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

Authors: Yali Dong; Jing Hao; Yonghong Yao; Huimin Wang
Journal: Complexity
 (2019)

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## 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.


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## Body

## 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 L2-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. Section2 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, Z+ is the set of nonnegative integers, Rn and Rn×m denote then-dimensional Euclidian space and the set of n×m real matrices, respectively. I and 0 represent the identity matrix and null matrix of appropriate dimensions, and diag⁡{…} 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 S(k) and a positive integerN, the matrix S(k) is denotedN-periodic if S(k+N)=S(k),∀k∈Z~. The superscriptT and (-1) 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:(1)xk+1=Aσkkxk+Adσkkxk-τσkk+Bσkkuk,xk=φk,k=-d2,-d2+1,⋯,0,where x(k)∈Rn is the state vector and u(k)∈Rm is the control. σ(k):Z+=0,1,2,⋯→N⌢=1,2,⋯,N- is the switching signal. Aσ(k)(k),Adσ(k)(k), and Bσ(k)(k) are uncertainN-periodic real matrices with appropriate dimensions that are assumed to be confined to the following polytope:(2)Εik=Υik:Υik=∑r=1vλrΥirk,λr≥0,∑r=1vλr=1,where (3)Υik=Aik,Adik,Bik,(4)Υirk=Airk,Adirk,Birk,with Air(k),Adir(k), and Bir(k),r=1,2,⋯,v,i=1,2,⋯,N-, being givenN-periodic real matrices.The sequenceφ(k) is the initial condition; the time-varying delay τi(k) satisfies the following condition:(5)0<d1≤τik≤d2,where d1 and d2 are nonnegative integers representing.In this paper, our aim is to design a stabilizing state feedback control(6)uk=Kσkkxk,where Kσ(k)(k) is anN-periodic matrix, such that the closed-loop system (7)xk+1=Aσkk+BσkkKσkkxk+Adσkkxk-τσkk,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:(8)xk+1=Aσkkxk+Adσkkxk-τσkk,xk=φk,k=-d2,-d2+1,⋯,0.The system (8) can be written in the equivalent descriptor form(9)I000xk+1yk+1=IIAσkk-I-Ixkyk+00Adσkk0xk-τσkkyk-τσkk,xk=φk,k=-d2,-d2+1,⋯,0.That is,(10)xk+1=xk+yk,yk=Aσkk-Ixk+Adσkkxk-τσkk,xk=φk,k=-d2,-d2+1,⋯,0,which can be further rewritten:(11)xk+1=xk+yk,yk=Aσkk+Adσkk-Ixk-Adσkk∑l=k-τσkkk-1yl,xk=φk,k=-d2,-d2+1,⋯,0.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 scalars0<d1≤d2, the switched periodic system (8) is robustly asymptotically stable, if there existN periodic matrices Ri(k),Ui(k),Ti(k),Si(k),i=1,2,⋯,N-,Q1r(k)>0,Q2r(k)>0,r=1,2,⋯,v, and matrices Q3>0,Z>0,Pi>0,i=1,2,⋯,N-, such that (12)Ξ~1Ξ~2Ξ~13Ξ~14-d2UikAdirk∗Ξ~3-TiTk-SiTk-d2RikAdirk∗∗-Q1rk-d1+Q30-d2TikAdirk∗∗∗-Q2rk-d2-d2SikAdirk∗∗∗∗-Z<0,k=1,2,⋯,N,r=1,2,⋯,v,i,j∈N⌢,where(13)Ξ~1=Pj-Pi+Q1rk+Q2rk+d12Q3+UikAirk+Adirk-I+Airk+Adirk-ITUiTk,Ξ~2=Pj+AirTk+AdirTk-IRiTk-Uik,Ξ~3=Pj+Z-Rik-RiTk,d12=d2-d1,Ξ~13=Airk+Adirk-ITTiTk,Ξ~14=Airk+Adirk-ITSiTk.Proof.
Construct a switching Lyapunov-Krasovskii functionalV(k) as(14)Vk=V1k+V2k+V3k+V4kwith(15)V1k=d2xTkPσkxk,V2k=d2∑s=k-d1k-1xTsQ1sxs+d2∑s=k-d2k-1xTsQ2sxs,V3k=d2∑s=-d2+1-d1∑m=k+sk-1xTmQ3xm,V4k=∑s=-d2+10∑l=k-1+sk-1yTlZyl,where Q1k=∑r=1vλrQ1rk and Q2(k)=∑r=1vλrQ2r(k).
Givenσ(k)=i and σ(k+1)=j, then the difference of Vi(k),i=1,2,3,4, along the solution of (11) is given by(16)ΔV1k=d2xTk+1Pjxk+1-d2xTkPixk=d2xk+ykTPjxk+yk-d2xTkPixk=d2xTkPj-Pixk+2d2xTkPjyk+d2yTkPjyk,(17)ΔV2k=d2xTkQ1k+Q2kxk-xTk-d1Q1k-d1xk-d1-xTk-d2Q2k-d2xk-d2,(18)ΔV3k=d2∑s=-d2+1-d1∑m=k+1+skxTmQ3xm-d2∑s=-d2+1-d1∑m=k+sk-1xTmQ3xm}=d2d12xTkQ3xk-∑m=k+1-d2k-d1xTmQ3xm,(19)ΔV4k=d2yTkZyk-∑l=k-d2k-1yTlZyl≤d2yTkZyk-∑l=k-τikk-1yTlZylFrom (11), we have(20)2d2xTkUik+yTkRik-yk+Aik+Adik-Ixk-Adik∑l=k-τikk-1yl=0,(21)2d2xTk-d1Tik+xTk-d2Sik-yk+Aik+Adik-Ixk-Adik∑l=k-τikk-1yl=0.From (16) to (21), we get(22)
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Whenτσ(k)(k)≡d, i.e., the lower and upper bounds of the time delay satisfy(27)d1=d2,the following corollary can be obtained.Corollary 3.
Switched periodic system (8) with τσ(k)(k)≡d is robustly asymptotically stable, if there existN periodic matrices Ri(k),Ui(k),Ti(k),i=1,2,⋯,N-,Qr(k)>0,r=1,2,⋯,v, and matrices Z>0,Pi>0,i=1,2,⋯,N- such that(28)Γ1Γ2Γ13-dUikAdirk∗Γ3-TiTk-dRikAdirk∗∗-Qrk-d-d2TikAdirk∗∗∗-Z<0,k=1,2,⋯,N,r=1,2,⋯,v,i,j∈N⌢,where(29)Γ1=Pj-Pi+Qrk+UikAirk+Adirk-I+Airk+Adirk-ITUiTk,Γ2=Pj+AirTk+AdirTk-IRiTk-Uik,Γ3=Pj+Z-Rik-RiTk,Γ13=Aik+Adik-ITTiTk.Proof.
Construct a switching Lyapunov-Krasovskii functionalV(k) as(30)Vk=V1k+V2k+V3k,where(31)V1k=dxTkPσkxk,V2k=d∑s=k-dk-1xTsQsxs,V3k=∑s=-d2+10∑l=k-1+sk-1yTlZyl.
The rest proofs are omitted as they are similar to the proof of Theorem1.


## 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 scalars0<d1≤d2, the switched periodic system (1) is robustly asymptotically stabilizable, if there existN periodic inverse matrices Gi(k),Oi(k),i=1,2,⋯,N-,N periodic matrices Yi(k),Hr(k)>0,r=1,2,⋯v, and matrices Γ>0,Xi>0,i=1,2,⋯,N- such that(32)Τ^1Τ^200d2AdirkΓGik02Gik∗Τ^300-d2AdirkΓOikOik0∗∗-Hrk-d100000∗∗∗-Hrk-d20000∗∗∗∗-Γ000∗∗∗∗∗-Xj00∗∗∗∗∗∗-Γ0∗∗∗∗∗∗∗-Hrk<0,k=1,2,⋯,N,i,j∈N⌢,r=1,2,⋯,v,where(33)Τ^1=Xi-2Gik-2GiTk+AirkGiTk+BirkYik+AdirkGiTk+GikAirTk+YiTkBirTk+GikAdirTk,Τ^2=GikAirTk+YiTkBirTk+GikAdirTk-Gik-Oi-Tk,Τ^3=-Oik-Oi-Tk.Particularly, the state feedback matrix is Ki(k)=Yi(k)Gi-T(k).Proof.
Define the Lyapunov-Krasovskii functional as follows:(34)Vk=d2xTkPσkxk+d2∑s=k-d2k-1xTsQsxs+∑s=-d2+10∑l=k-1+sk-1yTlZyl,where Q(s)=H-1(s),H(s)=∑r=1vλrHr(s).
Givenσ(k)=i and σ(k+1)=j, then using (20), similar to the proof of Theorem 1, the sufficient condition for asymptotic stability of system (1) with feedback u(k)=Ki(k)x(k) is(35)Ω^=Ξ^1Ξ^200-d2UikAdik∗Ξ^300-d2RikAdik∗∗-Qk-d100∗∗∗-Qk-d20∗∗∗∗-Z<0,k=1,2,⋯,N,i,j∈N⌢,where(36)Ξ^1=Pj-Pi+2Qk+UikAik+BikKik+Adik-I+Aik+BikKik+Adik-ITUiTk,Ξ^2=Pj+AiTk+KiTkBiTk+AdiTk-IRiTk-Uik,Ξ^3=Pj+Z-Rik-RiTk.By Schur complement, (35) holds if and only if(37)Ξ^1-PjΞ^2-Pj00-d2UikAdikI0∗Ξ^3-Pj-Z00-d2RikAdikII∗∗-Qk-d10000∗∗∗-Qk-d2000∗∗∗∗-Z00∗∗∗∗∗-Pj-10∗∗∗∗∗∗-Z-1<0,k=1,2,⋯,N,i,j∈N⌢,Before and after multiplying the left term of (37) by diag⁡{Ui-1(k),Ri-1(k),I,I,Z-1,I,I} and diag⁡{Ui-T(k),Ri-T(k),I,I,Z-1,I,I}, (37) can be cast as below:(38)Τ1Τ200d2AdikZ-1Ui-1k0∗Τ300-d2AdikZ-1Ri-1kRi-1k∗∗-Qk-d10000∗∗∗-Qk-d2000∗∗∗∗-Z-100∗∗∗∗∗-Pj-10∗∗∗∗∗∗-Z-1<0,k=1,2,⋯,N,i,j∈N⌢,where(39)Τ1=-Ui-1kPiUi-Tk+2Ui-1QkUi-Tk+Aik+BikKik+Adik-IUi-Tk+Ui-1kAik+BikKik+Adik-IT,Τ2=Ui-1kAiTk+KiTkBiTk+AdiTk-I-Ri-Tk,Τ3=-Ri-Tk-Ri-1k.We note that (40)-Ui-1PiUi-1T≤-Ui-1-Ui-1T+Pi-1,so (37) holds if(41)Τ~1Τ200d2AdikZ-1Ui-1k0∗Τ300-d2AdikZ-1Ri-1kRi-1k∗∗-Qk-d10000∗∗∗-Qk-d2000∗∗∗∗-Z-100∗∗∗∗∗-Pj-10∗∗∗∗∗∗-Z-1<0,k=1,2,⋯,N,i,j∈N⌢,where(42)Τ~1=Pi-1-Ui-1k-Ui-Tk+2Ui-1kQkUi-Tk+Aik+BikKik+Adik-IUi-Tk+Ui-1kAik+BikKik+Adik-IT.Let Xi=Pi-1,Gi(k)=Ui-1(k),Ki(k)GiT(k)=Yi(k),Γ=Z-1,Oi(k)=Ri-1(k). Using Schur complement, (41) holds if and only if(43)Τ-1Τ200d2AdikΓGik02Gik∗Τ300-d2AdikΓOikOik0∗∗-Qk-d100000∗∗∗-Qk-d20000∗∗∗∗-Γ000∗∗∗∗∗-Pj-100∗∗∗∗∗∗-Γ0∗∗∗∗∗∗∗-Q-1k<0,k=1,2,⋯,N,i,j∈N⌢,where(44)Τ-1=Xi-2Gik-2GiTk+AikGiT+BikYik+AdikGiTk+GikAiTk+YiTkBiTk+GikAdiTk,Τ2=GikAiTk+YiTkBiTk+GiAdiTk-Gik-Oi-Tk,Τ3=-Oik-Oi-Tk.Let H(k)=Q-1(k). We have (43) being equivalent to (45)Τ-1Τ200d2AdikΓGik02Gik∗Τ300-d2AdikΓOikOik0∗∗-Hk-d100000∗∗∗-Hk-d20000∗∗∗∗-Γ000∗∗∗∗∗-Xj00∗∗∗∗∗∗-Γ0∗∗∗∗∗∗∗-Hk<0,k=1,2,⋯,N,i,j∈N⌢.So (32) implies that (45) holds. Now the proof of Theorem 4 has been completed.Theorem 5.
For given scalars0<d1≤d2, the switched periodic system (1) is robustly asymptotically stabilizable, if there existN periodic inverse matrices Gi(k),Oi(k),i=1,2,⋯,N-,N periodic matrices Yi(k), and matrices Γ>0,Xi>0,i=1,2,⋯,N-, such that(46)Τ^1Τ^2d2AdirkΓGik0∗Τ^3-d2AdirkΓOikOik∗∗-Γ00∗∗∗-Xj0∗∗∗∗-Γ<0,k=1,2,⋯,N,i,j∈N⌢,r=1,2,⋯,v,where(47)Τ^1=Xi-2Gik-2GiTk+AirkGiTk+BirkYik+AdirkGiTk+GikAirTk+YiTkBirTk+GikAdirTk,Τ^2=GikAirTk+YiTkBirTk+GikAdirTk-Gik-Oi-Tk,Τ^3=-Oik-Oi-Tk.Moreover, the controller gains are Ki(k)=Yi(k)Gi-T(k).Proof.
Consider the following Lyapunov-Krasovskii functional candidate:(48)Vk=d2xTkPσkxk+∑s=-d2+10∑l=k-1+sk-1yTlZyl.
The rest of the proofs are similar to the proof of Theorem4, which are omitted here.Remark 6.
WhenGi(k),Oi(k), and Yi(k) are constant matrices, we have obtained constant gain matrices Ki. The following corollary can be immediately obtained.Corollary 7.
For given scalars0<d1≤d2, the switched periodic system (1) is robustly asymptotically stabilizable, if there exist matrices Γ>0,Xi>0,Yi,i=1,2,⋯,N-, and inverse matrices Gi,Oi,i=1,2,⋯,N-, such that (49)Π1Π2d2AdirkΓGi0∗Π3-d2AdirkΓOiOi∗∗-Γ00∗∗∗-Xj0∗∗∗∗-Γ<0,k=1,2,⋯,N,i,j∈N⌢,r=1,2,⋯,v,where(50)Π1=Xi-2Gi-2GiT+AirkGiT+BirkYi+AdirkGiTk+GiAirTk+YiTBirTk+GiAdirTk,Π2=GiAirTk+YiTBirTk+GiAdirTk-Gi-Oi-T,Π3=-Oi-Oi-T.Moreover, the controller gains are Ki=YiGi-T.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 N⌢={1,2} and with parameters given by(51)A111=0.80.100.7,A112=0.8000.9,Ad111=-0.20.11-0.2-0.1,Ad112=-0.20.0-0.1-0.1,A211=0.70.100.6,A212=0.720.100.74,Ad211=-0.10.11-0.1-0.1,Ad212=-0.10.0-0.1-0.1,A121=0.90.1300.8,A122=0.82000.92,Ad121=-0.220.11-0.23-0.1,Ad122=-0.220.0-0.13-0.11,A221=0.730.1200.6,A222=0.60.1200.64,Ad221=-0.120.11-0.1-0.11,Ad222=-0.120.0-0.13-0.1,Birk=01T,i=1,2,r=1,2,k=1,2,d2=3.Solving (46) gives feasible solutions: (52)G11=0.56750.07940.07941.1183,G12=0.8050-0.0020-0.00201.1276,G21=0.9955-0.2183-0.21831.2351,G22=1.0742-0.0927-0.09271.3958,O11=-0.1457-0.2262-0.22620.3703,O12=0.68080.01380.01381.0317,O21=0.7931-0.0788-0.07881.0210,O22=0.7122-0.0141-0.01411.0887,Γ=0.80620.11910.11912.6328X1=0.6113-0.0036-0.00360.5159,X2=0.5115-0.0761-0.07611.0831,Y11=-0.2002-0.6940,Y12=0.0297-1.7651,Y21=0.1521-1.4531,Y22=0.0965-1.8628.The state feedback gains are given by(53)K11=-0.2686-0.6015,K12=0.0330-1.5653,K21=-0.1094-1.1958,K22=-0.0255-1.3363.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.Figure 1
State trajectories of the closed-loop system in Example1.Example 2.
Consider the uncertain 2-periodic time-delay system (1) with N⌢={1,2} and with parameters given by(54)A11=-0.20.20-0.7,A12=-0.80.20-0.9,Ad11=-0.200.1-0.2,Ad12=-0.200.1-0.2,A21=-0.50.20-0.6,A22=-0.90.20-0.74,Ad21=-0.30.110.1-0.2,Ad22=-0.40.00.1-0.2,B11=0.100.10.1,B12=0.120.1100.11,B21=0.100.10.2,B22=0.30.200.12.Taking d2=3 and solving (46) gives feasible solutions:(55)G11=10-100.75930.06060.06060.8832,G12=10-100.20650.03590.03590.3104,G21=10-100.61800.10450.10450.6588,G22=10-100.57680.09380.09380.6349,O11=10-100.54020.08690.08690.7681,O12=10-100.20110.03670.03670.3249,O21=10-100.51300.06250.06250.6888,O22=10-100.57050.10620.10620.6681,X1=10-100.43690.02370.02370.4369,X2=10-100.45000.03460.03460.5109,Y11=10-9-0.0020-0.1017-0.10170.3025,Y12=10-9-0.27460.0404-0.0404-0.0255,Y21=10-100.5248-0.3385-0.33850.3209,Y22=10-90.1078-0.0076-0.00760.0203,Γ=10-90.10920.07350.07350.3025.The state feedback gains are given by(56)K11=0.0666-1.1566-1.74885.1201,K12=-13.80242.90102.1444-1.0694,K21=0.9619-0.6663-0.64740.5898,K22=1.9351-0.4061-0.18870.3473.Figure 2 shows the state responses of the corresponding closed-loop system. The simulation results reveal that the trajectories x1(k) and x2(k) are converging to the equilibrium point zero.Figure 2
State trajectories of the closed-loop system in Example2.


## 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.



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