Abstract
We are concerned with the stability problem for linear discrete-time switched systems with time delays. The problem is solved by using multiple Lyapunov functions to develop constructive tools for the exponential stability analysis of the switched time-delay system. Furthermore, the uncertainties of the switched systems are also taken into consideration. Sufficient delay-dependent conditions are derived in terms of the average dwell time for the exponential stability based on linear matrix inequalities (LMIs). Finally, numerical examples are provided to illustrate the effectiveness of the proposed method.
1. Introduction
Switched systems represent dynamical systems described by a collection of differential equations with both continuous-time dynamics and discrete-time elements [1]. In recent years, hybrid and switched dynamic systems have attracted much attention because of their wide applications in control of mechanical systems, electrical systems [2, 3], networked systems [4–7], and many other fields. One of the important topics in the study of switched systems is stability analysis, and many results have been reported for linear switched system. By exploiting average dwell time, Hespanha and Morse derived some sufficient conditions for the uniform exponential stability of the switched linear systems [8]. A concise and timely survey on analysis and synthesis of switched linear system is presented in [9]. In [10], the stability of switched linear system is analyzed by using multiple Lyapunov functions and Lyapunov-Metzler inequalities. Note that these results can not be extended to switched time-delay systems due to the infinite dimensionality of time-delay systems.
Most existing results in switched systems are based on finite dimensional systems free of time delays. However, time-delay phenomena are very common in most practical industrial control systems [11]. As a matter of fact, switched time-delay systems have often appeared in the mathematical models of networked systems, hereditary systems, Lotka-Volterra systems, and so on. More importantly, the controller design of time-delay systems sometimes requires switching controller when one single controller cannot meet the design requirements. Thus, it is of great importance to investigate switched systems with time delays. To investigate the time-delay problem for switched systems, some important research efforts have been conducted. Sufficient conditions for exponential stability and weighted -gain were developed for a class of switched systems with time-varying delays [12]. In [13], an average dwell time approach was used to analyze switched linear systems with time-varying delays. Furthermore, the literatures [14, 15] extended the average dwell time approach to switched singular time-delay systems. By using a Lyapunov functional and LMI approach, various delay-independent and delay-dependent stability results were provided for linear switched time-delay system in [16]. In [17, 18], the piecewise Lyapunov-Razumikhin functions were introduced for the stability analysis of the switched time-delay systems. It should be noticed that most of the aforementioned results do not consider the uncertainties of switched linear systems.
Due to the existence of model uncertainties in real applications, it is very desirable to consider the impact of uncertainties for the switched systems [19]. To the best of our knowledge, such problems for the switched systems with both uncertainties and time delay have rarely been studied till present. In [20], some sufficient conditions for the robust stabilization of a class of uncertain switched time-delay systems were developed based on average dwell time. Using a common Lyapunov function, several sufficient delay-independent conditions for the robust stability of the uncertain linear hybrid systems with time delay were given in [21]. Nevertheless, it may be hard to construct a common Lyapunov function for all the subsystems in most of the application cases. In addition, the research for the delay-dependent stability analysis is relatively a new topic. Generally speaking, the delay-dependent stability analysis is considered less conservative than the delay-independent case [22].
Motivated by the challenges discussed above, this paper considers generalized uncertain time-delay systems in a discrete domain where some sufficient delay-dependent conditions are derived by using multiple Lyapunov functions and average dwell time to guarantee global exponential stability of the closed loop systems. Compared with [21], stronger stability results are provided, that is, the exponential stability rather than the asymptotic stability.
The remainder of this paper is organized as follows. In Section 2, the mathematical model of the uncertain switched system with delay time is presented and some preliminaries are given. In Section 3, the stability of uncertain switched time-delay systems in the discrete-time domain is analyzed; some sufficient conditions with the dwell time for switching signal are given. In Section 4, some numerical examples are provided to illustrate the effectiveness of the results. Finally, some conclusions are drawn in Section 5.
The following notations will be used throughout this paper. Let and , let be the -dimensional Euclidean space, and denotes the set of all nonnegative integers and is the space of real matrices. means that matrix is symmetric and semipositive (seminegative) definite. means that matrix is symmetric and positive (negative) definite. denotes the real identity matrix. denotes the transpose of a square matrix . and denote the minimum and the maximum eigenvalue of the corresponding matrix, respectively. For , the norm of is . For , the norm of is .
2. Problem Definition and Preliminaries
In this section, we introduce an uncertain linear discrete-time switched system with delays of the following general form: and the initial condition where , , the state . Let , the switching signal , and the set denotes switching sequence, which is assumed to be a closed discrete subset of with , and . For any , are given constant matrices, and are the parameter uncertainties which satisfy the following assumptions: where , , , and are given constant matrices of appropriate dimensions. The uncertain matrix is assumed to satisfy the condition .
Definition 1. The discrete-time uncertain linear switched system with time delay (1) is robustly stable if there exist a positive definite scalar function for all and a switching signal such that
Definition 2. The induced norm of a matrix is denoted by where and satisfy the inequality
Definition 3. A switching signal is said to have an average dwell time if there exist two positive numbers and such that where is the number of switches in the interval .
Definition 4 (see [11, 23]). The switched system (1)-(2) is said to be exponentially stable if its solutions satisfy for any initial conditions , where , , and is the decay rate.
Lemma 5 (see [24]). Let , , and be given matrices such that , . Then, the linear matrix inequality (LMI) holds if and only if .
Lemma 6 (see [25, 26]). Let , , and , are constant matrices. If there exists such that , then where .
Lemma 7 (see [27]). For a quadratic positive definite , there exist and such that
3. Stability Analysis
In this section, we analyze the stability for uncertain switched time-delay systems, and some sufficient conditions are given. Firstly, let us consider switched time-delay systems in discrete-time domain instead.
3.1. Switched Systems with Time Delay
Consider the discrete-time switched system with state delays given by and the initial condition System (11)-(12) can be obtained from (1)-(2) as long as and . is the switching signal, for each ; we get the following sets:
Obviously, the switched linear system (11)-(12) has different subsystems, that is, and the initial condition where , , , and . Clearly, there exists such that and are constrained to jump to and among their own sets, respectively, where .
Considering discrete-time switched system (11)-(12), we choose the following Lyapunov function given by where, , and is a given constant.
Proposition 8. For a given scalar and the delay , there exist matrices , , and such that the following matrix inequality holds; then the function in (16) along any trajectory of system (14)-(15) guarantees the following growth estimation:
Proof. Applying the transformation , we obtain the following system from (14)-(15): and the initial condition Choose the following Lyapunov function for system (19)-(20): The forward difference of the Lyapunov function along the trajectory of system (19)-(20) is given by where is defined in (17) and . Using (17), we arrive at which implies From (16), we have and the fact that It follows that The proof is completed.
Theorem 9. For given scalars and and the delay , assume that there exist and such that equality (17) holds. Then, switched delay-time system (11)-(12) is exponentially stable if the following conditions hold:(A1)the positive definite matrices and satisfy (A2)there exists such that the average dwell time satisfies
Proof. Choose the following Lyapunov function for system (11)-(12): For , using Proposition 8, if equality (17) holds, we obtain Using condition (A1), then we have By virtue of (31) and (32), it follows that Iterating from to , we have Applying (29), we have Thus, there exists and such that It follows that which yields , where Hence, it is concluded from Definition 4 that switched delay-time system (11)-(12) is exponentially stable. The proof is completed.
3.2. Uncertain Discrete-Time System with Time Delay
Consider the following subsystem of switched system (1)-(2): and the initial condition . Choose the following Lyapunov function for (39)-(40): where , and is a given constant.
To derive the exponential stability of switched time-delay system (1)-(2), we give the decay estimation of the Lyapunov function along the trajectory (39)-(40) in the following proposition firstly.
Proposition 10. For a given scalar , , and any delay , if there exist matrices , , and such that the following inequalities, hold, then function in (41) along any trajectory of switched system (39)-(40) guarantees the decay estimation as follows: where
Proof. , by applying the transformation , we obtain the following system from (39)-(40): and the initial condition Choose the following Lyapunov function for switched system (45)-(46): Let and . The forward difference of Lyapunov function along any trajectory of system (45)-(46) is given by where Note that On the other hand, by virtue of Lemma 6, we have where Then, Now taking into account (42) and using Lemma 5, we have which implies that From (41), we have and the fact that It follows that The proof is completed.
Theorem 11. For given scalars and and any delay , if there exist matrices and , such that inequalities (42) and the conditions hold, then the uncertain switched time-delay system (1)-(2) is exponentially stable and guarantees a decay rate , where and is the average dwell time.
Proof. Considering system (1)-(2), choose the following Lyapunov function given by By virtue of (59), we have On the other hand, function ensures the decay estimation (43) under condition (42). Hence, we have Let ; by virtue of (60), we have where . It follows that which yields , where Hence, by virtue of Definition 4, the switched system (1)-(2) is exponentially stable. The proof is completed.
Remark 12. In literature [21], the common Lyapunov function was employed to derive the delay-independent conditions. From condition (59), it can be seen that the common Lyapunov function approach can be treated as a special case of Theorem 11 if and only if satisfies . In this sense, we get away from the common Lyapunov conditions as increases from , which indicates the conservativeness of the common Lyapunov function approach. In contrast, this paper presents the delay-dependent exponential stability conditions by constructing multiple Lyapunov functions.
Remark 13. It should be noted that [11] only considers the switched time-delay systems without the parameter uncertainties. The present paper extends the results in [11] to the uncertain switched time-delay system in discrete-time domain by constructing different Lyapunov functions and employing the concept of the average dwell time.
4. Numerical Example
In this section, we use an example to illustrate the results in Section 3.
Example 1. Consider the delay-time switched system (11)-(12) given by where . Let . Assuming that the average dwell time is and , then we have . If we take , then we can choose . From equalities (17) and (28), by using LMIs toolbox, we can obtain the feasible solutions for , and , given by From Theorem 9, it is concluded that the delay-time switched system (11)-(12) is exponentially stable.
Example 2. Consider the delay-time switched system (1)-(2), where , and where Let . Assuming that the average dwell time is and , then we have . If we take , then we can choose . By virtue of Theorem 11, the decay rate is . If we choose and the matrices we can verify the conditions From equalities (42) and (59), by using LMIs toolbox, we can obtain the feasible solutions for and given by where The initial states are given by . According to Theorem 11, the state curves can be obtained by simulation (as shown in Figure 1).
5. Conclusions
In this paper, we have investigated the stability for switched time-delay systems in discrete-time domain. Several sufficient conditions have been proposed by utilizing multiple Lyapunov functions with the average dwell time. On one hand, the exponential stability conditions have been derived for the switched systems in the presence of time delays. On the other hand, the exponential stability conditions have been developed for the uncertain switched linear system with time delays based on LMIs conditions. Finally, the illustrative examples have been given to verify the main theoretical results.
We are currently working on the switched systems with interval time delays as well as controller synthesis methods for such systems, in the hope that switching controller design can in the long run offer a new look of synthesis of systems with large and/or time-varying delays when a single controller cannot suffice the design requirements.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank the financial support from the NSFC under Grant no. 61327003 and China Fundamental Research Funds for the Central Universities under Grant no. 10062013YWF13-ZY-68.