Robust Stability Analysis for Uncertain Switched Discrete-Time Systems

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.

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, robust 𝐻_∞ 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 Section 2. 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; 𝑅^𝑛 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(𝐴^𝑇𝐴)]^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 𝑥(𝑘+1)=𝐴_{𝜎(𝑥(𝑘))}+Δ𝐴_{𝜎(𝑥(𝑘))}𝑥(𝑘),𝑥(0)=𝑥₀, (2.1) where 𝑥(𝑘)∈𝑅^𝑛  is the state, 𝐴_{𝜎(𝑥(𝑘))}∈𝑅^𝑛,𝑥₀  is the initial state, 𝜎(𝑥(𝑘))∶𝑅^𝑛→{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 𝐴_{𝜎(𝑥(𝑘))} switched between matrices 𝐴₁,𝐴₂,…,𝐴_𝑁  belonging to the set 𝐴={𝐴₁,𝐴₂,…,𝐴_𝑁}.  Δ𝐴_{𝜎(𝑥(𝑘))}  denotes the parameter uncertainty and is assumed to be in the following form:Δ𝐴_{𝜎(𝑥(𝑘))}=𝐹_{𝜎(𝑥(𝑘))}(𝑘)𝐸_{𝜎(𝑥(𝑘))}, (2.2) where 𝐸_{𝜎(𝑥(𝑘))} is real constant matrices of appropriate dimensions and 𝐹_{𝜎(𝑥(𝑘))} is unknown matrix, satisfying 𝐹^𝑇_{𝜎(𝑥(𝑘))}𝐹_{𝜎(𝑥(𝑘))}≤𝐼. The switched discrete-time system (2.1) can be described as follows:𝑥(𝑘+1)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘),𝑖=1,2,…,𝑁, (2.3) where Δ𝐴_𝑖=𝐹_𝑖(𝑘)𝐸_𝑖 with 𝐹^𝑇_𝑖𝐹_𝑖≤𝐼.

For the stability of switched discrete-time system (2.3), some helpful lemmas are given in the following.

Lemma 2.1. For any matrices 𝐴₁,𝐴₂,…,𝐴_𝑁  with the same dimensions and Δ𝐴_𝑖,𝑖=1,2,…,𝑁, which are given by (2.3), the following inequality holds for any positive constant c: ^𝑁_𝑖=1𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑁_𝑖=1𝐴_𝑖+Δ𝐴_𝑖≤(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖−1𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐^−1𝑁−1𝐴_𝑁𝐴^𝑇_𝑁≤(1+𝑐)^2𝑁−1_𝑖=11+𝑐^−1𝑖−1𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐^−1𝑁−1(1+𝑐)𝐴^𝑇_𝑁𝐴_𝑁+(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖𝐸^𝑇_𝑖𝐸_𝑖+1+𝑐^−1𝑁𝐸^𝑇_𝑁𝐸_𝑁, (2.4) where 𝐴_𝑖=𝐴_𝑖+Δ𝐴_𝑖,𝑖∈{1,2,…,𝑁}.

Proof. For a positive constant 𝑐, 𝐴 and 𝐵 with the same dimensions, it is an obvious fact that (𝐴+𝐵)^𝑇(𝐴+𝐵)=𝐴^𝑇𝐴+𝐵^𝑇𝐵+𝐴^𝑇𝐵+𝐵^𝑇𝐴≤𝐴^𝑇𝐴+𝐵^𝑇𝐵+𝑐𝐴^𝑇𝐴+𝑐⁻¹𝐵^𝑇𝐵=(1+𝑐)𝐴^𝑇𝐴+1+𝑐⁻¹𝐵^𝑇𝐵. (2.5) So, we have 𝐴_𝑖+Δ𝐴_𝑖^𝑇𝐴_𝑖+Δ𝐴_𝑖≤(1+𝑐)𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐⁻¹Δ𝐴^𝑇_𝑖Δ𝐴_𝑖=(1+𝑐)𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐⁻¹𝐹_𝑖(𝑘)𝐸_𝑖^𝑇𝐹_𝑖(𝑘)𝐸_𝑖=(1+𝑐)𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐⁻¹𝐸^𝑇_𝑖𝐹_𝑖(𝑘)^𝑇𝐹_𝑖(𝑘)𝐸_𝑖≤(1+𝑐)𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐⁻¹𝐸^𝑇_𝑖𝐸_𝑖. (2.6) Using (2.5) and (2.6), we have ^𝑁_𝑖=1𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑁_𝑖=1𝐴_𝑖+Δ𝐴_𝑖=^𝑁_𝑖=1𝐴_𝑖^𝑇𝑁_𝑖=1𝐴_𝑖=𝐴₁+^𝑁_𝑖=2𝐴_𝑖^𝑇𝐴₁+^𝑁_𝑖=2𝐴_𝑖≤(1+𝑐)𝐴₁𝐴^𝑇₁+1+𝑐^−1𝑁_𝑖=2𝐴_𝑖^𝑇𝑁_𝑖=2𝐴_𝑖≤(1+𝑐)𝐴₁𝐴^𝑇₁+1+𝑐⁻¹(1+𝑐)𝐴₂𝐴^𝑇₂+1+𝑐^−1𝑁_𝑖=3𝐴_𝑖^𝑇𝑁_𝑖=3𝐴_𝑖≤(1+𝑐)𝐴₁𝐴^𝑇₁+1+𝑐⁻¹(1+𝑐)𝐴₂𝐴^𝑇₂+1+𝑐⁻¹²×(1+𝑐)𝐴₃𝐴^𝑇₃+1+𝑐^−1𝑁_𝑖=4𝐴_𝑖^𝑇𝑁_𝑖=4𝐴_𝑖≤(1+𝑐)𝐴₁𝐴^𝑇₁+1+𝑐⁻¹(1+𝑐)𝐴₂𝐴^𝑇₂+1+𝑐⁻¹²(1+𝑐)𝐴₃𝐴^𝑇₃+⋯+1+𝑐^−1𝑁−2(1+𝑐)𝐴_𝑁−1𝐴^𝑇_𝑁−1+1+𝑐^−1𝑁−1𝐴_𝑁𝐴^𝑇_𝑁=(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖−1𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐^−1𝑁−1𝐴_𝑁𝐴^𝑇_𝑁≤(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖−1(1+𝑐)𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐⁻¹𝐸^𝑇_𝑖𝐸_𝑖+1+𝑐^−1𝑁−1(1+𝑐)𝐴^𝑇_𝑁𝐴_𝑁+1+𝑐⁻¹𝐸^𝑇_𝑁𝐸_𝑁=(1+𝑐)^2𝑁−1_𝑖=11+𝑐^−1𝑖−1𝐴^𝑇_𝑖𝐴_𝑖+1+𝑐^−1𝑁−1(1+𝑐)𝐴^𝑇_𝑁𝐴_𝑁+(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖𝐸^𝑇_𝑖𝐸_𝑖+1+𝑐^−1𝑁𝐸^𝑇_𝑁𝐸_𝑁. (2.7) This completes the proof of Lemma 2.1.

Set 𝛽_𝑖=1(1+𝑐)1+𝑐^−1𝑖−1,𝑖=1,2,…,𝑁−1,𝛽_𝑁=11+𝑐^−1𝑁−1. (2.8)

We have ^𝑁_𝑖=1𝛽_𝑖=^𝑁−1_𝑖=11(1+𝑐)1+𝑐^−1𝑖−1+11+𝑐^−1𝑁−11/(1+𝑐)−1/(1+𝑐)1+𝑐^−1𝑁−11−1/1+𝑐⁻¹+11+𝑐^−1𝑁−1=1. (2.9)

Considering the following system (2.10): 𝑥(𝑘+1)=^𝑁_𝑖=1𝛽_𝑖𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘), (2.10) where 𝛽_𝑖,𝑖=1,2,…,𝑁, is given by (2.8), we have the following result.

Lemma 2.2. If there exist 𝑐>0 and a symmetric matrix 𝑃>0 such that the following inequality holds: ^𝑛_𝑖=1𝛽_𝑖(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃<0, (2.11) 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 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘). (2.12) Since 𝑃 is a symmetric positive-definite matrix, there exists a matrix 𝑄 such that 𝑃=𝑄^𝑇𝑄.
We have Δ𝑉(𝑘)|_(2.10)=𝑉(𝑘+1)−𝑉(𝑘)=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)(𝑘)=𝑃𝑥_𝑁𝑖=1𝛽_𝑖𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘)^𝑇𝑃^𝑁_𝑖=1𝛽_𝑖𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)^𝑁_𝑖=1𝛽_𝑖𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑃^𝑁_𝑖=1𝛽_𝑖𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)^𝑁_𝑖=1𝛽_𝑖𝑄𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑁_𝑖=1𝛽_𝑖𝑄𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)≤𝑥^𝑇(𝑘)(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖−1𝛽_𝑖𝑄𝐴_𝑖+Δ𝐴_𝑖^𝑇𝛽_𝑖𝑄𝐴_𝑖+Δ𝐴_𝑖+1+𝑐^−1𝑁−1×𝛽_𝑁𝑄𝐴_𝑁+Δ𝐴_𝑁^𝑇𝛽_𝑁𝑄𝐴_𝑁+Δ𝐴_𝑁𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)≤𝑥^𝑇(𝑘)(1+𝑐)^𝑁−1_𝑖=11+𝑐^−1𝑖−1(1+𝑐)𝛽_𝑖²𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹𝛽_𝑖²Δ𝐴^𝑇_𝑖𝑃Δ𝐴_𝑖+1+𝑐^−1𝑁−1×(1+𝑐)𝛽²_𝑁𝐴^𝑇_𝑁𝑃𝐴_𝑁+1+𝑐⁻¹𝛽²_𝑁Δ𝐴^𝑇_𝑁𝑃Δ𝐴_𝑁𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)≤𝑥^𝑇(𝑘)(1+𝑐)^𝑁−1_𝑖=1𝛽_𝑖𝐴^𝑇_𝑖𝑃𝐴_𝑖+^𝑁−1_𝑖=11+𝑐⁻¹𝛽_𝑖Δ𝐴^𝑇_𝑖𝑃Δ𝐴_𝑖+(1+𝑐)𝛽_𝑁𝐴^𝑇_𝑁𝑃𝐴_𝑁𝑖+1+𝑐⁻¹𝛽_𝑁Δ𝐴^𝑇_𝑁𝑃Δ𝐴𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)≤𝑥^𝑇(𝑘)(1+𝑐)^𝑁−1_𝑖=1𝛽_𝑖𝐴_𝑖^𝑇𝑃𝐴_𝑖+^𝑁−1_𝑖=11+𝑐⁻¹𝛽_𝑖‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖+(1+𝑐)𝛽_𝑁𝐴^𝑇_𝑁𝑃𝐴_𝑁+1+𝑐⁻¹𝛽_𝑁‖𝑃‖𝐸^𝑇_𝑁𝐸_𝑁𝑥(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)(1+𝑐)^𝑁_𝑖=1𝛽_𝑖𝐴^𝑇_𝑖𝑃𝐴_𝑖+^𝑁_𝑖=11+𝑐⁻¹𝛽_𝑖‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)^𝑁_𝑖=1𝛽_𝑖(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃𝑥(𝑘)<0,𝑥(𝑘)≠0. (2.13) 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 Δ𝑉(𝑘)|_(2.3)=𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)(𝐴_𝑖+Δ𝐴_𝑖)^𝑇𝑃𝐴_𝑖+Δ𝐴_𝑖𝑥^𝑇(𝑘)−𝑥^𝑇(𝑘)𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)𝑄𝐴_𝑖+Δ𝐴_𝑖^𝑇𝑄𝐴_𝑖+Δ𝐴_𝑖−𝑃𝑥^𝑇(𝑘)≤𝑥^𝑇(𝑘)(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹Δ𝐴^𝑇_𝑖𝑃Δ𝐴_𝑖−𝑃𝑥^𝑇(𝑘)≤𝑥^𝑇(𝑘)(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃𝑥^𝑇(𝑘). (2.14) From (2.11), we know that, for any 𝑘, at least there exists an 𝑖∈{1,2,…,𝑁}such that Δ𝑉(𝑘)|_(2.3)≤𝑥^𝑇(𝑘)(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃𝑥^𝑇(𝑘)<0,𝑥(𝑘)≠0. (2.15)
The switching signal is given as follows. (1)If the following inequality holds: 𝑥^𝑇(𝑘)(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖−𝑃𝑥^𝑇(𝑘)<0, (2.16) then 𝜎(𝑥(𝑘))=𝑖. (2.17)(2)If 𝑥(𝑘)=0, then 𝜎(𝑥(𝑘))=𝛾,𝛾∈{1,2,…,𝑁}. (2.18)
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:𝑥(𝑘+1)=𝐴_𝑖+Δ𝐴_𝑖𝑥(𝑘),𝑖=1,2, (3.1) where𝐴_𝑖=𝐴_𝑖11𝐴_𝑖12𝐴_𝑖21𝐴_𝑖22∈𝑅^𝑛×𝑛,𝐴_𝑖11∈𝑅^𝑛1×𝑛1,𝐴_𝑖12∈𝑅^𝑛2×𝑛2. (3.2)

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 Lemma 2.2, the study of asymptotical stability for (3.1) can be transformed into the study for (3.3): 𝑥(𝑘+1)=𝛽𝐴₁+Δ𝐴₁+1−𝛽𝐴₂+Δ𝐴₂𝑥(𝑘), (3.3) where 𝛽=1/(1+𝑐).
We choose the Lyapunov function candidate as 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘), (3.4) where 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂]^𝑇,𝑥₁∈𝑅^𝑛1,𝑥₂∈𝑅^𝑛2,   and 𝑃=diag{𝑃₁,𝑃₂},𝑃₁∈𝑅^𝑛1×𝑛1,𝑃₂∈𝑅^𝑛2×𝑛2, are real symmetric positive-definite matrices.
Computing the product, we have 𝐴^𝑇_𝑖𝑃𝐴_𝑖=𝐴^𝑇_𝑖11𝐴^𝑇_𝑖21𝐴^𝑇_𝑖12𝐴^𝑇_𝑖22𝑃₁00𝑃₂𝐴_𝑖11𝐴_𝑖12𝐴_𝑖21𝐴_𝑖22=𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖21𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖22𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖21𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22. (3.5) So, 𝑥^𝑇𝐴^𝑇_𝑖𝑃𝐴_𝑖𝑥=𝑥^𝑇₁𝑥^𝑇₂𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖21𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖22𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖21𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₁𝑥₂=𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖21+𝑥^𝑇₂𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖21𝑥₁+𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖22+𝑥^𝑇₂𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂=𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖21𝑥₁+𝑥^𝑇₂𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖11+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖21𝑥₁+𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖21𝑃₂𝐴_𝑖22𝑥₂+𝑥^𝑇₂𝐴^𝑇_𝑖12𝑃₁𝐴_𝑖12+𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂. (3.6) Using the properties of matrix norm, we have 𝑥^𝑇𝐴^𝑇_𝑖𝑃𝐴_𝑖𝑥≤𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥₁²𝐴_𝑖21²𝑃₂+2𝑥₁𝑥₂𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑥₂²𝑃₁𝐴_𝑖12²+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂≤𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂+𝑥₁²+𝑥₂²×𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑥₁²𝐴_𝑖21²𝑃₂+𝑥₂²𝑃₁𝐴_𝑖12²=𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥₁²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝐴_𝑖21²𝑃₂+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂+𝑥₂²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑃₁𝐴_𝑖12². (3.7) It will be convenient throughout this section to use the following notations: 𝑒₁=𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝐴₁₂₁²𝑃₂,𝑒₂=𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝐴₂₂₁²𝑃₂,𝑑₁=𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝑃₁𝐴₁₁₂²,𝑑₂=𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝑃₁𝐴₂₁₂². (3.8)

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,𝜆_𝑖𝑗,𝑖,𝑗=1,2, such that the following inequalities hold: 𝐴^𝑇_𝑖𝑗𝑗𝑃_𝑗𝐴_𝑖𝑗𝑗−𝑃_𝑗≤−𝜆_𝑖𝑗𝑒𝐼,𝑖,𝑗=1,2,₁+𝑐𝑒₂−𝜆₁₁−𝑐𝜆₂₁+𝑐𝜆_max𝑃₁+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂𝑑<0,₁+𝑐𝑑₂−𝜆₁₂−𝑐𝜆₂₂+𝑐𝜆_max𝑃₂+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂<0. (3.9)(3.10)

Proof. By Lemma 2.1, the difference of the Lyapunov function (3.4) is as following Δ𝑉(𝑘)≤𝑥^𝑇(𝑘)𝛽(1+𝑐)𝐴^𝑇₁𝑃𝐴₁+1+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+1−𝛽(1+𝑐)𝐴^𝑇₂𝑃𝐴₂+1+𝑐⁻¹‖𝑃‖𝐸^𝑇₂𝐸₂−𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)𝛽(1+𝑐)𝐴₁^𝑇𝑃𝐴₁+𝛽1+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+1−𝛽(1+𝑐)𝐴^𝑇₂𝑃𝐴₂+1−𝛽1+𝑐⁻¹‖𝑃‖𝐸^𝑇₂𝐸₂−𝑃𝑥(𝑘)=𝑥^𝑇(𝑘)𝐴^𝑇₁𝑃𝐴₁+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+𝑐𝐴^𝑇₂𝑃𝐴₂+‖𝑃‖𝐸^𝑇₂𝐸₂−𝑃𝑥(𝑘). (3.11) Using (3.7), it follows Δ𝑉(𝑘)≤𝑥^𝑇₁𝐴^𝑇₁₁₁𝑃₁𝐴₁₁₁𝑥₁+𝑥^𝑇₂𝐴^𝑇₁₂₂𝑃₂𝐴₁₂₂𝑥₂+𝑥₁²𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝐴₁₂₁²𝑃₂+𝑥₂²𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝑃₁𝐴₁₁₂²+𝑐𝑥^𝑇₁𝐴^𝑇₂₁₁𝑃₁𝐴₂₁₁𝑥₁+𝑥^𝑇₂𝐴^𝑇₂₂₂𝑃₂𝐴₂₂₂𝑥₂+𝑥₁²𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝐴₂₂₁²𝑃₂+𝑥₂²𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝑃₁𝐴₂₁₂²+‖𝑥‖²𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂−𝑥^𝑇₁𝑃₁𝑥₁+𝑥^𝑇₂𝑃₂𝑥₂≤−𝜆₁₁𝑥₁²−𝜆₁₂𝑥₂²+𝑐𝑥^𝑇₁−𝜆₂₁𝐼+𝑃₁𝑥₁+𝑐𝑥^𝑇₂−𝜆₂₂𝐼+𝑃₂𝑥₂+𝑥₁²𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝐴₁₂₁²𝑃₂+𝑐𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝐴₂₂₁²𝑃₂+𝑥₂²𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝑃₁𝐴₁₁₂²+𝑐𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝑃₁𝐴₂₁₂²+‖𝑥‖²𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂≤𝑥₁²𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝐴₁₂₁²𝑃₂+𝑐𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝐴₂₂₁²𝑃₂−𝜆₁₁−𝑐𝜆₂₁+𝑐𝜆_max𝑃₁+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂+𝑥₂²𝐴₁₁₁𝑃₁𝐴₁₁₂+𝐴₁₂₁𝑃₂𝐴₁₂₂+𝑃₁𝐴₁₁₂²+𝑐𝐴₂₁₁𝑃₁𝐴₂₁₂+𝐴₂₂₁𝑃₂𝐴₂₂₂+𝑃₁𝐴₂₁₂²−𝜆₁₂−𝑐𝜆₂₂+𝑐𝜆_max𝑃₂+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂≤𝑥₁²𝑒₁+𝑐𝑒₂−𝜆₁₁−𝑐𝜆₂₁+𝑐𝜆_max𝑃₁+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂+𝑥₂²𝑑₁+𝑐𝑑₂−𝜆₁₂−𝑐𝜆₂₂+𝑐𝜆_max𝑃₂+𝑐⁻¹‖𝑃‖𝐸^𝑇₁𝐸₁+‖𝑃‖𝐸^𝑇₂𝐸₂. (3.12) From (3.10), we get that Δ𝑉(𝑘)<0,𝑥(𝑘)≠0. (3.13) 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 𝐴_𝑖=𝐴_𝑖11𝐴_𝑖12𝐴_𝑖21𝐴_𝑖22∈𝑅^𝑛×𝑛,𝐴_𝑖11∈𝑅^𝑛1×𝑛1,𝐴_𝑖12∈𝑅^𝑛2×𝑛2. (3.14) 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∈𝑅^𝑛1×𝑛1,𝑃₂∈𝑅^𝑛2×𝑛2, and positive constants 𝑐,𝜆_𝑖𝑗,𝑖,𝑗=1,2, satisfying the follow inequalities: 𝐴^𝑇_𝑖𝑗𝑗𝑃_𝑗𝐴_𝑖𝑗𝑗−1𝑃1+𝑐^𝑗≤−𝜆_𝑖𝑗𝐼,𝑖=1,2,…𝑁,𝑗=1,2,_𝑁𝑖=1𝛽_𝑖𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝐴_𝑖21²𝑃₂−𝜆_𝑖1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖<0,_𝑁𝑖=1𝛽_𝑖𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑃₁𝐴_𝑖12²−𝜆_𝑖2+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖<0, (3.15)(3.16) where 𝑃=diag{𝑃₁,𝑃₂}, and 𝛽_𝑖=1(1+𝑐)1+𝑐^−1𝑖−1,𝑖=1,2,…,𝑁−1,𝛽_𝑁=11+𝑐^−1𝑁−1. (3.17)

Proof. We choose the Lyapunov function candidate as 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘), (3.18) where 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂]^𝑇,𝑥₁∈𝑅^𝑛1,𝑥₂∈𝑅^𝑛2 and 𝑃=diag{𝑃₁,𝑃₂} are real symmetric positive-definite matrices.
Using Lemma 2.1 and (3.15), we get Δ𝑉(𝑘)|_(2.10)≤𝑥^𝑇(𝑘)^𝑁_𝑖=1𝛽_𝑖(1+𝑐)𝐴^𝑇_𝑖𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖𝑥(𝑘)≤−𝑃_𝑁𝑖=1𝛽_𝑖(1+𝑐)𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂+𝑥₁²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝐴_𝑖21²𝑃₂+𝑥₂²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑃₁𝐴_𝑖12²+^𝑁_𝑖=1𝛽_𝑖‖𝑥‖²1+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖−𝑥^𝑇(𝑘)(𝑘≤𝑃𝑥)_𝑁𝑖=1𝛽_𝑖(1+𝑐)𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11−1()𝑃1+𝑐¹𝑥₁+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22−1()𝑃1+𝑐²𝑥₂+𝑥₁²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝐴_𝑖21²𝑃₂+𝑥₂²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑃₁𝐴_𝑖12²+‖𝑥‖²1+𝑐⁻¹‖𝑃𝐸1+𝑐‖^𝑇_𝑖𝐸_𝑖≤^𝑁_𝑖=1𝛽_𝑖(1+𝑐)𝑥₁²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝐴_𝑖21²𝑃₂−𝜆_𝑖1+1+𝑐⁻¹‖𝑃𝐸1+𝑐‖^𝑇_𝑖𝐸_𝑖+𝑥₂²𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑃₁𝐴_𝑖12²−𝜆_𝑖2+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖=𝑥₁^2𝑁_𝑖=1𝛽_𝑖(1+𝑐)𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝐴_𝑖21²𝑃₂−𝜆_𝑖1+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖+𝑥₂^2𝑁_𝑖=1𝛽_𝑖(1+𝑐)𝐴_𝑖11𝑃₁𝐴_𝑖12+𝐴_𝑖21𝑃₂𝐴_𝑖22+𝑃₁𝐴_𝑖12²−𝜆_𝑖2+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖. (3.19) 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 𝐴_𝑖=𝐴_𝑖11𝐴_𝑖12𝐴_𝑖13𝐴_𝑖21𝐴_𝑖22𝐴_𝑖23𝐴_𝑖31𝐴_𝑖32𝐴_𝑖33∈𝑅^𝑛×𝑛,𝐴_𝑖11∈𝑅^𝑛1×𝑛1,𝐴_𝑖22∈𝑅^𝑛2×𝑛2,𝐴_𝑖33∈𝑅^𝑛3×𝑛3. (3.20)
According to Lemma 2.2, 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,𝜂_𝑖𝑗,𝑖,𝑗=1,2,   and symmetric positive-definite matrices  𝑃₁,𝑃₂,𝑃₃>0,𝑃₁∈𝑅^𝑛1×𝑛1,𝑃₂∈𝑅^𝑛2×𝑛2,𝑃₃∈𝑅^𝑛3×𝑛3 satisfying the following inequalities: 𝐴^𝑇_𝑖𝑗𝑗𝑃_𝑗𝐴_𝑖𝑗𝑗−1𝑃1+𝑐^𝑗≤−𝜂_𝑖𝑗𝐼,𝑖=1,2,…𝑁,𝑗=1,2,3,_3𝑗=2𝐴_𝑖𝑗1²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝜂_𝑖1<0,_{3𝑗=1𝑗≠2}𝐴_𝑖𝑗2²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝜂_𝑖2<0,_2𝑗=1𝐴_𝑖𝑗3²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑐⁻¹‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝜂_𝑖3<0, (3.21)(3.22) where 𝑃=diag{𝑃₁,𝑃₂,𝑃₃}, and 𝛽_𝑖=1(1+𝑐)1+𝑐^−1𝑖−1,𝑖=1,2,…,𝑁−1,𝛽_𝑁=11+𝑐^−1𝑁−1. (3.23)

Proof. We choose the Lyapunov function candidate as 𝑉(𝑘)=𝑥^𝑇(𝑘)𝑃𝑥(𝑘), (3.24) where 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂,𝑥^𝑇₃]^𝑇 and 𝑃=diag{𝑃₁,𝑃₂,𝑃₃} are real symmetric positive-definite matrices.
Computing the product, we have 𝐴^𝑇_𝑖𝑃𝐴_𝑖=𝐴^𝑇_𝑖11𝐴^𝑇_𝑖21𝐴^𝑇_𝑖31𝐴^𝑇_𝑖12𝐴^𝑇_𝑖22𝐴^𝑇_𝑖32𝐴^𝑇_𝑖13𝐴^𝑇_𝑖23𝐴^𝑇_𝑖33𝑃₁000𝑃₂000𝑃₃𝐴_𝑖11𝐴_𝑖12𝐴_𝑖13𝐴_𝑖21𝐴_𝑖22𝐴_𝑖23𝐴_𝑖31𝐴_𝑖32𝐴_𝑖33=^3𝑗=1𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴³_{𝑖𝑗1𝑗=1}𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴³_{𝑖𝑗2𝑗=1}𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴_{3𝑖𝑗3𝑗=1}𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴³_{𝑖𝑗1𝑗=1}𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴³_{𝑖𝑗2𝑗=1}𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴_{3𝑖𝑗3𝑗=1}𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴³_{𝑖𝑗1𝑗=1}𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴³_{𝑖𝑗2𝑗=1}𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴_𝑖𝑗3. (3.25) So, we get 𝑥^𝑇𝐴^𝑇_𝑖𝑃𝐴_𝑖𝑥=𝑥^𝑇₁𝑥^𝑇₂𝑥^𝑇₃^3𝑗=1𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴³_{𝑖𝑗1𝑗=1}𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴³_{𝑖𝑗2𝑗=1}𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴_{3𝑖𝑗3𝑗=1}𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴³_{𝑖𝑗1𝑗=1}𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴³_{𝑖𝑗2𝑗=1}𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴_{3𝑖𝑗3𝑗=1}𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴³_{𝑖𝑗1𝑗=1}𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴³_{𝑖𝑗2𝑗=1}𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴_𝑖𝑗3𝑥₁𝑥₂𝑥₃=𝑥^𝑇₁³_𝑗=1𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴_𝑖𝑗1+𝑥^𝑇₂³_𝑗=1𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴_𝑖𝑗1+𝑥^𝑇₃³_𝑗=1𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴_𝑖𝑗1𝑥₁+𝑥^𝑇₁³_𝑗=1𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴_𝑖𝑗2+𝑥^𝑇₂³_𝑗=1𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴_𝑖𝑗2+𝑥^𝑇₃³_𝑗=1𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴_𝑖𝑗2𝑥₂+𝑥^𝑇₁³_𝑗=1𝐴^𝑇_𝑖𝑗1𝑃_𝑗𝐴_𝑖𝑗3+𝑥^𝑇₂³_𝑗=1𝐴^𝑇_𝑖𝑗2𝑃_𝑗𝐴_𝑖𝑗3+𝑥^𝑇₃³_𝑗=1𝐴^𝑇_𝑖𝑗3𝑃_𝑗𝐴_𝑖𝑗3𝑥₃≤𝑥₁^𝑇𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥₁²³_𝑗=2𝐴_𝑖𝑗1²𝑃_𝑗+2𝑥₁𝑥₂³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+2𝑥₁𝑥₃³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂+𝑥₂²³_{𝑗=1𝑗≠2}𝐴_𝑖𝑗2²𝑃_𝑗+2𝑥₃𝑥₂³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑥₃²³_{𝑗=1𝑗≠3}𝐴_𝑖𝑗3²𝑃_𝑗+𝑥^𝑇₃𝐴^𝑇_𝑖33𝑃₃𝐴_𝑖33𝑥₃≤𝑥^𝑇₁𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥₁²³_𝑗=2𝐴_𝑖𝑗1²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+𝑥^𝑇₂𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂+𝑥₂²³_{𝑗=1𝑗≠2}𝐴_𝑖𝑗2²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑥^𝑇₃𝐴^𝑇_𝑖33𝑃₃𝐴_𝑖33𝑥₃+𝑥₃²²_𝑗=1𝐴_𝑖𝑗3²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗. (3.26) Using the properties of matrix norm, (3.24), and Lemma 2.2, we have Δ𝑉(𝑘)|_(2.10)≤𝑥^𝑇(𝑘)^𝑁_𝑖=1𝛽_𝑖(1+𝑐)𝐴_𝑖^𝑇𝑃𝐴_𝑖+1+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖𝑥(𝑘)≤−𝑃_𝑁𝑖=1𝛽_𝑖(1+𝑐)𝑥₁^𝑇𝐴^𝑇_𝑖11𝑃₁𝐴_𝑖11𝑥₁+𝑥₁²×³_𝑗=2𝐴_𝑖𝑗1²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+𝑥₂^𝑇𝐴^𝑇_𝑖22𝑃₂𝐴_𝑖22𝑥₂+𝑥₂²×³_{𝑗=1𝑗≠2}𝐴_𝑖𝑗2²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑥₃^𝑇𝐴^𝑇_𝑖33𝑃₃𝐴_𝑖33𝑥₃+𝑥₃²×³_𝑗=1𝐴_𝑖𝑗3²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑥^𝑇(𝑘)^𝑁_𝑖=1𝛽_𝑖1+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖𝑥(𝑘)≤−𝑃_𝑁𝑖=1𝛽_𝑖(1+𝑐)𝑥₁²³_𝑗=2𝐴_𝑖𝑗1²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖−𝜂_𝑖1+𝑥₂²³_{𝑗=1𝑗≠2}𝐴_𝑖𝑗2²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗2𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖−𝜂_𝑖2+𝑥₃²²_𝑗=1𝐴_𝑖𝑗3²𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗1𝐴_𝑖𝑗3𝑃_𝑗+³_𝑗=1𝐴_𝑖𝑗3𝐴_𝑖𝑗2𝑃_𝑗+𝑐⁻¹‖𝑃‖𝐸_𝑖^𝑇𝐸_𝑖−𝜂_𝑖3. (3.27) 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, Theorems 3.3 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. Mode1:𝑥(𝑘+1)=𝐴₁+Δ𝐴₁𝑥(𝑘),𝐴₁=0.40.101,Δ𝐴₁=𝐹₁(𝑘)𝐸₁,𝐸₁=[],()=𝐴0,0.1Mode2:𝑥𝑘+1₂+Δ𝐴₂𝑥(𝑘),𝐴₂=−0.20.110.3,Δ𝐴₂=𝐹₂(𝑘)𝐸₂,𝐸₂=[0,0.2]. (4.1)
Set 𝑃₁=𝑃₂=0.5,𝑐=1. It is easily obtained that 𝑒₁=0.02,𝑒₂=0.66,𝑑₁=0.025,𝑑₂𝜆=0.165,₁₁=0.84,𝜆₁₂=0.75,𝜆₂₁=0.96,𝜆₂₂=0.91. (4.2) It is validated that 𝐴^𝑇_𝑖𝑗𝑗𝑃_𝑗𝐴_𝑖𝑗𝑗−𝑃_𝑗≤−𝜆_𝑖𝑗𝑒𝐼,𝑖,𝑗=1,2,₁+𝑐𝑒₂−𝜆₁₁−𝑐𝜆₂₁+𝑐𝜆_max𝑃₁+𝑐⁻¹‖𝑃‖𝐸₁^𝑇𝐸₁+‖𝑃‖𝐸₂^𝑇𝐸₂𝑑<0,₁+𝑐𝑑₂−𝜆₁₂−𝑐𝜆₂₂+𝑐𝜆_max𝑃₂+𝑐⁻¹‖𝑃‖𝐸₁^𝑇𝐸₁+‖𝑃‖𝐸₂^𝑇𝐸₂<0. (4.3) 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: 𝑥^𝑇(𝑘)2𝐴^𝑇_𝑖𝑃𝐴_𝑖+2‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃𝑥^𝑇(𝑘)<0. (4.4) If 𝑥(𝑘)=0, then 𝜎(𝑥(𝑘))=𝛾,𝛾∈{1,2}. (4.5)

Example 4.2. Consider the switched discrete-time system composed of three individual systems given as follows. Mode1∶𝑥(𝑘+1)=𝐴₁+Δ𝐴₁𝑥(𝑘),Δ𝐴₁=𝐹₁(𝑘)𝐸₁,𝐸₁=[].()=𝐴0,0,0.1Mode2:𝑥𝑘+1₂+Δ𝐴₂𝑥(𝑘),Δ𝐴₂=𝐹₂(𝑘)𝐸₂,𝐸₂=[].()=𝐴0,0,0.2Mode3:𝑥𝑘+1₃+Δ𝐴₃𝑥(𝑘),Δ𝐴₃=𝐹₃(𝑘)𝐸₃,𝐸₃=[0,0.1,0], (4.6) where 𝐴₁=0.200.10.10.30.4000.7,𝐴₂=−0.20.1000.30.10.100.1,𝐴₃=−0.10.1000.40.300.10.2. (4.7) Taking 𝑐=1,𝑃₁=0.35,𝑃₂=0.45,𝑃₃=0.7, it is easily obtained that 𝜂₁₁=0.8691,𝜂₁₂=0.8190,𝜂₁₃=0.4191,𝜂₂₁=0.8691,𝜂₂₂𝜂=0.8191,₂₃=0.8991,𝜂₃₁=0.8991,𝜂₃₂=0.7491,𝜂₃₃=0.8691. (4.8) 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: 𝑥^𝑇(𝑘)2𝐴^𝑇_𝑖𝑃𝐴_𝑖+2‖𝑃‖𝐸^𝑇_𝑖𝐸_𝑖−𝑃𝑥^𝑇(𝑘)<0. (4.9) If 𝑥(𝑘)=0, then 𝜎(𝑥(𝑘))=𝛾,𝛾∈{1,2,3}. (4.10)

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.

Acknowledgments

This work was supported by the National Nature Science Foundation of China under Grant 50977047 and the Natural Science Foundation of Tianjin under Grant 11JCYBJC06800.