Z~ 11 X −1 ;

U ab ¼ X −1 U~ ab X −1 ;

Z 12 ¼ X −1 Z~ 12 X −1 ;

V ¼ X −1 V~ X −1 ;

Z 22 ¼ X −1 Z~ 22 X −1 ;

V α ¼ X −1 V~ α X −1 ;

~ l X −1 : W l ¼ X −1 W

ð41Þ ð42Þ

For hi ðθðtÞÞ≥0 for i ¼ 1; 2; …; r, and ∑ri ¼ 1 hi ðθðtÞÞ ¼ 1 for all t, once the condition Ξ 1j;α ≤0; 1≤j≤r, is verified, then Ξ 1α ðhÞ ¼ ∑β∈Sk π αβ V β −V−∑β∈Sk π αβ T 3α ðhÞ r

¼ ∑ hj ðθðtÞÞfX −1 Ξ 1j;α X −1 g≤0;

ð43Þ

j¼1

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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13

is assured for all j∈½1; r. On the other hand, from Eq. (38) and Schur's complement, we also have " Ξ 2α ðhÞ ¼

Pβ −τT 1α ðhÞ

−T 2α ðhÞ

−T 2α ðhÞ

V β −T 3α ðhÞ

#

(" ¼ ∑rj ¼ 1 hj ðθðtÞÞdiagfX −1 ; X −1 g

−τT~ 1j;α −T~ 2j;α

−T~ 2j;α V~ β −T~ 3j;α

#

    X −1 X ~ P þ diagfX −1 ; X −1 g 0 β 0 1≤j≤r; β∈S αuk ; β≠α;

≤0;

ð44Þ

and " Ξ 3α ðhÞ ¼

#

τT 1α ðhÞ−Pβ

T 2α ðhÞ

T 2α ðhÞ

T 3α ðhÞ−V β 2

−1 τT~ 1j;α −X P~ β X −1 −1 4 ¼ ∑ hj ðθðtÞÞdiagfX ; X g T~ 2j;α j¼1

T~ 2j;α

r

T~ 3j;α −V~ β

3 5diagfX −1 ; X −1 g

r

≤ ∑ hj ðθðtÞÞdiagfX −1 ; X −1 gΞ 3j;α diagfX −1 ; X −1 g≤0;

ð45Þ

j¼1

where the inequality in Eq. (45) holds due to the fact that for any positive definite matrices X and P~ β , −1

−X P~ β X≤P~ β −2X:

ð46Þ

Next, we will prove the LMIs in Eq. (15) hold according to Theorem 2. First, the set β∈S αk is divided into two parts, that is, α∈S αk ; β∈S αk , and α∈S αuk ; β∈S αk . When α∈S αk ; β∈S αk , noting that π αα o0 and −1 P~ α 40, it can be readily verified that −1 −1 π αα X P~ α X ¼ −ð−π αα ÞX P~ α X≤ð−π αα ÞðP~ α −2XÞ:

ð47Þ

Substituting this expression into Ω0ij;α ð1Þ, yields 2 6 6 6 0 6 ~ Ω ij;α ð1Þ ¼ 6 6 6 4

−1

ϒ 11 þ π αα X P~ α X

ϒ 12

ϒ 13

ϒ 13

n

ϒ 22

n

n

ϒ 23 ϒ 33

ϒ 24 0

n

n

n

ϒ 44

n

n

n

n

ϒ 15

3

7 ϒ 25 7 7 7 ϒ 35 7≤Ω0ij;α ð1Þ: 7 0 7 5 ϒ 55

ð48Þ

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

14

By employing Lemma 3, and using the Schur complement formula to Eqs. (35) and (36) when ϱ ¼ 1, we have 2 ~ 0 ð1Þ≤Ω ~ 0 ð1Þ þ Ω ii;α ii;α

∑

α≠β;β∈S αk

e~ 1

3

2

e~ 1

2

3T

e~ 1 X

3

2

e~ 1 X

3T

6 7 −1 6 7 6 7 −1 6 7 π αβ 4 0 5X P~ β X 4 0 5 þ 4 0 5P~ α 4 0 5 0 0 0 0

2

3 2 T 3T 2 T 3 2 T 3T Θ3 ΘT4ii Θ4ii Θ3 6 7 −1 6 7 6 7 6 7 þ4 0 5ðP~ α þ X −1 Þ4 0 5 þ 4 0 5X −1 4 0 5 o0; 1≤i≤r; 0 0 0 0

and

ð49Þ

2

~ 0 ð1Þ þ Ω ~ 0 ð1Þ≤Ω ~ 0 ð1Þ þ Ω ~ 0 ð1Þ þ 2 Ω ij;α ji;α ij;α ij;α 2

3

∑

α≠β;β∈S αk

2

3T

3 2 3T e~ 1 e~ 1 6 7 ~ −1 6 7 π αβ 4 0 5X P β X 4 0 5 2

0 3

2

3T

0

e~ 1 X e~ 1 X 6 7 ~ −1 6 7 6 7 6 7 þ24 0 5P α 4 0 5 þ 24 0 5X −1 4 0 5 0 0 0 0 2 T 3 2 T 3T 2 T 3 2 T 3T Θ4ij Θ4ji Θ4ij Θ4ji 6 7 −1 6 7 6 7 −1 6 7 þ4 0 5P~ α 4 0 5 þ 4 0 5P~ α 4 0 5 2

0 ΘT4ij

3

2

0 ΘT4ij

3T

2

ΘT3

0 ΘT4ji

3

2

ΘT3

0 ΘT4ji

3T

6 6 6 6 7 7 7 7 þ4 0 5X −1 4 0 5 þ 4 0 5X −1 4 0 5 o0; 0 0 0 0

1≤ioj≤r: ð50Þ

Thus, noting the parameters in Eqs. (40)–(42) and denoting J 4 ¼ diagfX; X; I; X; X; …; X; I; Ig, it is easy seen from Eqs. (49) and (50) that when α∈S αk ; β∈S αk , r

s−1

~ 0 ð1Þ þ ∑ J T4 Ξ 0α ðhÞJ 4 ¼ ∑ h2i ðθðtÞÞΩ ii;α i¼1

s

~ 0 ð1Þ þ Ω ~ 0 ð1ÞÞo0; ∑ hi ðθðtÞÞhj ðθðtÞÞðΩ ij;α ji;α

ð51Þ

i ¼ 1 j ¼ iþ1

When α∈S αuk , β∈S αk , which means that the π αα is unknown. In the case, we need to replace ϒ 11 ð1Þ, Θ1 ð1Þ, Θ5 ð1Þ, with ϒ 11 ð2Þ, Θ1 ð2Þ, Θ5 ð2Þ, and following a similar line as the proof of Eq. (51) in the case when α∈S αk , β∈S αk , we may also readily obtain that Eq. (51) is satisfied when α∈S αuk , β∈S αk . Thus, according to Theorem 1, we can conclude that the resulting closed-loop system is stochastically stable and strictly ðQ; R; SÞθdissipative. This completes the proof. □ Remark 2. In terms of coupled LMIs, Theorem 2 provides a sufficient condition for the existence of retarded feedback controllers guaranteeing that the resulting closed-loop system is stochastically stable and strictly ðQ; R; SÞθdissipative. The given condition is dependent on the time delay τ and dissipation parameters Q, R, S, θ, simultaneously. This means that the size of Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

15

time delay τ may effect the values of dissipation parameters Q, R, S, θ. More specifically, it may be verified that the maximum allowable value of θ is increasing as τ is decreasing for fixed parameters Q, R, S, k. Remark 3. If there is only one unknown element in the αth row of transition rate matrix Ψ , then according to Eq. (3), one can readily calculate all elements in the αth row of Ψ , i.e., when S ¼ S αk ; S αuk ¼ ∅. Therefore, the transition rate case considered here includes the completely known transition rate case (the usual assumption case) as a special case. Besides, Theorems 1 and 2 provide sufficient conditions to ensure the fuzzy Markovian jump system in Eqs. (7) and (8) to be stochastically mean-square stable and strictly ðQ; R; SÞθdissipative. It is noted that the augmented Markovian switched Lyapunov functional in Eq. (19) is proposed to derive delaydependent conditions. In view of Lemma 2, it can be verified that the computational complexity may increase as the integer k increases, although the conservatism may be reduced simultaneously. In this case, one can choose an appropriate k in accordance with different practical conditions. These illustrate the operability of the proposed method. 5. Numerical examples In this section, we give two examples to illustrate the usefulness of the presented method. Example 1. Consider the fuzzy Markovian jump system in Eqs. (1) and (2) with two plant rules ðr ¼ 2Þ and four operation modes ðα ¼ 1; 2; 3; 4Þ: 2

x_ ðtÞ ¼ ∑ hi ðθðtÞÞfAi;α xðtÞ þ Bi;α uðtÞ þ Di;α ωðtÞg;

ð52Þ

i¼1 2

zðtÞ ¼ ∑ hi ðθðtÞÞfC i;α xðtÞ þ C 1i;α uðtÞ þ C2i;α ωðtÞg; i¼1

and the following parameters   0 −1:1 ; A1;1 ¼ −1:2 0:9   0 −0:6 A1;4 ¼ ; −0:7 0:5   0 −0:7 A2;3 ¼ ; −0:4 0:6

are borrowed from [47]:   0 −0:8 A1;2 ¼ ; −1:1 0:9   0 −0:7 A2;1 ¼ ; −0:9 0:9   0 −0:6 A2;4 ¼ ; −0:7 0:5

sð0Þ ¼ s0;

 A1;3 ¼  A2;2 ¼

ð53Þ

0

−0:7

−0:4

1:2

0

−0:8

−0:9

0:8

 ;  ;

Table 1 Comparisons of minimum H ∞ performance level γ min for different π 43 . π 43

π 43 ¼ 0:28

π 43 ¼ 0:3

π 43 ¼ 0:32

γ min by [47] γ min by Theorem 2

3.6390 2.5745

3.6390 2.5778

3.6390 2.5780

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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 B1;1 ¼ B2;1 ¼ B1;4 ¼ B2;4 ¼

0:5



0   0:3

 ;

B1;2 ¼ B2;2 ¼

0:3



 ;

0   0:5

B1;3 ¼ B2;3 ¼

0:2



; 0   0:15

; D1;1 ¼ D2;1 ¼ ; D1;2 ¼ D2;2 ¼ ; 0 0:1 0       0 0:2 0 T D1;3 ¼ D2;3 ¼ ; D1;4 ¼ D2;4 ¼ ; C1;1 ¼ C2;1 ¼ ; 0:4 0:3 −0:1 " "   0:1 T 0 T 0:1 T  ; C 1;3 ¼ C 2;3 ¼  ; C 1;4 ¼ C 2;4 ¼ ; C 1;2 ¼ C2;2 ¼ 0 0:1 0

C ll;1 ¼ 0:1;

C ll;2 ¼ 0:3;

Cll;3 ¼ −0:2;

C ll;4 ¼ 0:4;

l ¼ 1; 2;

and the transition rates matrix is partly known as follows: 2 3 −0:5 ? 0:3 ? 6 ? −0:6 ? 0:3 7 6 7 6 7: π 32 ? ? 5 4 0:2 ? ? π 43 ? In [47], by the LMIs conditions in Theorem 4, the minimum H ∞ performance level is obtained as γ min ¼ 3:6390 for 0:295≤π 32 ≤0:31 and 0:28≤π 43 ≤0:32. To indicate the less conservatism of the result in Theorem 2 in this paper, we assume π 32 ¼ 0:3, Q ¼ −1; S ¼ 0; θ ¼ 0; R ¼ θI þ γ 2 I, τ ¼ 1; k ¼ 1. The comparison results of the minimum H ∞ performance level γ min from [47] and our results are tabulated in Table 1 for different values of π 43 . It is shown that our method generally produces less conservative results than that in [47] from Table 1. Example 2. Consider the fuzzy Markovian jump system in Eqs. (52) and (53), where     1 1 h1 ðθðtÞÞ ¼ 1−  ; 1 þ expð−3ðθðtÞ−0:5πÞÞ 1 þ expð−3ðθðtÞ þ 0:5πÞÞ h2 ðθðtÞÞ ¼ 1−h1 ðθðtÞÞ; 

0:6α−1:8 A1;α ¼ −0:3908 A11 2;1 ¼ −1:3;

θðtÞ ¼ x2 ðtÞ−10 11x1 ðtÞ;

 0:1α þ 0:0217 ; −0:1α−0:4618

A11 2;2 ¼ −1;

" A2;α ¼

A11 2;α

0:202 þ 0:01α

−0:4904

−0:5687

# ;

11 A11 2;3 ¼ A2;4 ¼ 0 for α ¼ 1; 2; 3; 4

Table 2 The maximum allowable value of θ for different k and τ. Cases

τ ¼ 0:3

τ ¼ 0:5

τ ¼ 0:7

τ¼1

k¼ 1 k¼ 2 k¼ 3

2.4574 2.4860 2.5012

2.3977 2.3411 2.4046

2.2104 2.2837 2.3155

1.6702 1.7081 1.7329

Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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and other parameters are given in Example 1. Now, we suppose Q ¼ −1, S ¼ 1, R ¼ 4, and the transition rates matrix is partly known as follows: 2

−1:7 6 ? 6 6 4 0:1

0:5 ?

0:8 0:4

1

−2:7

?

?

0:8

3 0:4 0:6 7 7 7: 1:6 5

ð54Þ

−10:8

System modes

Then, solving LMIs (35)–(39) in Theorem 2 by using the Matlab LMI Toolbox, the maximum allowable value of θ for a fixed ρ ¼ 0:9, different k and time delay τ can be computed, as shown in Table 2. It is easily seen from Table 2 that the larger time delay τ we set, the smaller maximum allowable value of θ for different k; moreover, the maximum allowable value of θ is increasing as k is increasing for a fixed τ. In other words, one can get better results by choosing an appropriate k. Hence, according to the aforementioned discussion, it is reasonable to study these parameters ðθ; k; τÞ together in some cases, and it is also significant to investigate these values as selection problem. 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

5

10

15

20

25 t/s

30

35

40

45

50

Fig. 1. System modes' evolutions sðtÞ.

1

x 1035 x1 x2

State responses

0.5 0 −0.5 −1 −1.5 −2 −2.5 −3

0

5

10

15

20

25 t/s

30

35

40

45

50

Fig. 2. State responses of open-loop system. Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

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Furthermore, we suppose Q ¼ −1; S ¼ 1; R ¼ 4; τ ¼ 0:1; k ¼ 1. Then it can be verified that there is a set of feasible solutions to LMIs (35)–(39) in Theorem 2, and a desired controller in the form of Eq. (6) with parameters as K 11;1 ¼ ½3:2391 0:5153;

K 11;2 ¼ ½1:2378 0:3761;

K 11;3 ¼ ½0:7644 0:8215;

K 11;4 ¼ ½0:7884 0:1626;

K 12;1 ¼ ½3:2719 0:5019;

K 12;2 ¼ ½1:9494 0:2805;

K 12;3 ¼ ½0:5891 0:9141;

K 12;4 ¼ ½−1:0040 0:3467;

K 21;1 ¼ ½−0:1121 0:9137;

K 21;2 ¼ ½2:3801 −1:2212;

K 21;3 ¼ ½−0:3934 0:2018;

K 21;4 ¼ ½0:7068 −0:0598;

K 22;1 ¼ ½0:2907 −0:5322;

K 22;2 ¼ ½−0:2153 0:1078;

K 22;3 ¼ ½0:0451 −0:0248;

K 22;4 ¼ ½0:5766 0:1682:

12 x1 x2

10 State response

8 6 4 2 0 −2 −4 −6 −8

0

5

10

15

20

25 t/s

30

35

40

45

50

Fig. 3. State responses of closed-loop system.

15

u

Control input

10 5 0 −5 −10 −15

0

5

10

15

20

25 t/s

30

35

40

45

50

Fig. 4. Responses of the control input. Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031

H. Shen et al. / Journal of the Franklin Institute ] (]]]]) ]]]–]]]

19

3 2.5

v (t)

2 1.5 1 0.5 0 0

5

10

15

20

25 t/s

30

35

40

45

50

Fig. 5. Responses of ηðtÞ for closed-loop system.

In the simulation, the giving system modes' evolutions are shown in Fig. 1, where “1”, “2”, “3”, and “4” in the y-axis denote the operation modes. The initial condition and the disturbance input are assumed to be xð0Þ ¼ ½−2:5π 3:75πT and ωðtÞ ¼ −1=ð2 þ tÞ, respectively. In this case, we get the state response of the open-loop system as depicted in Fig. 2, which shows that the openloop system is unstable. Now, we use the obtained controller to deal with the system, and thus the state responses of the closed-loop system and the response of the control input are depicted in Figs. 3 and 4, respectively. It is obvious from Figs. 3 and 4 that the resulting closed-loop system is stable. In addition, we let ηðtÞ ¼ Ef〈z; Qz〉ϝ þ 2〈z; Sω〉ϝ þ 〈ω; Rω〉g−θ〈ω; ω〉ϝ : Thus, the response of ηðtÞ for the resulting closed-loop system is given as plotted in Fig. 5. It is seen from Fig. 5 that ηðtÞ≥0 is satisfied, which implies that the resulting closed-loop system is strictly ðQ; R; SÞθdissipative. 6. Conclusions In this paper, we have investigated the problems of fuzzy dissipative control for a class of nonlinear Markovian jump systems. The information on transition rates of Markov process is assumed to be partially known. A retard feedback controller design method has been developed by using T–S fuzzy model approach. The desired fuzzy controller, which ensures that the corresponding closed-loop system is stochastically mean-square stable and strictly ðQ; R; SÞθ dissipative, can be constructed by solving a set of LMIs. A numerical example has been provided to illustrate the effectiveness of the proposed method.

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Please cite this article as: H. Shen, et al., Fuzzy dissipative control for nonlinear Markovian jump systems via retarded feedback, Journal of the Franklin Institute. (2013), http://dx.doi.org/10.1016/j.jfranklin.2013.02.031