Finite-time control of a switched discrete-time system with average dwell time
© Zhong et al.; licensee Springer 2013
Received: 3 March 2013
Accepted: 14 June 2013
Published: 28 June 2013
For switched discrete-time systems, switching behavior always affects the finite-time stability property, which was neglected by most previous research. This paper investigates the problem of control for a switched discrete-time system with average dwell time. Based on the results on finite-time boundned and average dwell time, sufficient conditions for finite-time bounded and finite-time control under arbitrary switching are derived, and the closed-loop system trajectory stays within a prescribed bound. Finally, an example is given to illustrate the efficiency of the proposed method.
During the last few decades, the problem of control for continuous time systems with time delays has been extensively investigated [1, 2]. In the discrete-time context, there is rapidly growing interest in control for a discrete-time system due to its being frequently encountered in many practical engineering systems such as chemical, electronics, process control systems and networked control systems [3–5]. Furthermore, in the state feedback case, the augmentation approach generally leads to a static output feedback control problem, which is non-convex . Control of stochastic systems is a research topic of both practical and theoretical importance, which has received much attention [7, 8]. Many results related to stochastic systems have been reported in the literature. For instance, an optimal stochastic linear-quadratic control problem was investigated by a stochastic algebraic Riccati equation approach in infinite-time horizon in , where the diffusion term in dynamics depends on both the state and the control variables. It is important to invest control problem.
In recent years, there has been increasing interest in the analysis of hybrid and switched systems due to their significance both in theory and applications. Switched linear control systems, as an important class of hybrid systems, comprise a collection of linear subsystems described by differential/difference equations and a switching law to specify the switching among these subsystems. A switched system is a type of a hybrid system which is a combination of discrete and continuous dynamical systems. These systems arise as models for phenomena which cannot be described by exclusively continuous or exclusively discrete processes. Moreover, control design remains open for the switched systems that exhibit switching jumps and subsystem models. Most recently, on the basis of Lyapunov functions and other analysis tools, the control design for linear switched systems has been further investigated and many valuable results have been obtained; for a recent survey on this topic and related questions one can refer to [10–25].
It well known that most of the existing literature has focused on Lyapunov asymptotic stability for switched systems, the behavior of which is over an infinite time interval. On the other hand, many concerned problems are the practical ones which described system state which does not exceed some bound over a time interval. To deal with the above problem, in 1961, Dorato proposed the concept of finite-time stability in .
Over the years, many research efforts have been devoted to the study of finite-time stability (FTS) of systems. In the study of the transient behavior of systems, FTS concerns the stability of a system over a finite interval of time and plays an important role. It is worth pointing out that finite-time stability and Lyapunov asymptotic stability are different concepts, and they are independent of each other. Therefore, it is important to emphasize the distinction between classical Lyapunov stability and finite-time stability. The problem of finite-time stability has been accordingly studied in the literature [26–40].
Recently, some papers related to finite-time stability for switched systems can be found. For example, based on the average dwell time technique, the problem of finite-time boundedness for a switched linear system with time-delay was investigated in , and  discussed the static-state feedback and dynamic output feedback finite-time stabilization. But to the best of our knowledge, the finite-time control problems for switched discrete-time systems with average dwell time have not been studied, and this motivates us to consider this interesting and challenging problem.
The main contribution of this paper lies in that we present a novel approach to finite-time stability of a switched system. Moreover, several sufficient conditions ensuring finite-time stability and boundness are proposed with different information from what we know about the switching signal. It is shown that less conservative results can be derived when more information about the switching signal is available. By selecting the appropriate Lyapunov-Krasovskii functional, the sufficient conditions are derived to guarantee finite-time boundness of the systems. The finite-time boundness (FTB) criteria can be tackled in the form of LMIs. Finally, an example is used to illustrate the effectiveness of the developed techniques.
Notations: Throughout this paper, we let (, , ) denote a symmetric positive definite matrix P (positive-semi definite, negative definite and negative-semi definite). For any symmetric matrix P, and denote the maximum and minimum eigenvalues of matrix P, respectively. denotes the n-dimensional Euclidean space and refers to the set of all real matrices. The identity matrix of order n is denoted as . ∗ represents the elements below the main diagonal of a symmetric matrix. The superscripts ⊺ and −1 stand for matrix transposition and matrix inverse, respectively.
In this section, we give a mathematical description of the problem under the study, followed by a definition of the average dwell time for a discrete switched system.
is called a switching law or switching signal, which is a piecewise constant function of discrete-time k and takes its values in the finite set . is the number of subsystems. For simplicity, at any arbitrary discrete-time , the switching signal is denoted by σ. Matrices , , , and are constant real matrices with appropriate dimensions for all . We denote the matrices associated with , , , and .
Definition 2.1 
holds for and , then is called the average dwell time and is the chatter bound.
Remark 1 The concept of average dwell time has been modified to fit the discrete-time ones in some existing literature [26–31]. The definition of average dwell time in Definition 2.1 is borrowed from these existing results. For simplicity, but without loss of generality, we choose in what follows.
Definition 2.2 
is said to be finite-time stable (FTS) with respect to , where is a positive definite matrix, and , if , .
Definition 2.3 
subject to an exogenous disturbance satisfying (2), is said to be finite-time bounded (FTB) with respect to , where is a positive definite matrix, and , if , .
Definition 2.4 
Under zero initial condition, it holds for all nonzero ω: .
3 Finite-time stability analysis
Then switched discrete-time system (5) with is finite-time bounded with respect to .
Thus we can conclude that switched discrete-time system (5) with is finite-time bounded with respect to . The proof is completed. □
4 Finite-time performance analysis
Then switched system (5) is finite-time bounded with performance level γ for any switching discrete-time signal with respect to .
According to Definition 2.4, we know that Theorem 4.1 holds. This completes the proof. □
5 Finite-time control design
Therefore, we can obtain (37). The proof is completed. □
Remark 2 In our paper, finite-time stability and Lyapunov asymptotic stability are independent concepts: a system which is finite-time stable maybe not Lyapunov asymptotically stable. On the contrary, a Lyapunov asymptotically stable system could be not finite-time stable, and during the transients, its state exceeds the prescribed bounds.
The optimized controller gains , can be derived by optimization procedure subject to (39) and (40) with fixed γ and minimum .
Remark 4 In this paper, if we can find a feasible solution with the parameter , through the discussion above, we know that the designed controller can ensure both finite-time and asymptotical stability of the delayed switched system. While in most situations we obtain controller with , and only finite-time stability can be established. Therefore, in real applications, asymptotical stabilizing controller for each subsystem should be designed to ensure asymptotical stability, which can be easily obtained by existing results for a non-switched system.
6 Illustrative example
In this paper, disturbance . The control objective is to find a feedback controller ensuring system (5) is finite-time bounded with respect to and minimum value of . Choose , and . According to Theorem 5.1, the optimal value of depends on parameter μ. Through LMI then we see that the feasible solution is , and .
In this paper, we have examined the problems of finite-time control of a switched discrete-time system with average dwell time. Based on the analysis result, the static state feedback control of finite-time boundness is given. Although the derived result is not in an LMIs form, we can turn it into the LMIs feasibility problem by fixing some parameters. A numerical example has also been given to demonstrate the effectiveness of the proposed approach. It should be noted that one of future research topics would be to investigate the problems of synchronous or asynchronous estimation for the switched neural network under the dwell time over a finite-time horizon.
The authors would like to thank the associate editor and the anonymous reviewers for their detailed comments and suggestions. This work was supported by the China Postdoctoral Science Foundation (2012M521683) and the Fundamental Research Funds for the Central Universities (103.1.2E022050205).
- Nagpal K, Ravi R: control and estimation problems with delayed measurements: state-space solutions. SIAM J. Control Optim. 1997, 35(4):1217-1243. 10.1137/S0363012994277499MathSciNetView ArticleGoogle Scholar
- Wu Z, Su H, Chu J: Delay-dependent H -infinity filtering for singular Markovian jump time-delay systems. Signal Process. 2010, 90(6):1815-1824. 10.1016/j.sigpro.2009.11.029View ArticleGoogle Scholar
- Zhao H, Chen Q, Xu S: guaranteed cost control for uncertain Markovian jump systems with mode-dependent distributed delays and input delays. J. Franklin Inst. 2009, 346(10):945-957. 10.1016/j.jfranklin.2009.05.007MathSciNetView ArticleGoogle Scholar
- Zhao H, Xu S, Lu J, Zhou J: Passivity based control for uncertain stochastic jumping systems with mode-dependent round-trip time delays. J. Franklin Inst. 2012, 349(5):1665-1680. 10.1016/j.jfranklin.2011.11.011MathSciNetView ArticleGoogle Scholar
- Lin J, Fei S, Gao Z: Stabilization of discrete-time switched singular time-delay systems under asynchronous switching. J. Franklin Inst. 2012, 349(5):1808-1827. 10.1016/j.jfranklin.2012.02.009MathSciNetView ArticleGoogle Scholar
- Wu Z, Shi P, Su H, Chu J: Stochastic synchronization of Markovian jump neural networks with time-varying delay using sampled-data. IEEE Trans. Cybern. 2012. 10.1109/TSMCB.2012.2230441Google Scholar
- Kushner H Math. Sci. Eng. 33. In Stochastic Stability and Control. Academic Press, New York; 1967.Google Scholar
- Kozin J: A survey of stability of stochastic systems. Automatica 1969, 5: 95-112. 10.1016/0005-1098(69)90060-0MathSciNetView ArticleGoogle Scholar
- Rami M, Zhou X: Linear matrix inequalities, Riccati equations, and indefinite stochastic linear quadratic controls. IEEE Trans. Autom. Control 2000, 45(6):1131-1143. 10.1109/9.863597MathSciNetView ArticleGoogle Scholar
- Wu Z, Shi P, Su H, Chu J: Delay-dependent stability analysis for switched neural networks with time-varying delay. IEEE Trans. Syst. Man Cybern., Part B, Cybern. 2011, 41(6):1522-1530.MathSciNetView ArticleGoogle Scholar
- Koenig D, Marx B:-filtering and state feedback control for discrete-time switched descriptor systems. IET Control Theory Appl. 2009, 3(6):661-670. 10.1049/iet-cta.2008.0132MathSciNetView ArticleGoogle Scholar
- Ma S, Zhang C, Wu Z:Delay-dependent stability and control for uncertain discrete switched singular systems with time-delay. Appl. Math. Comput. 2008, 206(1):413-424. 10.1016/j.amc.2008.09.020MathSciNetView ArticleGoogle Scholar
- Phat V, Botmart T, Niamsup P: Switching design for exponential stability of a class of nonlinear hybrid time-delay systems. Nonlinear Anal. Hybrid Syst. 2009, 3: 1-10. 10.1016/j.nahs.2008.10.001MathSciNetView ArticleGoogle Scholar
- Vu L, Morgansen K: Stability of time-delay feedback switched linear systems. IEEE Trans. Autom. Control 2010, 55(10):2385-2390.MathSciNetView ArticleGoogle Scholar
- Zhao X, Zhang L, Shi P, Liu M: Stability of switched positive linear systems with average dwell time switching. Automatica 2012, 48(6):1132-1137. 10.1016/j.automatica.2012.03.008MathSciNetView ArticleGoogle Scholar
- Xiang W, Xiao J: finite-time control for switched nonlinear discrete-time systems with norm-bounded disturbance. J. Franklin Inst. 2011, 348: 331-352. 10.1016/j.jfranklin.2010.12.001MathSciNetView ArticleGoogle Scholar
- Zhang L, Shi P:Stability, -gain and asynchronous control of discrete-time switched systems with average dwell-time. IEEE Trans. Autom. Control 2009, 54: 2193-2200.View ArticleGoogle Scholar
- Niu B, Zhao J:Stabilization and -gain analysis for a class of cascade switched nonlinear systems: an average dwell-time method. Nonlinear Anal. Hybrid Syst. 2011, 5(4):671-680. 10.1016/j.nahs.2011.05.005MathSciNetView ArticleGoogle Scholar
- Zhao X, Zhang L, Shi P, Liu M: Stability and stabilization of switched linear systems with mode-dependent average dwell time. IEEE Trans. Autom. Control 2012, 57(7):1809-1815.MathSciNetView ArticleGoogle Scholar
- Zhao X, Shi P, Zhang L: Asynchronously switched control of a class of slowly switched linear systems. Syst. Control Lett. 2012, 61(12):1151-1156. 10.1016/j.sysconle.2012.08.010MathSciNetView ArticleGoogle Scholar
- Wang D, Shi P, Wang W, Wang J: Delay-dependent exponential H-infinity filtering for discrete-time switched delay systems. Int. J. Robust Nonlinear Control 2012, 22(13):1522-1536. 10.1002/rnc.1764View ArticleGoogle Scholar
- Mahmoud M, Shi P:Asynchronous filtering of discrete-time switched systems. Signal Process. 2012, 92(10):2356-2364. 10.1016/j.sigpro.2012.02.007View ArticleGoogle Scholar
- Zhang G, Han C, Guan Y, Wu L: Exponential stability analysis and stabilization of discrete-time nonlinear switched systems with time delays. Int. J. Innov. Comput. Inf. Control 2012, 8(3):1973-1986.Google Scholar
- Attia S, Salhi S, Ksouri M: Static switched output feedback stabilization for linear discrete-time switched systems. Int. J. Innov. Comput. Inf. Control 2012, 8(5):3203-3213.Google Scholar
- Hou L, Zong G, Wu Y: Finite-time control for switched delay systems via dynamic output feedback. Int. J. Innov. Comput. Inf. Control 2012, 8(7):4901-4913.Google Scholar
- Dorato P: Short time stability in linear time-varying systems. Proc. IRE Int. Convention Record 1961, 4: 83-87.Google Scholar
- Cheng J, Zhu H, Zhong S, Zhang Y, Zeng Y:Finite-time stabilization of filtering for switched stochastic systems. Circuits Syst. Signal Process. 2012. 10.1007/s00034-012-9530-yGoogle Scholar
- Zhu L, Shen Y, Li C: Finite-time control of discrete-time systems with time-varying exogenous disturbance. Commun. Nonlinear Sci. Numer. Simul. 2009, 14: 361-370. 10.1016/j.cnsns.2007.09.013MathSciNetView ArticleGoogle Scholar
- Huang X, Lin W, Yang B: Global finite-time stabilization of a class of uncertain nonlinear systems. Automatica 2005, 41(5):881-888. 10.1016/j.automatica.2004.11.036MathSciNetView ArticleGoogle Scholar
- Qian C, Li J: Global finite-time stabilization by output feedback for planar systems without observable linearization. IEEE Trans. Autom. Control 2005, 50(6):549-564.MathSciNetGoogle Scholar
- Hong Y: Finite-time stabilization and stability of a class of controllable systems. Syst. Control Lett. 2002, 48(4):231-236.View ArticleGoogle Scholar
- Amato F, Ariola M: Finite-time control of discrete-time linear systems. IEEE Trans. Autom. Control 2005, 50(5):724-729.MathSciNetView ArticleGoogle Scholar
- Zhai G, Hu B, Yasuda K, Michel A: Qualitative analysis of discrete-time switched systems. Proceedings of the American Control Conference 2002, 1880-1885.Google Scholar
- Song Y, Fan J, Fei M, Yang T:Robust control of discrete switched system with time delay. Appl. Math. Comput. 2008, 205(1):159-169. 10.1016/j.amc.2008.05.046MathSciNetView ArticleGoogle Scholar
- Zhang L, Boukas E, Shi P: Exponential filtering for uncertain discrete-time switched linear systems with average dwell time: a μ -dependent approach. Int. J. Robust Nonlinear Control 2008, 18(11):1188-1207. 10.1002/rnc.1276MathSciNetView ArticleGoogle Scholar
- Yang Y, Li J, Chen G: Finite-time stability and stabilization of Markovian switching stochastic systems with impulsive effects. J. Syst. Eng. Electron. 2010, 21(2):2254-2260.Google Scholar
- Luan X, Liu F, Shi P: Finite-time filtering for non-linear stochastic systems with partially known transition jump rates. IET Control Theory Appl. 2010, 4(5):735-745. 10.1049/iet-cta.2009.0014MathSciNetView ArticleGoogle Scholar
- Amato F, Ariola M, Cosentino C: Finite-time control of discrete-time linear systems: analysis and design conditions. Automatica 2010, 46: 919-924. 10.1016/j.automatica.2010.02.008MathSciNetView ArticleGoogle Scholar
- Xiang W, Xiao J, Xiao C: On finite-time stability and stabilization for switched discrete linear systems. Control Intell. Syst. 2011, 39(2):122-128.MathSciNetGoogle Scholar
- Lin X, Du H, Li S:Finite-time boundedness and -gain analysis for switched delay systems with norm-bounded disturbance. Appl. Math. Comput. 2011, 217(12):982-993.MathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.