Robust Stabilization of Discrete-Time Switched Periodic Systems with Time Delays

This paper studies the problems of robust stability and robust stabilization for discrete-time switched periodic systems with timevarying 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.


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][4][5].Switched systems are involved in multiple applications, such as communication, computer, networked control systems, and flight and robot manipulators [6][7][8][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  2 -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 timeinvariant 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 2 Complexity of robust asymptotical stabilization for uncertain discretetime switched periodic systems with mode-dependent timevarying 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, Z is the set of integers,  + is the set of nonnegative integers,   and  × denote the ndimensional Euclidian space and the set of × real matrices, respectively. and 0 represent the identity matrix and null matrix of appropriate dimensions, and diag{. ..} is a blockdiagonal 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 ( + ) = (), ∀ ∈ Z.The superscript T and (−1) denote the matrix transposition and matrix inverse, respectively.

Robust Stability Analysis
First, we consider the following uncertain discrete-time switched periodic system: The system (8) can be written in the equivalent descriptor form That 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.

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.
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 .When   (),   (), and   () are constant matrices, we have obtained constant gain matrices   .The following corollary can be immediately obtained.
Remark .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 .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 .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 discretetime 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.

Numerical Examples
In this subsection, we give two numerical examples to show the high performance of the proposed approach.
Example .Consider the uncertain 2-periodic time-delay system (1) with   = {1, 2} and with parameters given by The state feedback gains are given by The 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 .Consider the uncertain 2-periodic time-delay system (1) with   = {1, 2} and with parameters given by (55) The state feedback gains are given by (56) Figure 2 shows the state responses of the corresponding closed-loop system.The simulation results reveal that the trajectories  1 () and  2 () are converging to the equilibrium point zero.

Conclusions
This paper has investigated the robust asymptotical stability and robust asymptotical stabilization for uncertain discretetime switched periodic systems with mode-dependent timevarying 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 modedependent 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.

Figure 2 :
Figure 2: State trajectories of the closed-loop system in Example 2.