# Robust Stability Analysis for Uncertain Switched Discrete-Time Systems

Authors: Yali Dong
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

This paper is concerned with the robust stability for a class of switched discrete-time systems with state parameter uncertainty. Firstly, a new matrix inequality considering uncertainties is introduced and proved. By means of it, a novel sufficient condition for robust stability of a class of uncertain switched discrete-time systems is presented. Furthermore, based on the result obtained, the switching law is designed and has been performed well, and some sufficient conditions of robust stability have been derived for the uncertain switched discrete-time systems using the Lyapunov stability theorem, block matrix method and inequality technology. Finally, some examples are exploited to illustrate the effectiveness of the proposed schemes.


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## Body

## 1. Introduction

A switched system is a hybrid dynamical system consisting of a finite number of subsystems and a logical rule that manages switching between these subsystems. Switched systems have drawn a great deal of attention in recent years, see [1–24] and references therein. The motivation for studying switched systems comes partly from the fact that switched systems and switched multi-controller systems have numerous applications in control of mechanical systems, process control, automotive industry, power systems, aircraft and traffic control, and many other fields. An important qualitative property of switched system is stability [1–3]. The challenge of analyzing the stability of switched system lies partly in the fact that, even if the individual systems are stable, the switched system might be unstable. Using a common quadratic Lyapunov function on all subsystems, the quadratic Lyapunov stability facilitates the analysis and synthesis of switched systems. However, the obtained results within this framework have been recognized to be conservative. In [10], various algorithms both for stability and performance analysis of discretetime piece-wise affine systems were presented. Different classes of Lyapunov functions were considered, and how to compute them through linear matrix inequalities was also shown. Moreover, the tradeoff between the degree of conservativeness and computational requirements was discussed. The problem of stability analysis and control synthesis of switched systems in the discrete-time domain was addressed in [11]. The approach followed in [11] looked at the existence of a switched quadratic Lyapunov function to check asymptotic stability of the switched system under consideration. Two different linear matrix inequality-based conditions allow to check the existence of such a Lyapunov function. These two conditions have been proved to be equivalent for stability analysis.There are many methodologies and approaches developed in the switched systems theory: approaches of looking for an appropriate switching strategy to stabilize the system [4], dwell-time and average dwell-time approaches for stability analysis and stabilization problems [5], approaches of studying stability and control problems under a specific class of switching laws [1] or under arbitrary switching sequences [6–9]. Reference [19] investigated the quadratic stability and linear state feedback and output feedback stabilization of switched delayed linear dynamic systems with, in general, a finite number of non-commensurate constant internal point delays. The results were obtained based on Lyapunov’s stability analysis via appropriate Krasovskii-Lyapunov’s functionals, and the related stability study was performed to obtain both delay-independent and delay-dependent results. The problem of fault estimation for a class of switched nonlinear systems of neutral type was considered in [20]. Sufficient delay-dependent existence conditions of the H∞ fault estimator were given in terms of certain matrix inequalities based on the average dwell-time approach. The problem of robust reliable control for a class of uncertain switched neutral systems under asynchronous switching was investigated in [21]. A state feedback controller was proposed to guarantee exponential stability and reliability for switched neutral systems, and the dwell-time approach was utilized for the stability analysis and controller design. The exponential stability for a class of nonlinear hybrid time-delay systems was addressed in [24]. The delay-dependent stability conditions were presented in terms of the solution of algebraic Riccati equations, which allows computing simultaneously the two bounds that characterize the stability rate of the solution.On another research front line, it has been recognized that parameter uncertainties, which often occur in many physical processes, are main sources of instability and poor performance. Therefore, much attention has been devoted to the study of various systems with uncertainties, and a great number of useful results have been reported in the literature on the issues of robust stability, robustH∞ control, robust H∞ filtering, and so on, by considering different classes of parameter uncertainties [12–14].Recently, some stability condition and stabilization approaches have been proposed for the switched discrete-time system [15–18]. In [15], the quadratic stabilization of discrete-time switched linear systems was studied, and quadratic stabilization of switched systems with norm bounded time varying uncertainties was investigated. In [16], the stability property for the switched systems which were composed of a continuous-time LTI subsystem and a discretetime LTI subsystem was studied. There existed a switched quadratic Lyapunov function to check asymptotic stability of the switched discrete-time system in [17].The objective of this paper is to present novel approaches for the asymptotical stability of switched discrete-time system with parametric uncertainties. The parameter uncertainties are time-varying but norm-bounded. Firstly, a new inequality is given. Using the new result, a new sufficient condition for robust stability of a class of uncertain switched discrete-time systems is proposed. Furthermore, using the block matrix method, inequality technology, and the Lyapunov stability theorem, some sufficient conditions for robust stability have been presented for the uncertain switched discrete-time systems, and the switched law design has been performed. Comparing with [22, 23], the uncertainty in system was not considered in [22, 23], but we consider the uncertainty in systems and the design switching law is simple and easy for application.The rest of this paper is organized as follows. The problem is formulated in Section2. Section 3 deals with robust stability criteria for a class of discrete-time switched system with uncertainty. Numerical examples are provided to illustrate the theoretical results in Section 4, and the conclusions are drawn in Section 5.Notation 1.
The notation used in this paper is fairly standard. The superscript “T” stands for matrix transposition; Rn denotes the n-dimensional Euclidean space. diag{⋯} stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. A symmetric matrix P>0(≥0) means P is positive (semipositive) definite. ∥·∥ stands for the Euclidean vector norm of the vector. λ(A) stands for the eigenvalues of matrix A.  ∥A∥ denotes the norm of matrix A, that is, ∥A∥=Max[λ(ATA)]1/2.  I and 0 represent, respectively, identity matrix and zero matrix.


## 2. Systems Description and Problem Statement

Consider a class of uncertain switched discrete-time systems given by(2.1)x(k+1)=(Aσ(x(k))+ΔAσ(x(k)))x(k),x(0)=x0,
where x(k)∈Rn  is the state, Aσ(x(k))∈Rn,x0  is the initial state, σ(x(k)):Rn→{1,2,…,N},N≥2,  is a piecewise constant scalar function, called a switch signal, and N  is the number of the individual systems, that is, the matrix Aσ(x(k)) switched between matrices A1,A2,…,AN  belonging to the set A={A1,A2,…,AN}.  ΔAσ(x(k))  denotes the parameter uncertainty and is assumed to be in the following form:(2.2)ΔAσ(x(k))=Fσ(x(k))(k)Eσ(x(k)),
where Eσ(x(k)) is real constant matrices of appropriate dimensions and Fσ(x(k)) is unknown matrix, satisfying  Fσ(x(k))TFσ(x(k))≤I. The switched discrete-time system (2.1) can be described as follows:(2.3)x(k+1)=(Ai+ΔAi)x(k),i=1,2,…,N,
where ΔAi=Fi(k)Ei with FiTFi≤I.For the stability of switched discrete-time system (2.3), some helpful lemmas are given in the following.Lemma 2.1.
For any matricesA1,A2,…,AN  with the same dimensions and ΔAi,i=1,2,…,N, which are given by (2.3), the following inequality holds for any positive constant c:
(2.4)(∑i=1N(Ai+ΔAi))T(∑i=1N(Ai+ΔAi))≤(1+c){∑i=1N-1(1+c-1)i-1A̅iTA̅i}+(1+c-1)N-1A̅NA̅NT≤(1+c)2{∑i=1N-1(1+c-1)i-1AiTAi}+(1+c-1)N-1(1+c)ANTAN+(1+c){∑i=1N-1(1+c-1)iEiTEi}+(1+c-1)NENTEN,
where A̅i=Ai+ΔAi,i∈{1,2,…,N}.Proof.
For a positive constantc, A and B with the same dimensions, it is an obvious fact that
(2.5)(A+B)T(A+B)=ATA+BTB+ATB+BTA≤ATA+BTB+cATA+c-1BTB=(1+c)ATA+(1+c-1)BTB.
So, we have
(2.6)(Ai+ΔAi)T(Ai+ΔAi)≤(1+c)AiTAi+(1+c-1)ΔAiTΔAi=(1+c)AiTAi+(1+c-1)(Fi(k)Ei)T(Fi(k)Ei)=(1+c)AiTAi+(1+c-1)EiTFi(k)T(Fi(k)Ei)≤(1+c)AiTAi+(1+c-1)EiTEi.
Using (2.5) and (2.6), we have
(2.7)(∑i=1N(Ai+ΔAi))T(∑i=1N(Ai+ΔAi))=(∑i=1NA̅i)T(∑i=1NA̅i)=(A̅1+∑i=2NA̅i)T(A̅1+∑i=2NA̅i)≤(1+c)A̅1A̅1T+(1+c-1)(∑i=2NA̅i)T(∑i=2NA̅i)≤(1+c)A̅1A̅1T+(1+c-1){(1+c)A̅2A̅2T+(1+c-1)(∑i=3NA̅i)T(∑i=3NA̅i)}≤(1+c)A̅1A̅1T+(1+c-1)(1+c)A̅2A̅2T+(1+c-1)2×{(1+c)A̅3A̅3T+(1+c-1)(∑i=4NA̅i)T(∑i=4NA̅i)}≤(1+c)A̅1A̅1T+(1+c-1)(1+c)A̅2A̅2T+(1+c-1)2(1+c)A̅3A̅3T+⋯+(1+c-1)N-2(1+c)A̅N-1A̅N-1T+(1+c-1)N-1A̅NA̅NT=(1+c){∑i=1N-1(1+c-1)i-1A̅iTA̅i}+(1+c-1)N-1A̅NA̅NT≤(1+c){∑i=1N-1(1+c-1)i-1((1+c)AiTAi+(1+c-1)EiTEi)}+(1+c-1)N-1((1+c)ANTAN+(1+c-1)ENTEN)=(1+c)2{∑i=1N-1(1+c-1)i-1AiTAi}+(1+c-1)N-1(1+c)ANTAN+(1+c){∑i=1N-1(1+c-1)iEiTEi}+(1+c-1)NENTEN.
This completes the proof of Lemma 2.1.Set(2.8)βi=1(1+c)(1+c-1)i-1,i=1,2,…,N-1,βN=1(1+c-1)N-1.We have(2.9)∑i=1Nβi=∑i=1N-11(1+c)(1+c-1)i-1+1(1+c-1)N-11/(1+c)-1/((1+c)(1+c-1)N-1)1-1/(1+c-1)+1(1+c-1)N-1=1.Considering the following system (2.10):
(2.10)x(k+1)=∑i=1Nβi(Ai+ΔAi)x(k),
where βi,i=1,2,…,N, is given by (2.8), we have the following result.Lemma 2.2.
If there existc>0 and a symmetric matrix P>0 such that the following inequality holds:
(2.11)∑i=1nβi[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]<0,
then system (2.10) is asymptotically stable, and there exists a switching law for the uncertain switched discrete-time system (2.3) such that the system (2.3) is asymptotically stable.Proof.
We choose the Lyapunov function candidate as(2.12)V(k)=xT(k)Px(k).
Since P is a symmetric positive-definite matrix, there exists a matrix Q such that P=QTQ.
We have(2.13)ΔV(k)|(2.10)=V(k+1)-V(k)=xT(k+1)Px(k+1)-xT(k)Px(k)=(∑i=1Nβi(Ai+ΔAi)x(k))TP(∑i=1Nβi(Ai+ΔAi)x(k))-xT(k)Px(k)=xT(k)(∑i=1Nβi(Ai+ΔAi))TP(∑i=1Nβi(Ai+ΔAi))x(k)-xT(k)Px(k)=xT(k)(∑i=1NβiQ(Ai+ΔAi))T(∑i=1NβiQ(Ai+ΔAi))x(k)-xT(k)Px(k)≤xT(k){(1+c)∑i=1N-1(1+c-1)i-1(βiQ(Ai+ΔAi))T(βiQ(Ai+ΔAi))+(1+c-1)N-1×(βNQ(AN+ΔAN))T(βNQ(AN+ΔAN))∑i=1N-1(1+c-1)i-1}x(k)-xT(k)Px(k)≤xT(k){(1+c)∑i=1N-1(1+c-1)i-1[(1+c)βi2AiTPAi+(1+c-1)βi2ΔAiTPΔAi]+(1+c-1)N-1×[(1+c)βN2ANTPAN+(1+c-1)βN2ΔANTPΔAN]∑i=1N-1(1+c-1)i-1}x(k)-xT(k)Px(k)≤xT(k){(1+c)∑i=1N-1βiAiTPAi+∑i=1N-1(1+c-1)βiΔAiTPΔAi+(1+c)βNANTPANi+(1+c-1)βNΔANTPΔA∑i=1N-1(1+c-1)i-1}x(k)-xT(k)Px(k)≤xT(k){(1+c)∑i=1N-1βiAiTPAi+∑i=1N-1(1+c-1)βi‖P‖EiTEi+(1+c)βNANTPAN+(1+c-1)βN‖P‖ENTEN∑i=1N-1(1+c-1)i-1}x(k)-xT(k)Px(k)=xT(k){(1+c)∑i=1NβiAiTPAi+∑i=1N(1+c-1)βi‖P‖EiTEi-P}x(k)=xT(k){∑i=1Nβi[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]}x(k)<0,x(k)≠0.
Equation (2.11) implies that ΔV(k)|(2.10)<0,x(k)≠0. Hence the system (2.10) is asymptotically stable.
For system (2.3), we have
(2.14)ΔV(k)|(2.3)=xT(k+1)Px(k+1)-xT(k)Px(k)=xT(k)(Ai+ΔAi)TP(Ai+ΔAi)xT(k)-xT(k)Px(k)=xT(k){[Q(Ai+ΔAi)]T[Q(Ai+ΔAi)]-P}xT(k)≤xT(k)[(1+c)AiTPAi+(1+c-1)ΔAiTPΔAi-P]xT(k)≤xT(k)[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]xT(k).
From (2.11), we know that, for any k, at least there exists an i∈{1,2,…,N}such that
(2.15)ΔV(k)|(2.3)≤xT(k)[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]xT(k)<0,x(k)≠0.
The switching signal is given as follows.(1)
If the following inequality holds:(2.16)xT(k)[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]xT(k)<0,
then
(2.17)σ(x(k))=i.(2)
Ifx(k)=0, then
(2.18)σ(x(k))=γ,γ∈{1,2,…,N}.
Thus, the switched discrete-time system (2.3) is asymptotically stable. This completes the proof of Lemma 2.2.


## 3. Stability Analysis of Uncertain Switched Discrete-Time System

Consider the asymptotical stability of the following system:(3.1)x(k+1)=(Ai+ΔAi)x(k),i=1,2,
where(3.2)Ai=[Ai11Ai12Ai21Ai22]∈Rn×n,Ai11∈Rn1×n1,Ai12∈Rn2×n2.Remark 3.1.
The motivation of considering (3.1) is that when the diagonal blocks satisfy Lyapunov matrix inequalities (3.9), system (3.1) has some good property.
According to Lemma2.2, the study of asymptotical stability for (3.1) can be transformed into the study for (3.3):
(3.3)x(k+1)=[β(A1+ΔA1)+(1-β)(A2+ΔA2)]x(k),
where β=1/(1+c).
We choose the Lyapunov function candidate as(3.4)V(k)=xT(k)Px(k),
where x=[x1T,x2T]T,x1∈Rn1,x2∈Rn2,   and P=diag{P1,P2},P1∈Rn1×n1,P2∈Rn2×n2, are real symmetric positive-definite matrices.
Computing the product, we have(3.5)AiTPAi=[Ai11TAi21TAi12TAi22T][P100P2][Ai11Ai12Ai21Ai22]=[Ai11TP1Ai11+Ai21TP2Ai21Ai11TP1Ai12+Ai21TP2Ai22Ai12TP1Ai11+Ai22TP2Ai21Ai12TP1Ai12+Ai22TP2Ai22].
So,
(3.6)xTAiTPAix=[x1Tx2T][Ai11TP1Ai11+Ai21TP2Ai21Ai11TP1Ai12+Ai21TP2Ai22Ai12TP1Ai11+Ai22TP2Ai21Ai12TP1Ai12+Ai22TP2Ai22][x1x2]=[x1T(Ai11TP1Ai11+Ai21TP2Ai21)+x2T(Ai12TP1Ai11+Ai22TP2Ai21)]x1+[x1T(Ai11TP1Ai12+Ai21TP2Ai22)+x2T(Ai12TP1Ai12+Ai22TP2Ai22)]x2=x1T(Ai11TP1Ai11+Ai21TP2Ai21)x1+x2T(Ai12TP1Ai11+Ai22TP2Ai21)x1+x1T(Ai11TP1Ai12+Ai21TP2Ai22)x2+x2T(Ai12TP1Ai12+Ai22TP2Ai22)x2.
Using the properties of matrix norm, we have
(3.7)xTAiTPAix≤x1TAi11TP1Ai11x1+‖x1‖2‖Ai21‖2‖P2‖+2‖x1‖‖x2‖(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖)+‖x2‖2‖P1‖‖Ai12‖2+x2TAi22TP2Ai22x2≤x1TAi11TP1Ai11x1+x2TAi22TP2Ai22x2+(‖x1‖2+‖x2‖2)×(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖)+‖x1‖2‖Ai21‖2‖P2‖+‖x2‖2‖P1‖‖Ai12‖2=x1TAi11TP1Ai11x1+‖x1‖2(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖Ai21‖2‖P2‖)+x2TAi22TP2Ai22x2+‖x2‖2(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖P1‖‖Ai12‖2).
It will be convenient throughout this section to use the following notations:
(3.8)e1=‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖A121‖2‖P2‖,e2=‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖A221‖2‖P2‖,d1=‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖P1‖‖A112‖2,d2=‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖P1‖‖A212‖2.Theorem 3.2.
There exists a switched law such that the discrete-time system (3.1) is asymptotically stable, if there exist symmetric positive-definite matrices P1,P2>0 and positive constants c,λij,i,j=1,2, such that the following inequalities hold:
(3.9)AijjTPjAijj-Pj≤-λijI,i,j=1,2,(3.10)e1+ce2-λ11-cλ21+cλmax⁡(P1)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖<0,d1+cd2-λ12-cλ22+cλmax⁡(P2)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖<0.Proof.
By Lemma2.1, the difference of the Lyapunov function (3.4) is as following
(3.11)ΔV(k)≤xT(k){β[(1+c)A1TPA1+(1+c-1)‖P‖E1TE1]+(1-β)[(1+c)A2TPA2+(1+c-1)‖P‖E2TE2]-P}x(k)=xT(k){[β(1+c)A1TPA1+β(1+c-1)‖P‖E1TE1]+[(1-β)(1+c)A2TPA2+(1-β)(1+c-1)‖P‖E2TE2]-P}x(k)=xT(k){[A1TPA1+c-1‖P‖E1TE1]+[cA2TPA2+‖P‖E2TE2]-P}x(k).
Using (3.7), it follows
(3.12)ΔV(k)≤x1TA111TP1A111x1+x2TA122TP2A122x2+‖x1‖2(‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖A121‖2‖P2‖)+‖x2‖2(‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖P1‖‖A112‖2)+c{x1TA211TP1A211x1+x2TA222TP2A222x2+‖x1‖2(‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖A221‖2‖P2‖)+‖x2‖2(‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖P1‖‖A212‖2)}+‖x‖2(c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖)-(x1TP1x1+x2TP2x2)≤-λ11‖x1‖2-λ12‖x2‖2+cx1T(-λ21I+P1)x1+cx2T(-λ22I+P2)x2+‖x1‖2[(‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖A121‖2‖P2‖)+c(‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖A221‖2‖P2‖)]+‖x2‖2[(‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖P1‖‖A112‖2)+c(‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖P1‖‖A212‖2)]+‖x‖2(c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖)≤‖x1‖2[(‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖A121‖2‖P2‖)+c(‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖A221‖2‖P2‖)-λ11-cλ21+cλmax⁡(P1)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖]+‖x2‖2[(‖A111‖‖P1‖‖A112‖+‖A121‖‖P2‖‖A122‖+‖P1‖‖A112‖2)+c(‖A211‖‖P1‖‖A212‖+‖A221‖‖P2‖‖A222‖+‖P1‖‖A212‖2)-λ12-cλ22+cλmax⁡(P2)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖]≤‖x1‖2[e1+ce2-λ11-cλ21+cλmax⁡(P1)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖]+‖x2‖2[d1+cd2-λ12-cλ22+cλmax⁡(P2)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖].
From (3.10), we get that
(3.13)ΔV(k)<0,x(k)≠0.
According to Lemma 2.2, the discrete-time system (3.1) is asymptotically stable. Proof of Theorem 3.2 is completed.Consider the system (2.3) with
(3.14)Ai=[Ai11Ai12Ai21Ai22]∈Rn×n,Ai11∈Rn1×n1,Ai12∈Rn2×n2.
According to Lemma 2.2, we have the following theorem for the asymptotically stability of (2.3).Theorem 3.3.
There exists a switched law such that the discrete-time system (2.3) is asymptotically stable, if there exist symmetric positive-definite matrices P1,P2>0,P1∈Rn1×n1,P2∈Rn2×n2, and positive constants c,λij,i,j=1,2, satisfying the follow inequalities:
(3.15)AijjTPjAijj-11+cPj≤-λijI,i=1,2,…N,j=1,2,(3.16)∑i=1Nβi(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖Ai21‖2‖P2‖-λi1+c-1‖P‖‖EiTEi‖)<0,∑i=1Nβi(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖P1‖‖Ai12‖2-λi2+c-1‖P‖‖EiTEi‖)<0,
where P=diag{P1,P2}, and
(3.17)βi=1(1+c)(1+c-1)i-1,i=1,2,…,N-1,βN=1(1+c-1)N-1.Proof.
We choose the Lyapunov function candidate as(3.18)V(k)=xT(k)Px(k),
where x=[x1T,x2T]T,x1∈Rn1,x2∈Rn2 and P=diag{P1,P2} are real symmetric positive-definite matrices.
Using Lemma2.1 and (3.15), we get
(3.19)ΔV(k)|(2.10)≤xT(k){∑i=1Nβi[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]}x(k)≤∑i=1Nβi(1+c)[x1TAi11TP1Ai11x1+x2TAi22TP2Ai22x2+‖x1‖2(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖Ai21‖2‖P2‖)+‖x2‖2(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖P1‖‖Ai12‖2)]+∑i=1Nβi[‖x‖2(1+c-1)‖P‖‖EiTEi‖-xT(k)Px(k)]≤∑i=1Nβi(1+c)[x1T(Ai11TP1Ai11-1(1+c)P1)x1+x2T(Ai22TP2Ai22-1(1+c)P2)x2+‖x1‖2(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖Ai21‖2‖P2‖)+‖x2‖2(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖P1‖‖Ai12‖2)+‖x‖21+c-11+c‖P‖‖EiTEi‖]≤∑i=1Nβi(1+c)[‖x1‖2(1+c-11+c‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖Ai21‖2‖P2‖-λi1+1+c-11+c‖P‖‖EiTEi‖)+‖x2‖2(‖EiTEi‖‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖P1‖‖Ai12‖2-λi2+c-1‖P‖‖EiTEi‖)∑i=1Nβi(1+c)]=‖x1‖2∑i=1Nβi(1+c)(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖Ai21‖2‖P2‖-λi1+c-1‖P‖‖EiTEi‖)+‖x2‖2∑i=1Nβi(1+c)(‖Ai11‖‖P1‖‖Ai12‖+‖Ai21‖‖P2‖‖Ai22‖+‖P1‖‖Ai12‖2-λi2+c-1‖P‖‖EiTEi‖).
From (3.16), we have ΔV(k)<0,x(k)≠0.
Switching Law. The switching law is given by (2.17) and (2.18). In the light of Lemma 2.2, the discrete-time system (2.3) is asymptotically stable. Proof of Theorem 3.3 is completed.Remark 3.4.
Conditions (3.15) and (3.16) in Theorem 3.3 are nonlinear with respect to unknown c>0. If c is a given positive constant, then (3.15) is a linear matrix inequality.  In order to find the solution for (3.15) and (3.16), c>0 is given in advance, then we solve (3.15) and (3.16).
Consider the system (2.3) with
(3.20)Ai=[Ai11Ai12Ai13Ai21Ai22Ai23Ai31Ai32Ai33]∈Rn×n,Ai11∈Rn1×n1,Ai22∈Rn2×n2,Ai33∈Rn3×n3.
According to Lemma2.2, we have the following theorem for the asymptotical stability of (2.3).Theorem 3.5.
There exists a switched law such that the discrete-time system (2.3) is asymptotically stable, if there exist positive constants  c,ηij,i,j=1,2,   and symmetric positive-definite matrices  P1,P2,P3>0,P1∈Rn1×n1,P2∈Rn2×n2,P3∈Rn3×n3 satisfying the following inequalities:
(3.21)AijjTPjAijj-11+cPj≤-ηijI,i=1,2,…N,j=1,2,3,(3.22)∑j=23‖Aij1‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖+c-1‖P‖‖EiTEi‖-ηi1<0,∑j≠2j=13‖Aij2‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖+c-1‖P‖‖EiTEi‖-ηi2<0,∑j=12‖Aij3‖2‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖+c-1‖P‖‖EiTEi‖-ηi3<0,
where P=diag{P1,P2,P3}, and
(3.23)βi=1(1+c)(1+c-1)i-1,i=1,2,…,N-1,βN=1(1+c-1)N-1.Proof.
We choose the Lyapunov function candidate as(3.24)V(k)=xT(k)Px(k),
where x=[x1T,x2T,x3T]T and P=diag{P1,P2,P3} are real symmetric positive-definite matrices.
Computing the product, we have(3.25)AiTPAi=[Ai11TAi21TAi31TAi12TAi22TAi32TAi13TAi23TAi33T][P1000P2000P3][Ai11Ai12Ai13Ai21Ai22Ai23Ai31Ai32Ai33]=[∑j=13Aij1TPjAij1∑j=13Aij1TPjAij2∑j=13Aij1TPjAij3∑j=13Aij2TPjAij1∑j=13Aij2TPjAij2∑j=13Aij2TPjAij3∑j=13Aij3TPjAij1∑j=13Aij3TPjAij2∑j=13Aij3TPjAij3].
So, we get
(3.26)xTAiTPAix=[x1Tx2Tx3T][∑j=13Aij1TPjAij1∑j=13Aij1TPjAij2∑j=13Aij1TPjAij3∑j=13Aij2TPjAij1∑j=13Aij2TPjAij2∑j=13Aij2TPjAij3∑j=13Aij3TPjAij1∑j=13Aij3TPjAij2∑j=13Aij3TPjAij3][x1x2x3]=(x1T∑j=13Aij1TPjAij1+x2T∑j=13Aij2TPjAij1+x3T∑j=13Aij3TPjAij1)x1+(x1T∑j=13Aij1TPjAij2+x2T∑j=13Aij2TPjAij2+x3T∑j=13Aij3TPjAij2)x2+(x1T∑j=13Aij1TPjAij3+x2T∑j=13Aij2TPjAij3+x3T∑j=13Aij3TPjAij3)x3≤x1TAi11TP1Ai11x1+‖x1‖2∑j=23‖Aij1‖2‖Pj‖+2‖x1‖‖x2‖∑j=13‖Aij1‖‖Aij2‖‖Pj‖+2‖x1‖‖x3‖∑j=13‖Aij1‖‖Aij3‖‖Pj‖+x2TAi22TP2Ai22x2+‖x2‖2∑j=1j≠23‖Aij2‖2‖Pj‖+2‖x3‖‖x2‖∑j=13‖Aij3‖‖Aij2‖‖Pj‖+‖x3‖2∑j=1j≠33‖Aij3‖2‖Pj‖+x3TAi33TP3Ai33x3≤x1TAi11TP1Ai11x1+‖x1‖2[∑j=23‖Aij1‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖]+x2TAi22TP2Ai22x2+‖x2‖2[∑j=1j≠23‖Aij2‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖]+x3TAi33TP3Ai33x3+‖x3‖2[∑j=12‖Aij3‖2‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖].
Using the properties of matrix norm, (3.24), and Lemma 2.2, we have
(3.27)ΔV(k)|(2.10)≤xT(k){∑i=1Nβi[(1+c)AiTPAi+(1+c-1)‖P‖EiTEi-P]}x(k)≤∑i=1Nβi(1+c){∑i=1Nβix1TAi11TP1Ai11x1+‖x1‖2×[∑j=23‖Aij1‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖]+x2TAi22TP2Ai22x2+‖x2‖2×[∑j=1j≠23‖Aij2‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖]+x3TAi33TP3Ai33x3+‖x3‖2×[∑j=13‖Aij3‖2‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖]}+xT(k)∑i=1Nβi[(1+c-1)‖P‖EiTEi-P]x(k)≤∑i=1Nβi(1+c){‖x1‖2[∑j=23‖Aij1‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖+c-1‖P‖‖EiTEi‖-ηi1]+‖x2‖2[∑j=1j≠23‖Aij2‖2‖Pj‖+∑j=13‖Aij1‖‖Aij2‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖+c-1‖P‖‖EiTEi‖-ηi2∑j=1j≠23‖Aij2‖2‖Pj‖]+‖x3‖2[∑j=12‖Aij3‖2‖Pj‖+∑j=13‖Aij1‖‖Aij3‖‖Pj‖+∑j=13‖Aij3‖‖Aij2‖‖Pj‖+c-1‖P‖‖EiTEi‖-ηi3]}.
From (3.22), we have ΔV(k)<0,x(k)≠0.
Switching Law. The switching law is given by (2.17) and (2.18). In the light of Lemma 2.2, the discrete-time system (2.3) is asymptotically stable. Proof of Theorem 3.5 is completed.Remark 3.6.
According to the system matrix that is divided into block matrices of the different dimension, Theorems3.3 and 3.5 are obtained, respectively. When the system matrix is divided into 2×2 block matrix, Theorem 3.3 can be used. When the system matrix is divided into 3×3 block matrix, Theorem 3.5 can be used.


## 4. Numerical Examples

Example 4.1.
Consider the switched discrete-time system composed of two individual systems given as follows.(4.1)Mode1:x(k+1)=(A1+ΔA1)x(k),A1=[0.40.101],ΔA1=F1(k)E1,E1=[0,0.1],Mode2:x(k+1)=(A2+ΔA2)x(k),A2=[-0.20.110.3],ΔA2=F2(k)E2,E2=[0,0.2].
SetP1=P2=0.5,c=1. It is easily obtained that
(4.2)e1=0.02,e2=0.66,d1=0.025,d2=0.165,λ11=0.84,λ12=0.75,λ21=0.96,λ22=0.91.
It is validated that
(4.3)AijjTPjAijj-Pj≤-λijI,i,j=1,2,e1+ce2-λ11-cλ21+cλmax⁡(P1)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖<0,d1+cd2-λ12-cλ22+cλmax⁡(P2)+c-1‖P‖‖E1TE1‖+‖P‖‖E2TE2‖<0.
According to Theorem 3.2, the switched discrete-time system (4.1) is asymptotically stable by the following switching law.
Switching Law. σ(x(k))=i, if the following inequality holds:
(4.4)xT(k)[2AiTPAi+2‖P‖EiTEi-P]xT(k)<0.
If x(k)=0, then
(4.5)σ(x(k))=γ,γ∈{1,2}.Example 4.2.
Consider the switched discrete-time system composed of three individual systems given as follows.(4.6)Mode1:x(k+1)=(A1+ΔA1)x(k),ΔA1=F1(k)E1,E1=[0,0,0.1].Mode2:x(k+1)=(A2+ΔA2)x(k),ΔA2=F2(k)E2,E2=[0,0,0.2].Mode3:x(k+1)=(A3+ΔA3)x(k),ΔA3=F3(k)E3,E3=[0,0.1,0],
where
(4.7)A1=[0.200.10.10.30.4000.7],A2=[-0.20.1000.30.10.100.1],A3=[-0.10.1000.40.300.10.2].
Taking c=1,P1=0.35,P2=0.45,P3=0.7, it is easily obtained that
(4.8)η11=0.8691,η12=0.8190,η13=0.4191,η21=0.8691,η22=0.8191,η23=0.8991,η31=0.8991,η32=0.7491,η33=0.8691.
It is validated that (3.21) and (3.22) hold. According to Theorem 3.5, the switched discrete-time system (4.6) is asymptotically stable by the following switching law.
Switching Law. σ(x(k))=i, if the following inequality holds:
(4.9)xT(k)[2AiTPAi+2‖P‖EiTEi-P]xT(k)<0.
If x(k)=0, then
(4.10)σ(x(k))=γ,γ∈{1,2,3}.


## 5. Conclusion

The robust stability is investigated for a class of switched discrete-time system with state parameter uncertainty.  The switched law design method is proposed. By a simple switching law, some sufficient conditions for robust stability have been derived for the uncertain switched discrete-time systems and are presented in terms of inequalities. The present results are straightforward. Two examples are given to show the effectiveness of our approach.



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*Source: 523020-2011-11-30.xml*