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Guaranteed cost stabilization control of discrete‐time switched systems

2020; Institution of Engineering and Technology; Volume: 15; Issue: 3 Linguagem: Inglês

10.1049/cth2.12051

1751-8652

Shengli Du, Xudong Zhao, Junfei Qiao, Guangdeng Zong,

Control and Stability of Dynamical Systems

IET Control Theory & ApplicationsVolume 15, Issue 3 p. 404-415 ORIGINAL RESEARCH PAPEROpen Access Guaranteed cost stabilization control of discrete-time switched systems Shengli Du, Corresponding Author Shengli Du shenglidu@bjut.edu.cn orcid.org/0000-0001-7372-1608 College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing, P. R. China Correspondence Shengli Du, College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing 100124, P. R. China. Email: shenglidu@bjut.edu.cnSearch for more papers by this authorXudong Zhao, Xudong Zhao School of Control Science and Engineering, Dalian University of Technology, Dalian, P. R. ChinaSearch for more papers by this authorJunfei Qiao, Junfei Qiao College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing, P. R. ChinaSearch for more papers by this authorGuangdeng Zong, Guangdeng Zong School of Engineering, Qufu Normal University, Rizhao, P. R. ChinaSearch for more papers by this author Shengli Du, Corresponding Author Shengli Du shenglidu@bjut.edu.cn orcid.org/0000-0001-7372-1608 College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing, P. R. China Correspondence Shengli Du, College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing 100124, P. R. China. Email: shenglidu@bjut.edu.cnSearch for more papers by this authorXudong Zhao, Xudong Zhao School of Control Science and Engineering, Dalian University of Technology, Dalian, P. R. ChinaSearch for more papers by this authorJunfei Qiao, Junfei Qiao College of Automation, Faculty of Information Technology, Beijing University of Technology, Beijing, P. R. ChinaSearch for more papers by this authorGuangdeng Zong, Guangdeng Zong School of Engineering, Qufu Normal University, Rizhao, P. R. ChinaSearch for more papers by this author First published: 20 December 2020 https://doi.org/10.1049/cth2.12051Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This paper is concerned with the guaranteed cost control problem of a class of linear discrete-time switched systems with parametric uncertainties. Both state-based and dynamic output-based robust stabilization problems are investigated by using mode-dependent dwell time technique. Different from the existing results, here, the designed controllers are both mode-dependent and quasi-time-dependent, which are less conservative. On the basis of the designed quasi-time-dependent Lyapunov functions, robust stabilization criteria are presented for the system under consideration. The solvable conditions are formulated as linear matrix inequalities, which are thus easy to be solved. Some comparative simulations are given to show the effectiveness of the proposed method. 1 INTRODUCTION Switched systems are special hybrid systems that consist of a series of discrete/continuous subsystems and are controlled by a switching signal. In the last decades, much attention has been paid to switched systems due to their wide applications in the flight control systems, power systems and social network systems (see, e.g. [1-7]). It is well known that stability is an essential issue for switched systems. Until now, fruitful results on stability analysis for switched systems have been reported over the last decades [8-12]. The paper [8] did a pioneer work regarding to the stability of switched systems. Specifically, in [8], a novel switching framework, namely, average dwell time switching is proposed to investigate the stability problem of switched systems with all subsystems being stable. It is shown that as long as some average dwell time properties are satisfied, the stability of the considered system can always be guaranteed. Based on this result, many excellent results on stability problem of switched systems have been presented [13-18]. By introducing the activated length rate of stable and unstable subsystems, the authors of [15] extend the result in [8] to the case where stable and unstable subsystems exist simultaneously. Though these results provide effective methods for the stability analysis of switched systems, there still has space for improvement in the switching signal design. Inspired by [16], Zhao et al. propose a mode-dependent average dwell time technique for the stability analysis of switched systems, which is less conservative than the previous average dwell time method. Since the mode-dependent average dwell time is more effective than the average dwell time, researchers develop fruitful results based on such a switching strategy. For example, based on the mode-dependent average dwell time approach, stability problem of switched systems with unstable subsystems is studied in [19]. It should be noted that besides the stability, stabilisation problem also plays an important role in the research of switched systems, and have also attracted increasing attentions recently (see, e.g. [20-27] and the references therein). In [20], stabilisation problem of switched systems with unstable subsystems is investigated by designing switching signals with mode-dependent average dwell time property. In [24], the stabilisation problem of positive switched systems under asynchronous switching is studied by adopting the average dwell time method. A switching signal and also the desired controller are developed such that the closed-loop system is asymptotically stable. Note that all the aforementioned literature consider only the stabilisation problem just by designing switching signal or (and) controllers such that the closed-loop system is stable. In the design of a switched system, it is usually expected that the closed-loop dynamic is not only robustly stable, but also owns an adequate level of robust performance. The guaranteed cost control (GCC) problem, which is first proposed in [28], has received significant attention recently (see, e.g. [29-34] and the references therein). The main objective of the GCC is to design a control strategy such that the closed-loop dynamic system is stable and an upper bound of a quadratic cost is minimised for all admissible uncertainties. In [29], GCC problem for switched systems is studied by designing some state-based switching laws. In [31], the robust stabilisation problem of switched systems with time delays is studied, in which the controllers are designed by implementing an iterative algorithm and only suboptimal values can be obtained. Since the state information of dynamic systems may not be always available, dynamic controller with output feedback has become an alternative to solve the stability/stabilisation problem [35, 36]. In these two papers, excellent results of dynamic compensator for second-order linear and high-order quasi-linear systems have been presented. However, the GCC problem has not been investigated in these two references. In [30], the GCC problem of switched continuous-time systems is investigated by using the dynamic output feedback control , and the sufficient conditions are obtained based on the average dwell time switching strategy. However, most of the above mentioned and other existing results are developed for continuous-time switched systems, and the stability and the controller design for discrete-time switched systems have not been fully addressed yet, which is the first motivation of current research. On the other hand, the Lyapunov method is usually adopted for the stability analysis and stabilisation synthesis of switched systems. However, it should be noted that the Lyapunov function in each activated time interval [ t i , t i + 1 ) is designed in the continuous form, which may give rise to conservative results. It still leaves space for improvement in terms of the Lyapunov function design. This is the second motivation of this study. We are intended to investigate the stability and GCC problem of a class of linear discrete-time switched systems with parametric uncertainties. For the purposes of stability analysis and controller design, a mode-dependent scheduler is introduced, and then some quasi-time-dependent Lyapunov functions are constructed based on such a scheduler. With the help of such a scheduler, the constructed Lyapunov function in time interval [ t i , t i + 1 ) is not continuous any more and thus reduce the conservativeness of the stability analysis. Based on these functions, sufficient conditions for the desired controllers under state-based and dynamic output-based feedbacks are developed, respectively. These conditions are all presented in the form of linear matrix inequalities (LMIs), which are easy to be verified. Compared with some papers concerning on the GCC problem, the obtained solvable conditions avoid the implementation of some iterative algorithms and thus have a lower computation burden. The main contribution of the paper can be summarised as follows. First, some novel and interesting quasi-time-dependent Lyapunov functions are designed for solving the considered robust stabilisation problem, based on which some less conservative stability criteria are derived. Then, both state-based and dynamic output-based solvable conditions are developed for the studied systems, and the solvable conditions are given in the form of LMIs, which can be solved easily. At last, some simulations and comparisons are provided to demonstrate the effectiveness of the proposed method. The rest of this paper is organised as follows. In Section 2, some preliminaries and the problem statement are presented. The GCC problems for a class of linear discrete-time switched systems under the state-based and dynamic output-based feedbacks are studied in Sections 3 and 4, respectively. The simulations and some comparisons are given in Section 5, and some conclusions are presented in the last section. Notations: The n-dimensional Euclidean space is denoted by R n . Z + denotes the set of non-negative integers. The notation S > 0 ( S < 0 , S ≤ 0 ) means that S is a symmetric positive definite (negative definite, negative semi-definite) matrix with appropriate dimensions. diag { a 1 , … , a n } denotes a diagonal matrix with a 1 , … , a n being its diagonal elements. I denotes a unitary matrix with compatible dimensions. The symbol ⋆ in a symmetric matrix stands for the symmetric block. 2 PRELIMINARIES AND PROBLEM STATEMENT Consider a class of discrete-time switched systems with the following form x ( t + 1 ) = ( A σ ( t ) + Δ A σ ( t ) ) x ( t ) + ( B σ ( t ) + Δ B σ ( t ) ) u ( t ) y ( t ) = C σ ( t ) x ( t ) (1)where t ∈ Z + ; x ( t ) ∈ R n , u ( t ) ∈ R m and y ( t ) ∈ R p are the state, control input and output, respectively; σ : Z + → P is a switching signal defined on Z + with P being the set of all the possible modes. We let P = { 1 , 2 , … , ℓ } . A i ∈ R n × n , B i ∈ R m × n , C i ∈ R p × n ( i ∈ P ) are known constant matrices. Δ A i , Δ B i , Δ C i ( i ∈ P ) are some matrix-valued functions denoting the parametric uncertainties, and have the following form [ Δ A i Δ B i ] = D i F ( t ) [ E 1 i E 2 i ] , i ∈ P (2)where D i , E 1 i , E 2 i are constant matrices and F ′ ( t ) F ( t ) ≤ I . Definition 1. ([[16]])For a given switching signal σ and a bounded time interval [ p , q ) . Let N i ( p , q ) denote the switching numbers and T i ( p , q ) be the total running time of the i-th subsystem over such a bounded time interval. If there exist two parameters ω i , N 0 i ≥ 0 ( i ∈ P ) such that N i ( p , q ) ≤ N 0 i + T i ( p , q ) ω i , i ∈ P (3) holds true, then switching signal σ is said to be a slow switching signal and ω i ( i ∈ P ) is called a mode-dependent average dwell time, and N 0 i is called a mode-dependent chattering bound. Remark 1.As pointed out in [16], Definition 1 provides a new type of switching signal σ. Compared with the switching signal with average dwell time property, the switching signal here has a close relationship with the modes. T i ( p , q ) consists of several disjoint time intervals related to only the i-th subsystem, thus each mode possesses its own average dwell time under this framework. Definition 2.Consider the discrete-time switched system (1). For a given switching signal σ, if there are two positive scalars a , γ 0 with a ≥ 1 , 0 < γ 0 < 1 such that ∥ x ( t ) ∥ ≤ a γ 0 t ∥ x ( 0 ) ∥ , (4) then system (1) is said to be γ0-exponentially stable with respect to the switching signal σ for any initial condition x(0). The objective of this paper is to design robust controllers (state-based and dynamic output-based) such that the closed-loop of system (1) is exponentially stable with respect to the uncertainties while attains an expected performance. Next, we provide the GCC problem for the system under consideration. The cost function associated with system (1) is given as follows J = ∑ t = 0 + ∞ κ t x ( t ) ′ Q x ( t ) + u ( t ) ′ R u ( t ) (5) where κ > 0 , Q , R are positive definite matrices with compatible dimensions given in prior. Definition 3.For the uncertain switched system (1), if there exist a control law u ∘ ( t ) and a positive scalar J ∘ such that the closed-loop dynamic of system (1) is exponentially stable with respect to all uncertainties and furthermore, the cost function (5) satisfies J ≤ J ∘ , then the control input u ∘ is called a GCC law and J ∘ is said to be a weighted guaranteed cost. Lemma 1. ([[37]])Given matrices Q , S , H with Q symmetrical, then Q + S F ( t ) H + H ′ F ′ ( t ) S ′ < 0 holds for all F ( t ) satisfying F ′ ( t ) F ( t ) ≤ I , if and only if there exists a scalar ε > 0 such that Q + ε S S ′ + 1 / ε H ′ H < 0 . Lemma 2. ([[38]])For given matrices W , X , Y , Z and a real scalar γ, the following two conditions are equivalent 1). There exist a real scalar γ and real matrices X , Y , Z , W such that W ★ Y − γ Z X γ Z + γ Z ′ < 0 ; 2). There exist real matrices W , X and Y such that W < 0 , W + X ′ Y + Y ′ X < 0 . For the given switching signal σ, let t i , i ∈ Z + , be the switching instants among each mode. Suppose that during the bounded interval [ t i , t i + 1 ) , the j-th subsystem is activated with t i being the time instant of the i-th switching. Though some results concerning on robust stabilisation have been reported, there still leaves room for improvement. In order to solve the stabilisation problem of switched systems, Lyapunov functions usually are designed to finish such a task. However, it is noted that the Lyapunov function designed in existing literature is continuous over time interval [ t i , t i + 1 ) , which will result in conservative results. In order to overcome such an issue, we will construct a quasi-time-dependent Lyapunov function over each activated time interval by introducing a scheduler. Similar to [39], the scheduler δ ( t ) corresponding to the activated subsystem j is defined as δ ( t ) = t − t i , t ∈ [ t i , t i + ϕ j ) ϕ j , t ∈ [ t i + ϕ j , t i + 1 ) (6)where ϕ j is a mode-dependent parameter denoting the minimum dwell time. Let S j = { 0 , 1 , … , ϕ j } , j ∈ P and T i be the total length of the [ t i , t i + 1 ) . Remark 2.In order to reduce the conservativeness caused by the continuous Lyapunov function adopted in time interval [ t i , t i + 1 ) , we introduce a scheduler. With such a scheduler, such a time interval will be partitioned into several subintervals, and we provide a new piecewise continuous Lyapunov function on such a time interval for the stabilisation synthesis. It will be shown in the simulation that the introduction of the scheduler can reduce the conservatism of the results. Two different control schemes, namely, the static state-based feedback control and the dynamic output-based control, will be presented in the following two sections. 3 GCC UNDER THE STATIC STATE-BASED FEEDBACK In the state feedback control case, the control law is designed as u ( t ) = K i ( s i ) x ( t ) , i ∈ P , (7)where K i ( s i ) ( i ∈ P , s i = 0 , … , ϕ i ) is a both quasi-time-dependent and mode-dependent controller gain to be determined. Remark 3.In (7), the designed controller is not only mode-dependent but also quasi-time-dependent, which is far different from some existing controllers [29]. The following theorem states a main result of this section for the studied system (1). Theorem 1.Consider system (1) and the cost function (5), for given constants 0 < α j < 1 , μ j > 1 , j ∈ P , if there exists a set of positive matrices P j ( τ ) ( j ∈ P , τ = 0 , … , ϕ j ) such that for all τ ∈ S j ∖ ϕ j ( j ∈ P ) and ∀ ( j , k ) ∈ P × P , j ≠ k , Π j ( τ + 1 , τ ) < 0 , (8) Π j ( ϕ j , ϕ j ) < 0 , (9) P j ( 0 ) − μ j P k ( τ 0 + 1 ) ≤ 0 , (10) where Π j ( τ + 1 , τ ) = A ̂ j ′ ( τ ) P j ( τ + 1 ) A ̂ j ( τ ) − α ¯ j P j ( τ ) + Q + K j ′ ( τ ) R K j ( τ ) A ̂ j ( τ ) = A ¯ j + B ¯ j K j ( τ ) with A ¯ j = A j + Δ A j , B ¯ j = B j + Δ B j , α ¯ j = 1 − α j , ∀ τ 0 ∈ S k ∖ ϕ k , k , j ∈ P . Then, system (1) is exponentially stable for any switching signal σ with the following mode-dependent average dwell time ω i > ω i ★ = − ln μ i ln α ¯ i , i ∈ P , (11)and the cost function satisfies J ≤ γ 1 ( 1 − α ¯ m ) 1 − α ¯ M x ( t 0 ) ′ P σ ( t 0 ) ( 0 ) x ( t 0 ) , (12)where γ 1 , α ¯ m , α ¯ M are defined in the proof part below. Proof.For system (1), we construct the following quasi-time-dependent Lyapunov function V ( x ( t ) , δ ( t ) ) = x ( t ) ′ P σ ( δ ( t ) ) x ( t ) , (13) where δ ( t ) is defined in (6) and P i ( τ ) , τ = 0 , … , ϕ i , i ∈ P , is a set of positive matrices satisfying (8)–(10). Let Δ V i ( x ( t ) , δ ( t ) ) = V i ( x ( t + 1 ) , δ ( t + 1 ) ) − V i ( x ( t ) , δ ( t ) ) . For any t ∈ [ t i , t i + ϕ j ) (the j-th subsystem is assumed to be activated in time interval [ t i , t i + 1 ) ), along the solution of system (1), for the above designed Lyapunov (13), one can obtain that Δ V j ( x ( t ) , t − t i ) + α j V j ( x ( t ) , t − t i ) = x ( t ) ′ A ̂ j ′ ( t − t i ) P j ( t − t i + 1 ) A ̂ j ( t − t i ) − α ¯ P j ( t − t i ) x ( t ) < − x ( t ) ′ Q + K j ( t − t i ) ′ R K j ( t − t i ) x ( t ) , < 0 (14)where the penultimate inequality is supported by (8) with τ ≜ t − t i . It is obvious that for any t ∈ [ t i , t i + ϕ j ) , Δ V j ( x ( t ) , t − t i ) + α j V j ( x ( t ) , t − t i ) + W ( t ) < 0 , where W ( t ) = x ( t ) ′ Q x ( t ) + u ( t ) ′ R u ( t ) . Similarly, condition (9) implies that Δ V j ( x ( t ) , ϕ j ) + α j V j ( x ( t ) , ϕ j ) + W ( t ) < 0 for t ∈ [ t i + ϕ j , t i + 1 ) . Combining these two cases together yields that V j ( x ( t + 1 ) , δ ( t + 1 ) ) ≤ α ¯ j V j ( x ( t ) , δ ( t ) ) − W ( t ) . (15)Furthermore, based on the definition of δ ( t ) , one can obtain that ∀ σ ( t i ) = j ≠ k = σ ( t i − 1 ) V j ( x ( t i ) , 0 ) ≤ μ j V k ( x ( t i ) , ϕ k ) , (16)where the inequality is ensured by (10).On the basis of (15)–(16), one can get that V σ ( t ) ( x ( t ) , δ ( t ) ) ≤ α ¯ σ ( t ) V σ ( t ) ( x ( t ) , δ ( t ) ) − W ( t ) ≤ α ¯ σ ( t i ) ( t − t i ) V σ ( t i ) ( x ( t i ) , 0 ) − ∑ s = t i t − 1 α ¯ σ ( t i ) ( t − s − 1 ) W ( s ) ≤ α ¯ σ ( t i ) ( t − t i ) μ σ ( t i ) V σ ( t i − 1 ) ( x ( t i ) , ϕ σ ( t i − 1 ) ) − ∑ s = t i t − 1 α ¯ σ ( t i ) ( t − s − 1 ) W ( s ) ≤ α ¯ σ ( t i ) ( t − t i ) … α ¯ σ ( t 0 ) ( t 1 − t 0 ) μ σ ( t i ) … μ σ ( t 1 ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) − α ¯ σ ( t i ) ( t − t i ) … α ¯ σ ( t 1 ) ( t 2 − t 1 ) μ σ ( t i ) … μ σ ( t 1 ) ∑ s = t 0 t 1 − 1 α ¯ σ ( t 0 ) ( t 1 − s − 1 ) W ( s ) − … − ∑ s = t i t − 1 α ¯ σ ( t i ) ( t − s − 1 ) W ( s ) . (17)From (17), it is easy to verify that V σ ( t ) ( x ( t ) , δ ( t ) ) ≤ α ¯ σ ( t i ) ( t − t i ) … α ¯ σ ( t 0 ) ( t 1 − t 0 ) μ σ ( t i ) … μ σ ( t 1 ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) = ∏ j = 1 p μ j N j ( t 0 , t ) α ¯ j T j ( t 0 , t ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) (18) ≤ exp ∑ j = 1 p N j ( t 0 , t ) ln μ j exp ∑ j = 1 p T j ( t 0 , t ) ln α ¯ j + ln μ j ω j × V σ ( t 0 ) ( x ( t 0 ) , 0 ) . (19)Let a 0 = exp ( ∑ j = 1 p N j ( t 0 , t ) ln μ j ) and γ min = min j ∈ P ( − ln α ¯ j − ln μ j ω j ) , it follows from (18) that δ 0 ∥ x ( t ) ∥ 2 ≤ V σ ( t ) ( x ( t ) , δ ( t ) ) ≤ a 0 b 0 e − γ min t ∥ x ( t 0 ) ∥ 2 (20)where δ 0 = min τ ∈ S σ ( t ) ( P σ ( t ) ( τ ) ) , b 0 = P σ ( t 0 ) ( 0 ) . Furthermore, one can get that ∥ x ( t ) ∥ ≤ a γ 0 k ∥ x ( t 0 ) ∥ , (21)where a = a 0 b 0 / δ 0 and γ 0 = exp ( − γ min / 2 ) . Based on Definition 2, the system under consideration is γ0-exponentially stable. In what follows, we will show that the weighted cost function satisfies (12).Multiplying ∏ j = 1 p μ j − N j ( t 0 , t ) on both sides of (17) gives that ∏ j = 1 p μ j − N j ( t 0 , t ) V σ ( t ) ( x ( t ) , δ ( t ) ) ≤ ∏ j = 1 p α ¯ j T j ( t 0 , t ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) − ∑ s = t 0 t − 1 α ¯ m ( t − s − 1 ) ∏ j = 1 p μ j − N j ( t 0 , s ) W ( s ) (22)where α ¯ m = min j ∈ P ( α ¯ j ) .Based on (3) and (11), it holds that ∑ s = t 0 t − 1 α ¯ m ( t − s − 1 ) ∏ j = 1 p μ j − N j ( t 0 , s ) W ( s ) ≥ ∏ j = 1 p μ j − N 0 j α ¯ m ( t − s − 1 ) exp ∑ j = 1 p T j ( t 0 , s ) ln α ¯ j W ( s ) ≥ ∏ j = 1 p μ j − N 0 j ∑ s = t 0 t − 1 α ¯ m ( t − s − 1 ) α ¯ m ( s − t 0 ) W ( s ) . (23)Combining the above inequality together with (22) gives that ∑ s = t 0 t − 1 α ¯ m ( t − s − 1 ) α ¯ m ( s − t 0 ) W ( s ) ≤ γ 1 ∏ j = 1 p α ¯ j T j ( t 0 , t ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) (24)where the property V σ ( t ) ( x ( t ) , δ ( t ) ) ≥ 0 is adopted and γ 1 = ∏ j = 1 p μ j N 0 j . Summing both sides of (24) from t0 to ∞ gives ∑ t = t 0 ∞ ∑ s = t 0 t − 1 α ¯ m ( t − s − 1 ) α ¯ m ( s − t 0 ) W ( s ) ≤ ∑ t = t 0 ∞ γ 1 ∏ j = 1 p α ¯ j T j ( t 0 , t ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) (25)which further gives ∑ s = t 0 ∞ ∑ t = s + 1 ∞ α ¯ m ( t − s − 1 ) α ¯ m s W ( s ) ≤ γ 1 ∑ t = t 0 ∞ α ¯ M ( t − t 0 ) V σ ( t 0 ) ( x ( t 0 ) , 0 ) (26)where α ¯ M = max j ∈ P ( α ¯ j ) . It then follows from (26) that ∑ t = t 0 ∞ α ¯ m t ( x ( t ) ′ Q x ( t ) + u ( t ) ′ R u ( t ) ) ≤ γ 1 ( 1 − α ¯ m ) 1 − α ¯ M x ( t 0 ) ′ P σ ( t 0 ) ( 0 ) x ( t 0 ) (27)The proof is thus completed. □ Theorem 1 provides sufficient conditions for the guaranteed cost controller design. However, the obtained conditions are not in the form of LMIs due to the presence of the uncertainties. We will provide LMI conditions for solving the guaranteed cost controllers hereinafter, which is stated by the following theorem. Theorem 2.Consider system (1) and the cost function (5), for given constants ε , 0 < α j < 1 , μ j > 1 , j ∈ P , if there exists a set of matrices P j ( τ ) > 0 , Z j ( τ ) , L j ( τ ) ( j ∈ P , τ = 0 , … , ϕ j ) such that for all τ ∈ S j ∖ ϕ j ( j ∈ P ) and ∀ ( j , k ) ∈ P × P , j ≠ k , Π ¯ j ( τ + 1 , τ ) < 0 , (28) Π ¯ j ( ϕ j , ϕ j ) < 0 , (29)and (10) holds true, where Π ¯ j ( κ + 1 , κ ) = Π ¯ j 1 ( κ ) ★ ★ ★ ★ ★ Π ¯ j 2 ( κ ) − P j ( κ + 1 ) ★ ★ ★ ★ Z j ( κ ) 0 − R − 1 ★ ★ ★ Π ¯ j 3 ( κ ) 0 0 − I ★ ★ 0 Π ¯ j 4 ( κ ) 0 0 − I 2 0 Π ¯ j 5 ( κ ) Π ¯ j 6 ( κ ) Π ¯ j 7 ( κ ) Π ¯ j 8 ( κ ) 0 Π ¯ j 9 ( κ ) with Π ¯ j 1 ( κ ) = − α ¯ j P j ( κ ) + Q + E 1 j ′ E 1 j , Π ¯ j 2 ( κ ) = P j ( κ + 1 ) A j + B j Z j ( κ ) , Π ¯ j 3 ( κ ) = E 2 j Z j ( κ ) , Π ¯ j 4 ( κ ) = P j ( κ + 1 ) D j , Π ¯ j 5 ( κ ) = ε Z j ( κ ) , Π ¯ j 6 ( κ ) = B j ′ P j ( κ + 1 ) − L j ′ ( κ ) B j ′ , Π ¯ j 7 ( κ ) = I − L j ′ ( κ ) , Π ¯ j 8 ( κ ) = E 2 j ′ − L j ′ ( κ ) E 2 j ′ , Π ¯ j 9 ( κ ) = − ε L j ( κ ) − ε L j ′ ( κ ) . Then system (1) is exponentially stable under switching signal with the mode-dependent average dwell time (11) and the cost function satisfies (12). Meanwhile, the controllers can be determined as K j ( τ ) = L j − 1 ( τ ) Z j ( τ ) , τ = 0 , … , ϕ j , j ∈ P . Proof.Pursuant to the Schur complement lemma, inequality (8) is equivalent to the following inequality − α ¯ j P j ( τ ) + Q ★ ★ P j A ̂ j ( τ ) − P j ( τ + 1 ) ★ K j ( τ ) 0 − R − 1 < 0 . (30)It follows from (30) that − α ¯ j P j ( τ ) + Q ★ ★ P j ( τ + 1 ) [ A j + B j K j ( τ ) ] − P j ( τ + 1 ) ★ K j ( τ ) 0 − R − 1 + 0 ★ 0 P j ( τ + 1 ) [ Δ A j + Δ B j K j ( τ ) ] 0 ★ 0 0 0 < 0 . (31)By using condition (2) and Lemma 1, one has that 0 ★ ★ P j ( s ) Δ A j 0 ★ 0 0 0 = E 1 j ′ 0 0 F ′ ( t ) 0 D j ′ P j ( s ) 0 + 0 P j ( s ) D j 0 F ( t ) E 1 j 0 0 ≤ E 1 j ′ E 1 j ★ ★ 0 P j ( s ) D j D j ′ P j ( s ) ★ 0 0 0 . (32)Similarly, one can also get 0 ★ ★ P j ( t ) Δ B j K j ( s ) 0 ★ 0 0 0 ≤ K j ′ ( s ) E 2 j ′ E 2 j K j ( s ) ★ ★ 0 P j ( t ) D j D j ′ P j ( t ) ★ 0 0 0 (33) Combining (30)–(33) together and using Schur complement lemma again yields the following inequality N ¯ j ( κ ) = Π ¯ j 1 ( κ ) ★ ★ ★ ★ N ¯ j 1 ( κ ) N ¯ j 2 ( κ ) ★ ★ ★ K j ( κ ) 0 − R − 1 ★ ★ N ¯ j 3 ( κ ) 0 0 − I ★ 0 Π ¯ j 4 ( κ ) 0 0 − I 2 < 0 , (34)where N ¯ j 1 ( κ ) = P j ( κ + 1 ) A j + P j ( κ + 1 ) B j K j ( κ ) , N ¯ j 2 ( κ ) = − P j ( κ + 1 ) , N ¯ j 3 ( κ ) = E 2 j K j ( κ ) . Define T ¯ j ( κ ) = 0 Π ¯ j 6 ( κ ) Π ¯ j 7 ( κ ) Π ¯ j 8 ( κ ) 0 , H ¯ ( κ ) = L j − 1 Z j ( κ ) 0 0 0 0 . By using Lemma 2, i