Abstract
This paper studies the problems of robust stability and robust stabilization for discrete-time switched periodic systems with time-varying delays and parameter uncertainty. We obtain the novel sufficient conditions to ensure the switched system is robustly asymptotically stable in terms of linear matrix inequalities. To obtain these conditions, we utilize a descriptor system method and introduce a switched Lyapunov-Krasovskii functional. The robust stability results are then extended to solve problems of robust stabilization via periodic state feedback. Novel sufficient conditions are established to ensure that the uncertain switched periodic system is robustly asymptotically stabilizable. Finally, we give two numerical examples to illustrate the effectiveness of our method.
1. Introduction
Cyclic processes exist in many nature and engineering phenomenon. Therefore, periodic systems can be applied in many fields, such as economics, population dynamics, and signal processing where cyclostationary noise, control of multirate plants, and multiplexed systems are present (see [1, 2] and the references therein).
Switched system consisted of several subsystems and a switching law which determines which one of these subsystem is activated. Switched system is considered as a particular kind of hybrid systems [3–5]. Switched systems are involved in multiple applications, such as communication, computer, networked control systems, and flight and robot manipulators [6–9]. In [9], Liu et al. studied the problems of stability and stabilization for a class of switched nonlinear systems via an average dwell time approach. Pérez et al. [10] proposed a new approach to stabilize switched linear systems. In [11], Li et al. studied the exponential stability of time-controlled switching systems with time delay. Dong et al. [12] considered the problems of exponential stabilization and -gain for a class of uncertain switched nonlinear systems. In [13], using the adaptive distributed observer method, the containment control problem of nonidentical networks was considered. In [14], the cooperative containment control problem for heterogeneous discrete-time linear multiagent systems was investigated.
Periodic linear systems can model many practical control systems such as sample data systems and systems that operate periodically [15]. The closed-loop system consisting of a time-invariant plant and a periodic controller is considered as another important source of periodic systems [16]. In [17], the stability of linear periodic systems with time-delay was considered. Some results on stabilization of linear periodic discrete-time systems were presented in [18]. In [19], Dong et al. considered the robustly exponential stability and stabilization for uncertain linear discrete-time periodic systems.
The problems of robust stability and stabilization for uncertain discrete-time switched periodic systems with mode-dependent time-varying delays have been barely studied. Motivated by this consideration, in this paper, we firstly consider the problem of robust stability for uncertain discrete-time switched periodic systems with time-varying delays and polytopic-type parameter uncertainty. By using uncertainty-dependent switched Lyapunov-Krasovskii functional, we established robust asymptotical stability criterion for uncertain discrete-time switched periodic systems without control. The results of robust asymptotical stability are then adapted to solve problems of robust stabilization via static periodic state feedback. We proposed novel criteria of robust asymptotical stabilization for uncertain discrete-time switched periodic systems with mode-dependent time-varying delays, and we designed the periodic state feedback control.
The following paper is consisted of 5 sections. Section 2 formulates the problem and gives the preliminaries. Methods of robust stability analysis are developed in Section 3. The techniques for designing robustly stabilizing periodic state feedback controller are derived in Section 4. In Section 5, two examples are given to show the performances of our method. Finally, in Section 6, conclusions are drawn.
Notations. Throughout this paper, is the set of integers, is the set of nonnegative integers, and denote the n-dimensional Euclidian space and the set of real matrices, respectively. and 0 represent the identity matrix and null matrix of appropriate dimensions, and is a block-diagonal matrix. For symmetric block matrices, stands for the transpose of the blocks outside of the main diagonal block. Given a matrix and a positive integer N, the matrix is denoted N-periodic if . The superscript T and denote the matrix transposition and matrix inverse, respectively.
2. Problem Formulation
In this paper, we consider a class of uncertain discrete-time switched periodic systems: where is the state vector and is the control. is the switching signal. , and are uncertain N-periodic real matrices with appropriate dimensions that are assumed to be confined to the following polytope:where with , and , being given N-periodic real matrices.
The sequence is the initial condition; the time-varying delay satisfies the following condition:where and are nonnegative integers representing.
In this paper, our aim is to design a stabilizing state feedback controlwhere is an N-periodic matrix, such that the closed-loop system is asymptotically stable for all system matrices be confined by (2).
3. Robust Stability Analysis
First, we consider the following uncertain discrete-time switched periodic system:
The system (8) can be written in the equivalent descriptor formThat is,which can be further rewritten:
In this section, we deal with the problem of robust stability for system (8). The LMI-based stability conditions will be established.
Theorem 1. For given scalars , the switched periodic system (8) is robustly asymptotically stable, if there exist N periodic matrices , and matrices , such that where
Proof. Construct a switching Lyapunov-Krasovskii functional aswithwhere and
Given and , then the difference of , along the solution of (11) is given byFrom (11), we haveFrom (16) to (21), we getwhere From (12) we know thatThus, using again, by (22), (24), and (12) we getwhere Now the proof of Theorem 1 has been completed.
Remark 2. When , i.e., the lower and upper bounds of the time delay satisfythe following corollary can be obtained.
Corollary 3. Switched periodic system (8) with is robustly asymptotically stable, if there exist N periodic matrices , and matrices such thatwhere
Proof. Construct a switching Lyapunov-Krasovskii functional aswhereThe rest proofs are omitted as they are similar to the proof of Theorem 1.
4. Robust Stabilization of Switched Periodic System
In this section, we study the problem of robust stabilization for system (1). We derive the LMI-based stabilization conditions.
Theorem 4. For given scalars , the switched periodic system (1) is robustly asymptotically stabilizable, if there exist N periodic inverse matrices , N periodic matrices , and matrices such thatwhereParticularly, the state feedback matrix is
Proof. Define the Lyapunov-Krasovskii functional as follows:where
Given and , then using (20), similar to the proof of Theorem 1, the sufficient condition for asymptotic stability of system (1) with feedback iswhereBy Schur complement, (35) holds if and only ifBefore and after multiplying the left term of (37) by and , (37) can be cast as below:whereWe note that so (37) holds ifwhereLet Using Schur complement, (41) holds if and only ifwhereLet . We have (43) being equivalent to So (32) implies that (45) holds. Now the proof of Theorem 4 has been completed.
Theorem 5. For given scalars , the switched periodic system (1) is robustly asymptotically stabilizable, if there exist N periodic inverse matrices , N periodic matrices , and matrices , such thatwhereMoreover, the controller gains are
Proof. Consider the following Lyapunov-Krasovskii functional candidate:The rest of the proofs are similar to the proof of Theorem 4, which are omitted here.
Remark 6. When , and are constant matrices, we have obtained constant gain matrices The following corollary can be immediately obtained.
Corollary 7. For given scalars , the switched periodic system (1) is robustly asymptotically stabilizable, if there exist matrices , and inverse matrices , such that whereMoreover, the controller gains are
Remark 8. In [20], the robust stability and robust control for linear discrete-time periodic systems were investigated. But the switched periodic systems were not considered in [20]. In [21], Sakly and Kermanit analyse the stability and stabilization with a state feedback controller for a class of switched systems. But the switched system in [21] was not a time delay system, and it was not a periodic system. Compared with [20, 21], the results obtained in this paper have a greater range of applications.
Remark 9. Deaecto et al. [22] considered stability analysis and control design for discrete-time switched linear system. But the system in [22] does not contain uncertainty and periodic systems were not considered in [22]. Li et al. [23] proposed conditions of stability and stabilization for periodic piecewise linear systems but they neither considered discrete-time periodic systems nor taken uncertainty into consideration. Compared with [22, 23], the results obtained in this paper have a greater range of applications.
Remark 10. Dong et al. [19] investigated the problems of robustly exponential stability and exponential stabilization for uncertain linear discrete-time periodic systems. But the switched periodic systems were not considered in [19]. In this paper, we deal with uncertain discrete-time switched periodic system with time delay. We establish the novel criterion of robust asymptotically stabilization for uncertain discrete-time switched periodic systems with time-varying delays via periodic switched state feedback. Compared with [19], the system structure is more complex, and the results obtained in this paper have a greater range of applications.
5. Numerical Examples
In this subsection, we give two numerical examples to show the high performance of the proposed approach.
Example 1. Consider the uncertain 2-periodic time-delay system (1) with and with parameters given bySolving (46) gives feasible solutions: The state feedback gains are given byThe simulation results in Figure 1 show for the state responses of the resulting closed-loop system. It can be observed that the closed-loop system in Example 1 is asymptotically stable.
Example 2. Consider the uncertain 2-periodic time-delay system (1) with and with parameters given byTaking and solving (46) gives feasible solutions:The state feedback gains are given byFigure 2 shows the state responses of the corresponding closed-loop system. The simulation results reveal that the trajectories and are converging to the equilibrium point zero.
6. Conclusions
This paper has investigated the robust asymptotical stability and robust asymptotical stabilization for uncertain discrete-time switched periodic systems with mode-dependent time-varying delays and polytopic-type parameter uncertainty in the matrices of the state-space model. We have obtained the novel sufficient conditions in terms of LMIs to ensure that the system is robustly asymptotically stable. Furthermore, the robust asymptotical stability results are extended to solve problems of robust asymptotically stabilization for uncertain discrete-time switched periodic systems with mode-dependent time-varying delays via periodic state feedback. Novel sufficient conditions are established to guarantee the switched periodic system is robustly asymptotically stabilizable. Finally, to illustrate the results we have obtained, we give two examples.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Natural Science Foundation of Tianjin under Grant no. 18JCYBJC88000 and the National Nature Science Foundation of China under Grant nos. 61873186 and 61703307.