Switching Signal Design for Exponential Stability of Uncertain Discrete-Time Switched Time-Delay Systems

Abstract

The switching signal design for exponential stability with 𝐻_∞ performance of uncertain switched linear discrete-time systems with interval time-varying delay is considered. Systems with norm-bounded parameter uncertainties are considered. By taking a new Lyapunov-Krasovskii (LK) function, sufficient conditions for the existence of a class of stabilizing switching laws are derived in terms of linear matrix inequalities (LMIs) to guarantee the considered switched time-delay system to be exponentially stable. The resulting stability criteria are of fewer matrix variables and are less conservative than some existing ones. In addition, numerical examples are illustrated to show the main improvement.

1. Introduction

Switched system is represented as the family of subsystems with switching rule which is concerned with various environmental factors and different controllers. During the last decades, there has been increasing interest in the stability analysis and control design for the switched systems (see, e.g., [1–6]). The design of switching signal is one of the three basic problems in stability analysis and the design of switched systems [2, 3]. Switching signal included time-driven switching and state-driven switching. In the first case, time-driven switching depends on time, and many effective methods have been developed such as dwell time method [7, 8], average dwell time method [9, 10], and mode-dependent average dwell time method [11, 12]. In the case of state-driven switching, a switching action takes place when the system state hits a switching surface, and the study of the stability is based on a number of methods, including piecewise Lyapunov function [13] and convex combination [14].

On the other hand, time delay is one of the instability sources for dynamical systems and is a common phenomenon in many industrial and engineering systems. During the last two decades, much attention has been paid to the problem of stability analysis and controller synthesis for time-delay systems (see, e.g., [15–18]). A switched system with time-delay individual subsystems is called a switched time-delay system [19–28]. Switched time-delay systems have various applications in practical engineering systems, such as power systems and power electronics [1, 21]. Many important results on the dynamical behavior have been reported for switched time-delay system [8–12].

Recently, increasing attention has been devoted to the problem of delay-dependent stability of switched delay systems [22–26]. In [23–25], a switching signal design technique is proposed to guarantee the asymptotic stability of switched systems with interval time-varying delay. Based on a discrete LK functional, in [26] a switching rule for the asymptotic stability and stabilization for a class of discrete-time switched systems with interval time-varying delays is designed via linear matrix inequalities. However, the term ∑^{𝑘−𝑑𝑚}_{𝑠=𝑘+1−𝑑(𝑘)}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)=∑^{𝑘−𝑑𝑚}_{𝑠=𝑘+1−𝑑𝑀}𝑥^𝑇(𝑠)𝑄𝑥(𝑠)−∑^{𝑘−𝑑(𝑘)}_{𝑠=𝑘+1−𝑑𝑀}𝑥^𝑇(𝑠)𝑄𝑥(𝑠) is estimated as ∑^{𝑘−𝑑𝑚}_{𝑠=𝑘+1−𝑑𝑀}𝑥^𝑇(𝑠)𝑄𝑥(𝑠) for any 0<𝑑_𝑚≤𝑑(𝑘)≤𝑑_𝑀,  𝑄=𝑄^𝑇>0, which may lead to considerable conservativeness. Following the work, free weighting matrix and additional nonnegative inequality approaches have been used to improve the conservativeness for the obtained results in [27]. However, many free weighting matrices were introduced, which made the stability result complicated. 𝐻_∞ concept was proposed to reduce the effect of the disturbance input on the regulated output to within a prescribed level. In [29], 𝐻_∞ controls were proposed for uncertain discrete switched systems under arbitrary switching signal. In [30], a switching signal design for 𝐻_∞ performance of uncertain discrete switched systems with interval delay and linear fractional perturbations is considered. Nevertheless, the criteria still leave some room for improvement in accuracy as well as complexity due to the method used and offer motivation for further investigation.

Motivated by the above literatures, in this paper, by using an improved discrete LK function combined with LMIs technique, a new switching signal design approach is developed to guarantee the 𝐻_∞ performance for uncertain discrete switched systems with interval time-varying delay to be exponentially stable. Numerical examples are given to show the effectiveness of the proposed method which can be easily solved by using MATLAB LMI control toolbox.

The remaining part of the paper is organized as follows. In Section 2, some preliminaries are introduced. In Section 3, the main results for uncertain switched linear discrete-time systems with interval time-varying delay are presented to be exponential stability. In Section 4, some numerical examples are given to illustrate the effectiveness and the merit of the proposed method. The last section concludes the work.

Notations. We use standard notations throughout the paper. 𝜆_min(𝑀)(𝜆_max(𝑀)) stands for the minimal (maximum) eigenvalue of 𝑀.  𝑀^𝑇 is the transpose of the matrix 𝑀. The relation 𝑀>𝑁(𝑀<𝑁) means that the matrix 𝑀−𝑁 is positive (negative) definite. ‖𝑥‖ denotes the Euclidian-norm of the vector 𝑥∈𝑅^𝑛. 𝑅^𝑛 represents the 𝑛-dimensional real Euclidean space. 𝑅^𝑛×𝑚 is the set of all real 𝑛×𝑚 matrices. diag{⋯} stands for a block-diagonal matrix. In symmetric block matrices or long matrix expressions, we use an asterisk “∗” to represent a term that is induced by symmetry. 𝐼 denotes the identity matrix.

2. Problem Description and Preliminaries

Consider the following uncertain linear discrete-time switched time-delay system: 𝑥(𝑘+1)=𝐴_𝜎+Δ𝐴_𝜎(𝑘𝑥(𝑘)+𝐴_𝑑𝜎)+Δ𝐴_𝑑𝜎(𝑘𝑥((𝑘+𝐵)𝑘−𝑑))_𝜎+Δ𝐵_𝜎(𝑘𝑤(𝑘),𝑦(𝑘)=𝐶_𝜎)+Δ𝐶_𝜎(𝑘𝑥(𝑘)+𝐶_𝑑𝜎)+Δ𝐶_𝑑𝜎(𝑘𝑥((𝑘+𝐷)𝑘−𝑑))_𝜎+Δ𝐷_𝜎(𝑘𝑤(𝑘),𝑥(𝑙)(𝑙))=𝜙,𝑙=𝑘₀−𝑑_𝑀,…,𝑘₀, (1) where 𝑥(𝑘)∈𝑅^𝑛 denotes the system state vector. 𝑦(𝑘)∈𝑅^𝑚 is the measured output. 𝑤(𝑘)∈𝑅^𝑝 is the disturbance input which belongs to 𝑙₂[0,∞). 𝜙(𝑙)∈𝑅^𝑛 is a vector-valued initial function. The switching signal 𝜎∶𝑍→℧={1,2,…,𝑁} is a piecewise constant function. 𝜎=𝑖 means that the 𝑖th subsystem is activated. 𝑁 is the number of subsystems of the switched system. The system matrices 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐶_𝑑𝑖, and 𝐷_𝑖 are a set of known real matrices with appropriate dimensions. Δ𝐴_𝑖(𝑘), Δ𝐴_𝑑𝑖(𝑘), Δ𝐵_𝑖(𝑘), Δ𝐶_𝑖(𝑘), Δ𝐶_𝑑𝑖(𝑘), and Δ𝐷_𝑖(𝑘) are real-valued unknown matrices representing time-varying parameter uncertainties, and are assumed to be of the form Δ𝐴_𝑖(𝑘)Δ𝐴_𝑑𝑖(𝑘)Δ𝐵_𝑖(𝑘)Δ𝐶_𝑖(𝑘)Δ𝐶_𝑑𝑖(𝑘)Δ𝐷_𝑖(𝑘)=𝑀_1𝑖𝑀_2𝑖Δ_1𝑖(𝑘)𝑁_1𝑖𝑁_2𝑖𝑁_3𝑖, (2) where 𝑀_1𝑖, 𝑀_2𝑖, 𝑁_1𝑖, 𝑁_2𝑖, and 𝑁_3𝑖 are known real constant matrices and Δ_1𝑖(𝑘)∶𝑁→𝑅^𝑙1×𝑙2 is unknown time-varying matrix function satisfying Δ^𝑇_1𝑖(𝑘)Δ_1𝑖(𝑘)≤𝐼. (3) The parameter uncertainties Δ𝐴_𝑖(𝑘),  Δ𝐴_𝑑𝑖(𝑘),  Δ𝐵_𝑖(𝑘),  Δ𝐶_𝑖(𝑘),  Δ𝐶_𝑑𝑖(𝑘), and Δ𝐷_𝑖(𝑘) are said to be admissible if both (2) and (3) hold.

For given finite positive integer 𝑑_𝑚 and 𝑑_𝑀, time-varying delay 𝑑(𝑘) satisfying 0<𝑑_𝑚≤𝑑(𝑘)≤𝑑_𝑀,∀𝑘∈𝑁⁺. (4)

Now we present the following definitions and lemmas that are useful in deriving the principal contribution of this paper.

Definition 1 (see [20]). The system (1) is said to be exponentially stable if there exist a switching function 𝜎(⋅) and positive number 𝑐 such that any solution 𝑥(𝑘,𝜙) of the system satisfies ‖𝑥(𝑘)‖≤𝑐𝜆^𝑘−𝑘0𝜙_𝑠,∀𝑘≥𝑘₀, (5) for any initial conditions (𝑘₀,𝜙)∈𝑅⁺×𝐶^𝑛. 𝑐>0 is the decay coefficient, 0<𝜆≤1 is the decay rate, and ‖𝜙‖_𝑠=sup{‖𝜙(𝑙)‖,𝑙=𝑘₀−𝑑_𝑀,𝑘₀−𝑑_𝑀+1,…,𝑘₀}.

Definition 2 (see [26]). The system of matrices {𝐿_𝑖}(𝑖∈℧) is said to be strictly complete if, for every 𝑥∈𝑅^𝑛⧵{0}, there is 𝑖∈℧ such that 𝑥^𝑇𝐿_𝑖𝑥<0.
It is easy to see that the system of matrices {𝐿_𝑖}(𝑖∈℧) is strictly complete if and only if ⋃^𝑁_𝑖=1Ω_𝑖=𝑅^𝑛⧵{0}, where Ω_𝑖={𝑥∈𝑅^𝑛∶𝑥^𝑇𝐿_𝑖𝑥<0}(𝑖∈℧).

Lemma 3 (see [31]). The system of matrices {𝐿_𝑖}(𝑖∈℧) is strictly complete if there exist 𝛼_𝑖≥0, ∑^𝑁_𝑖=1𝛼_𝑖=1 such that ∑^𝑁_𝑖=1𝛼_𝑖𝐿_𝑖<0. If 𝑁=2; then the above condition is also necessary for the strict completeness.

Lemma 4 (see [17]). For any matrix 𝑅=𝑅^𝑇>0 integers 𝑎≤𝑏, if vector function 𝜉(𝑘)∶{−𝑏,−𝑏+1,…,−𝑎}→𝑅^𝑛, then (𝑎−𝑏)^{𝑘−𝑎−1}_{𝑠=𝑘−𝑏}𝑧^𝑇(𝑠)𝑅𝑧(𝑠)≤𝜉^𝑇(𝑘)𝜉(𝑘)−𝑅𝑅∗−𝑅, (6) where 𝑧(𝑘)()(𝑘),𝜉=𝑥𝑘+1−𝑥_𝑇(𝑘)=𝑥^𝑇(𝑘−𝑎)𝑥^𝑇(𝑘−𝑏). (7)

Lemma 5 (see [32],  Schur’s complement). Let 𝑀,  𝑃,  and 𝑄 be given matrices such that 𝑄>0. Then 𝑃𝑀∗−𝑄<0⟺𝑃+𝑀𝑄⁻¹𝑀^𝑇<0. (8)

The objective of this paper is to design a reasonable switching rule for discrete-time switched system (1) with time-varying delay satisfying (4) to guarantee that the system while be exponentially stable.

3. Main Result

In this section, we will divide the whole state space 𝑅^𝑛 into 𝑁 subregions and then define a particular quadratic function in each sub region. Via an appropriate designed switching rule, the particular quadratic function in each sub region will decrease along the system trajectory with the corresponding subsystem. Consequently, the whole switched system remains exponentially stable.

Firstly, we will introduce the switching regions and the corresponding switching law. Given 𝑃>0 and 𝑈>0, define the domains by Ω_𝑖𝑃,𝑈,𝐴_𝑖=𝑥(𝑘)∈𝑅^𝑛∶𝑥^𝑇(𝑘)𝑌_𝑖𝑥(𝑘)<0, (9) where 𝑌_𝑖=𝐴^𝑇_𝑖𝑃𝐴_𝑖−𝑈, 𝑖∈℧.

From the similar proof of [26, 27], it can be easily obtained that ^𝑁_𝑖=1Ω_𝑖=𝑅^𝑛⧵{0}. (10) Construct the following switching region: Ω₁=Ω₁,Ω₂=Ω₂⧵Ω₁,Ω₃=Ω₃⧵Ω₁⧵Ω₂,…,Ω_𝑁=Ω_𝑁⧵Ω₁⧵⋯⧵Ω_𝑁−1. (11) We can obtain ⋃^𝑁_𝑖=1Ω_𝑖=𝑅^𝑛⧵{0} and Ω_𝑖∩Ω_𝑗=𝜙, for all 𝑖≠𝑗, where 𝜙 is an empty set.

After dividing the whole state space 𝑅^𝑛 into 𝑁 sub regions, we construct the switching signal as follows: 𝜎(𝑥(𝑘))=𝑖,∀𝑥(𝑘)∈Ω_𝑖(𝑖∈℧). (12)

In order to discuss robust stability of system (1), first, we consider the following nominal system without parametric uncertainties: 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝐵_𝑖𝑤(𝑘),𝑦(𝑘)=𝐶_𝑖𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝐷_𝑖𝑤(𝑘). (13) The following theorem gives a sufficient condition for the existence of an admissible reasonable switching rule for system (13) with disturbance input 𝑤(𝑘)=0 to be exponentially stable.

Theorem 6. For some constants 𝛾∈(0,1] and 0≤𝛼_𝑖≤1, 𝑖∈℧, ∑^𝑁_𝑖=1𝛼_𝑖=1, if there exist positive definite symmetric matrices 𝑃,𝑈,𝑄_𝑗,𝑅_𝑗,𝑗=1,2,3, such that the following LMIs hold: 𝜙Ξ=¹¹𝜙₁₂𝜙₁₃0𝜙₁₅∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅∗∗𝜙₃₃00∗∗∗𝜙₄₄0∗∗∗∗−𝑊<0,_𝑁𝑖=1𝛼_𝑖𝐴^𝑇_𝑖𝑃𝐴_𝑖<𝑈, (14)(15) then the system (13) with time-varying delay satisfying (4) and 𝑤(𝑘)=0 is globally exponentially stable with convergence rate 𝜆=√𝛾 by the switching signal designed by (12).
Here 𝜙₁₁=𝑈−𝛾𝑃+𝑄₁+𝑄₃−𝛾^𝑑𝑚𝑑_𝑚𝑅₁+𝑅₃𝜙₁₂=𝐴^𝑇_𝑖𝑃𝐵_𝑖,𝜙₁₃=𝛾^𝑑𝑚𝑑_𝑚𝑅₁+𝑅₃,𝜙₁₅=𝐴_𝑖−𝐼^𝑇𝑊^𝑇,𝜙₂₂=𝐵^𝑇_𝑖𝑃𝐵_𝑖−𝛾^𝑑𝑀𝑄₃−𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚2𝑅₂+𝑅₃,𝜙₂₃=𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚𝑅₂+𝑅₃,𝜙₂₄=𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚𝑅₂,𝜙₂₅=𝐵^𝑇_𝑖𝑊^𝑇,𝜙₃₃=𝛾^𝑑𝑚𝑄₂−𝑄₁−𝛾^𝑑𝑚𝑑_𝑚𝑅₁+𝑅₃−𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚𝑅₂+𝑅₃,𝜙₄₄=−𝛾^𝑑𝑀𝑄₂−𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚𝑅₂,𝑊=𝑑_𝑀−𝑑_𝑚𝑅₂+𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₃. (16)

Proof. Consider the following LK function for system (1): 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘). (17) Here 𝑉₁(𝑘)=𝑥^𝑇(𝑘)(𝑘),𝑉𝑃𝑥₂(𝑘)=^𝑘−1_{𝑠=𝑘−𝑑𝑚}𝛾^{𝑘−1−𝑠}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+^{𝑘−𝑑𝑚−1}_{𝑠=𝑘−𝑑𝑀}𝛾^{𝑘−1−𝑠}𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)+^𝑘−1_{𝑠=𝑘−𝑑(𝑘)}𝛾^{𝑘−1−𝑠}𝑥^𝑇(𝑠)𝑄₃𝑥(𝑠),𝑉₃(𝑘)=⁻¹_{𝜃=−𝑑𝑚}^𝑘−1_{𝑠=𝑘+𝜃}𝛾^{𝑘−1−𝑠}𝑧^𝑇(𝑠)𝑅₁𝑧(𝑠)+^{−𝑑𝑚−1}_{𝜃=−𝑑𝑀}^𝑘−1_{𝑠=𝑘+𝜃}𝛾^{𝑘−1−𝑠}𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠)+⁻¹_{𝜃=−𝑑(𝑘)}^𝑘−1_{𝑠=𝑘+𝜃}𝛾^{𝑘−1−𝑠}𝑧^𝑇(𝑠)𝑅₃𝑧(𝑠), (18)(19)(20) where 𝑃,𝑄_𝑗,𝑅_𝑗,𝑗=1,2,3 are positive definite symmetric matrices and 𝛾∈(0,1].
Now, we will show the decay estimation of 𝑉(𝑘) in (17) along the state trajectory of system (1). To this end, define 𝑉(𝑘+1)−𝛾𝑉(𝑘)=³_𝑗=1Δ𝑉_𝑗(𝑘). (21) Here Δ𝑉₁(𝑘)=𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃𝐴_𝑖−𝛾𝑃𝑥(𝑘)+2𝑥^𝑇(𝑘)𝐴^𝑇_𝑖𝑃𝐵_𝑖𝑥(𝑘−𝑑(𝑘))+𝑥^𝑇(𝑘−𝑑(𝑘))𝐵^𝑇_𝑖𝑃𝐵_𝑖𝑥(𝑘−𝑑(𝑘)),Δ𝑉₂(𝑘)=𝑥^𝑇(𝑘)𝑄₁+𝑄₃𝑥(𝑘)−𝛾^𝑑𝑚𝑥^𝑇𝑘−𝑑_𝑚𝑄₁−𝑄₂𝑥𝑘−𝑑_𝑚−𝛾^𝜏(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑄₃𝑥(𝑘−𝑑(𝑘))−𝛾^𝑑𝑀𝑥^𝑇𝑘−𝑑_𝑀𝑄₂𝑥𝑘−𝑑_𝑀≤𝑥^𝑇(𝑘)𝑄₁+𝑄₃𝑥(𝑘)−𝛾^𝑑𝑀𝑥^𝑇(𝑘−𝑑(𝑘))𝑄₃𝑥(𝑘−𝑑(𝑘))−𝛾^𝑑𝑚𝑥^𝑇𝑘−𝑑_𝑚𝑄₁−𝑄₂𝑥𝑘−𝑑_𝑚−𝛾^𝑑𝑀𝑥^𝑇𝑘−𝑑_𝑀𝑄₂𝑥𝑘−𝑑_𝑀,Δ𝑉₃(𝑘)=𝑧^𝑇(𝑘)𝑑_𝑀−𝑑_𝑚𝑅₂+𝑑_𝑚𝑅₁+𝑑(𝑘)𝑅₃𝑧(𝑘)−^𝑘−1_{𝑠=𝑘−𝑑𝑚}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₁𝑧(𝑠)−^{𝑘−𝑑𝑚−1}_{𝑠=𝑘−𝑑𝑀}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠−_{𝑘−1(𝑘)𝑠=𝑘−𝑑})𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₃𝑧(𝑠), (22)(23)(24) while −^𝑘−1_{𝑠=𝑘−𝑑(𝑘)}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₃𝑧(𝑠)=−^𝑘−1_{𝑠=𝑘−𝑑𝑚}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₃𝑧(𝑠−_{𝑘−𝑑𝑚−1𝑠=𝑘−𝑑(𝑘)})𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₃𝑧(𝑠),−^{𝑘−𝑑𝑚−1}_{𝑠=𝑘−𝑑𝑀}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠)=−^{𝑘−𝑑𝑚−1}_{𝑠=𝑘−𝑑(𝑘)}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠−_{𝑘−𝑑(𝑘)−1𝑠=𝑘−𝑑𝑀})𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠). (25) So (24) could be Δ𝑉₃(𝑘)≤𝑧^𝑇(𝑘)𝑑_𝑀−𝑑_𝑚𝑅₂+𝑑_𝑚𝑅₁+𝑑_𝑀𝑅₃𝑧(𝑘)−^𝑘−1_{𝑠=𝑘−𝑑𝑚}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₁+𝑅₃𝑧(𝑠)−^{𝑘−𝑑𝑚−1}_{𝑠=𝑘−𝑑(𝑘)}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂+𝑅₃𝑧(𝑠)−^{𝑘−𝑑(𝑘)−1}_{𝑠=𝑘−𝑑𝑀}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠). (26) From Lemma 4, we have −^𝑘−1_{𝑠=𝑘−𝑑𝑚}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₁+𝑅₃𝑧(𝑠)≤𝛾^𝑑𝑚𝑑_𝑚𝜉^𝑇₁(𝑘)−𝑅₁−𝑅₃𝑅₁+𝑅₃∗−𝑅₁−𝑅₃𝜉₁(𝑘),−^{𝑘−𝑑𝑚−1}_{𝑠=𝑘−𝑑(𝑘)}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂+𝑅₃𝑧(𝑠)≤𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚𝜉^𝑇₂(𝑘)−𝑅₂−𝑅₃𝑅₂+𝑅₃∗−𝑅₂−𝑅₃𝜉₂(𝑘),−^{𝑘−𝑑(𝑘)−1}_{𝑠=𝑘−𝑑𝑀}𝛾^𝑘−𝑠𝑧^𝑇(𝑠)𝑅₂𝑧(𝑠)≤𝛾^𝑑𝑀𝑑_𝑀−𝑑_𝑚𝜉^𝑇₃(𝑘)−𝑅₂𝑅₂∗−𝑅₂𝜉₃(𝑘), (27) where 𝜉^𝑇₁(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇𝑘−𝑑_𝑚,𝜉^𝑇₂(𝑘)=𝑥^𝑇𝑘−𝑑_𝑚𝑥^𝑇((𝑘,𝜉𝑘−𝑑))_𝑇3(𝑘)=𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇𝑘−𝑑_𝑀. (28) Combining (21)–(27), it yields 𝑉(𝑘+1)−𝛾𝑉(𝑘)≤𝜉^𝑇(𝑘)×𝜙₁₁+𝐴^𝑇_𝑖𝑊𝐴_𝑖𝜙₁₂+𝐴^𝑇_𝑖𝑊𝐴_𝑖𝜙₁₃0∗𝜙₂₂+𝐵^𝑇_𝑖𝑊𝐵_𝑖𝜙₂₃𝜙₂₄∗∗𝜙₃₃0∗∗∗𝜙₄₄𝜉(𝑘)+𝑥^𝑇(𝑘)𝑌_𝑖𝑥(𝑘), (29) where 𝜉^𝑇(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇𝑘−𝑑_𝑚𝑥^𝑇𝑘−𝑑_𝑀. (30)By Lemma 3 and condition (15), we know that the system of matrices 𝑌_𝑖=𝐴^𝑇_𝑖𝑃𝐴_𝑖−𝑈, 𝑖∈℧ is strictly complete, and the sets Ω_𝑖 and Ω_𝑖 by (5) and (6) are well defined such that ^𝑁_𝑖=1Ω_𝑖=𝑅^𝑛⧵{0},^𝑁_𝑖=1Ω_𝑖=𝑅^𝑛⧵{0},Ω_𝑖∩Ω_𝑗=Φ,𝑖≠𝑗. (31) Therefore, for any 𝑥(𝑘)∈𝑅^𝑛, 𝑘>0, there always exists an 𝑖∈{1,2,…,𝑁} such that 𝑥(𝑘)∈Ω_𝑖. Choosing the switching rule (12), by Schur’s complement of [32] with condition (14), leads to 𝑉(𝑘+1)−𝛾𝑉(𝑘)≤0⟺𝑉(𝑘)≤𝛾^𝑘𝑉(0). (32) By the system LK function (17), there always exist two positive constants 𝑐₁, 𝑐₂ such that 𝑐₁‖𝑥(𝑘)‖²≤𝑉(𝑘),𝑉(0)≤𝑐₂‖𝑥(0)‖²_𝑠. (33) Here 𝑐₁=𝜆_min(𝑃),𝑐₂=𝜆_max(𝑃)+³_𝑗=1𝜆_max𝑄_𝑗+𝜆_max𝑅_𝑗. (34) From (32) and (33), one obtains ‖𝑥(𝑘)‖≤𝑐₂𝑐₁𝛾^𝑘/2‖𝑥(0)‖_𝑠. (35) By Definition 1, we know that the system (1) is exponentially stable with decay rate 𝜆=√𝛾. This completes the proof.

In order to determine the exponentially stability with 𝐻_∞ performance 𝜅 of system (1), we need to introduce the following definition.

Definition 7 (see [30]). Consider system (1) with the switching signal in (12) and the following conditions.(i) With 𝑤(𝑘)=0, the system (1) is exponentially stable with convergence rate 0<𝛾<1. (ii) With zero initial conditions, the signals 𝑤(𝑘) and 𝑦(𝑘) are bounded by ^∞_𝑘=0𝛾^−2𝑘𝑦^𝑇(𝑘)𝑦(𝑘)≤𝜅^2∞_𝑘=0𝛾^−2𝑘𝑤^𝑇(𝑘)𝑤(𝑘), (36) for all 𝑤∈𝐿₂(𝛾,0,∞), 𝑤≠0 for constants 𝜅>0 and 0<𝛾<1. In the above conditions, the system (1) is exponentially stabilizable with 𝐻_∞ performance 𝜅 and convergence rate 𝛾 by switching signal in (12).

Theorem 8. For some constants 𝛾∈(0,1] and 0≤𝛼_𝑖≤1, ∑^𝑁_𝑖=1𝛼_𝑖=1(𝑖∈℧), if there exist positive definite symmetric matrices 𝑃,𝑈,𝑄_𝑗,𝑅_𝑗(𝑗=1,2,3) such that (15) and the following LMIs hold: 𝜙₁₁𝜙₁₂𝜙₁₃0𝜙₁₅𝐶^𝑇_𝑖𝜙₁₅∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝐶^𝑇_𝑑𝑖𝜙₂₅∗∗𝜙₃₃0000∗∗∗𝜙₄₄𝜙000∗∗∗∗₅₅𝐷^𝑇_𝑖𝐵^𝑇_𝑖𝑊^𝑇∗∗∗∗∗−𝐼0∗∗∗∗∗∗−𝑊<0, (37) then the system (13) with time-varying delay satisfying (4) is globally exponentially stable with convergence rate 𝜆=√𝛾 and 𝐻_∞ performance 𝜅 by the switching signal designed by (12).
Here 𝜙₁₅=𝐴^𝑇_𝑖𝑃𝐵_𝑖,𝜙₂₅=𝐴^𝑇_𝑑𝑖𝑃𝐵_𝑖,𝜙₅₅=𝐵^𝑇_𝑖𝑃𝐵_𝑖−𝜅². (38)

Proof. The proof is similar to that of Theorem 6. From Theorem 6, one can easily obtain 𝑉(𝑘+1)−𝛾𝑉(𝑘)+𝑦^𝑇(𝑘)𝑦(𝑘)−𝜅²𝑤^𝑇(𝑘)𝑤(𝑘)≤𝜂^𝑇₁(𝑘)𝜙₁₁𝜙₁₂𝜙₁₃0𝜙₁₅∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅∗∗𝜙₃₃00∗∗∗𝜙₄₄0∗∗∗∗𝜙₅₅𝜂₁(𝑘)+𝑦^𝑇(𝑘)𝑦(𝑘)+𝑧^𝑇(𝑘)𝑊𝑧(𝑘), (39) where𝜂^𝑇₁(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇𝑘−𝑑_𝑚𝑥^𝑇𝑘−𝑑_𝑀𝑤^𝑇(𝑘). (40)Combining (37) and (15), by Schur’s complement, we have 𝑉(𝑘+1)−𝛾𝑉(𝑘)+𝑦^𝑇(𝑘)𝑦(𝑘)−𝜅²𝑤^𝑇(𝑘)𝑤(𝑘)≤0. (41) By Definition 7, the system (13) with time-varying delay satisfying (4) is globally exponentially stable with convergence rate 𝜆=√𝛾 and 𝐻_∞ performance 𝜅 by the switching signal designed by (12).

Now, we extend Theorems 6 and 8 to obtain the corresponding results for uncertain switched systems (1). Set 𝑝(𝑘,𝑖)=Δ_𝑖(𝑘)𝑁_1𝑖𝑥(𝑘)+𝑁_2𝑖𝑥(𝑘−𝑑(𝑘))+𝑁_3𝑖𝑤(𝑘). (42) Combining (3), we have 𝑝^𝑇(𝑘)𝑝(𝑘)≤𝑁_1𝑖𝑥(𝑘)+𝑁_2𝑖𝑥(𝑘−𝑑(𝑘))+𝑁_3𝑖𝑤(𝑘)^𝑇×𝑁_1𝑖𝑥(𝑘)+𝑁_2𝑖𝑥(𝑘−𝑑(𝑘))+𝑁_3𝑖𝑤(𝑘). (43) Then, for any 𝜀>0, the following inequalities hold: 𝜀𝑁_1𝑖𝑥(𝑘)+𝑁_2𝑖𝑥(𝑘−𝑑(𝑘))+𝑁_3𝑖𝑤(𝑘)^𝑇×𝑁_1𝑖𝑥(𝑘)+𝑁_2𝑖𝑥(𝑘−𝑑(𝑘))+𝑁_3𝑖𝑤(𝑘)−𝜀𝑝^𝑇(𝑘,𝑖)𝑝(𝑘,𝑖)≥0. (44)

By (2), the system (1) using (42) can be expressed as follows: 𝑥(𝑘+1)=𝐴_𝑖𝑥(𝑘)+𝐴_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝐵_𝑖𝑤(𝑘)+𝑀_1𝑖𝑝(),𝑦(𝑘)𝑘,𝑖=𝐶_𝑖𝑥(𝑘)+𝐶_𝑑𝑖𝑥(𝑘−𝑑(𝑘))+𝐷_𝑖𝑤(𝑘)+𝑀_2𝑖𝑝(𝑘,𝑖). (45) The following theorem provides the robust exponential stability conditions for uncertain switched systems (45) with 𝑤(𝑘)=0.

Theorem 9. For some constants 𝛾∈(0,1] and 0≤𝛼_𝑖≤1, 𝑖∈℧,∑^𝑁_𝑖=1𝛼_𝑖=1, if there exist positive definite symmetric matrices 𝑃,𝑈,𝑄_𝑗,𝑅_𝑗(𝑗=1,2,3) such that (15) and the following LMIs hold: 𝜙₁₁𝜙₁₂00𝜙₁₆𝜙₁₅∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₆𝜙₂₅∗∗𝜙₃₃000∗∗∗𝜙₄₄𝜙00∗∗∗∗₆₆𝑀^𝑇_1𝑖𝑊^𝑇∗∗∗∗∗−𝑊<0, (46) then the system (45) with time-varying delay satisfying (4) and 𝑤(𝑘)=0 is globally exponentially stable with convergence rate 𝜆=√𝛾 by the switching signal designed by (12).
Here 𝜙₁₁=𝜙₁₁+𝜀𝑁^𝑇_1𝑖𝑁_1𝑖,𝜙₁₂=𝜙₁₂+𝜀𝑁^𝑇_1𝑖𝑁_2𝑖,𝜙₁₆=𝐴^𝑇_𝑖𝑃𝑀_1𝑖,𝜙₂₂=𝜙₂₂+𝜀𝑁^𝑇_2𝑖𝑁_2𝑖,𝜙₂₆=𝐴^𝑇_𝑑𝑖𝑃𝑀_1𝑖,𝜙₆₆=𝑀^𝑇_1𝑖𝑃𝑀_1𝑖−𝜀. (47)

Proof. The result is carried out using the techniques employed for proving Theorem 6. Combining (44) with 𝑤(𝑘)=0, one can obtain the following 𝑉(𝑘+1)−𝛾𝑉(𝑘)≤𝑧^𝑇(𝑘)𝑊𝑧(𝑘)+𝜂^𝑇₂(𝑘)𝜙₁₁𝜙₁₂00𝜙₁₆∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₆∗∗𝜙₃₃00∗∗∗𝜙₄₄0∗∗∗∗𝜙₆₆𝜂^𝑇₂(𝑘), (48) where𝜂^𝑇₂(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇𝑘−𝑑_𝑀𝑥^𝑇𝑘−𝑑_𝑀𝑝^𝑇(𝑘,𝑖). (49)From (46) and (15), by Schur Complement, we have 𝑉(𝑘+1)−𝛾𝑉(𝑘)≤0.

Now, we are going to analyze the 𝐻_∞ performance for the uncertain system (1). The main result is given as follows.

Theorem 10. For some constants 𝛾∈(0,1] and 0≤𝛼_𝑖≤1, 𝑖∈℧, ∑^𝑁_𝑖=1𝛼_𝑖=1, if there exist positive definite symmetric matrices 𝑃,𝑈,𝑄_𝑗,𝑅_𝑗(𝑗=1,2,3) such that (15) and the following LMIs hold 𝜙₁₁𝜙₁₂𝜙₁₃0𝜙₁₅𝜙₁₆𝐶^𝑇_𝑖𝜙₁₅∗𝜙₂₂𝜙₂₃𝜙₂₄𝜙₂₅𝜙₂₆𝐶^𝑇_𝑑𝑖𝜙₂₅∗∗𝜙₃₃00000∗∗∗𝜙₄₄0000∗∗∗∗𝜙₅₅𝜙₅₆𝐷^𝑇_𝑖𝜙₅₈∗∗∗∗∗𝜙₆₆𝑀^𝑇_2𝑖𝜙₆₈∗∗∗∗∗∗−𝐼0∗∗∗∗∗∗∗−𝑊<0, (50) then the system (1) with time-varying delay satisfying (4) is globally exponentially stable with convergence rate 𝜆=√𝛾 and 𝐻_∞ performance 𝜅 by the switching signal designed by (12).
Here 𝜙₁₅=𝜙₁₅+𝜀𝑁^𝑇_1𝑖𝑁_3𝑖,𝜙₂₅=𝜙₂₅+𝜀𝑁^𝑇_2𝑖𝑁_3𝑖,𝜙₅₈=𝐵^𝑇_𝑖𝑊^𝑇,𝜙₅₅=𝜙₅₅+𝜀𝑁^𝑇_3𝑖𝑁_3𝑖,𝜙₅₆=𝐵^𝑇_𝑖𝑃𝑀_1𝑖,𝜙₆₈=𝑀^𝑇_1𝑖𝑊^𝑇. (51)

Proof. The result is carried out using the techniques employed for proving Theorems 6, 8, and 9. There exists matrix Θ satisfying 𝑉(𝑘+1)−𝛾𝑉(𝑘)+𝑦^𝑇(𝑘)𝑦(𝑘)−𝜅²𝑤^𝑇(𝑘)𝑤(𝑘)≤𝜂^𝑇₃(𝑘)Θ𝜂^𝑇₃(𝑘)+𝑧^𝑇(𝑘)𝑊𝑧(𝑘)+𝑦^𝑇(𝑘)𝑦(𝑘), (52) where 𝜂^𝑇₃(𝑘)=𝑥^𝑇(𝑘)𝑥^𝑇(𝑘−𝑑(𝑘))𝑥^𝑇𝑘−𝑑_𝑚𝑥^𝑇𝑘−𝑑_𝑀𝑤^𝑇(𝑘)𝑝^𝑇(𝑘,𝑖). (53)Combining (50) and (15), by Schur Complement, we have 𝑉(𝑘+1)−𝛾𝑉(𝑘)+𝑦^𝑇(𝑘)𝑦(𝑘)−𝜅²𝑤^𝑇(𝑘)𝑤(𝑘)≤0.

4. Examples

In this section, we consider some numerical examples in [26, 27] to show that the proposed theorem in this paper provides less conservativeness with comparison to some existing results.

Example 1 (see [27]). Consider system (13) with 𝑤(𝑘)=0 and the following parameters: 𝐴₁=0.541.02−0.17−0.31,𝐴_𝑑1=,𝐴0.180.36−0.06−0.12₂=−0.01−0.060.010.04,𝐴_𝑑2=.0.110.18−0.03−0.04 (54) Assume the minimum delay bound 𝑑_𝑚=1. By Theorem 6, via solving LMIs (14) and (15) with 𝜆=0.8 and 𝛼₁=𝛼₂=0.5, we have 𝑑_𝑀=10 and 𝑃=13.201930.1428∗72.992,𝑈=0.44931.1242∗3.098. (55) From (9) and (11), one can obtain the switching regions Ω₁=𝑥₁𝑥₂^𝑇∈𝑅∶−0.0244𝑥²₁−0.5572𝑥₁𝑥₂−1.4106𝑥²₂<0,Ω₂=𝑅²⧵Ω₁. (56) The switching regions Ω₁ and Ω₂ of the system are shown in Figure 1. Select the switching signal by 𝜎(𝑥(𝑘))=𝑖,𝑥(𝑘)∈Ω_𝑖,𝑖=1,2. (57) With the initial condition 𝜙(𝜃)=[1−1]^𝑇, 𝜃=−10,−9,…,0, and 𝑑(𝑘)=10, the state responses of system are shown in Figure 2.

Figure 1 

The switching regions.

Figure 2 

The state response.

In order to show the improvement of our results over other recent results, some comparisons are made in Table 1. From Table 1, one can see that the results of this paper provide a larger allowable upper bound for time delay to guarantee the asymptotical or exponentially stability of system (1) with (54) by switching signal in (12).

Table 1 

Allowable delay upper bound for different decay rate with 𝑑_𝑚=1, 𝛼₁=𝛼₂=0.5.

Example 2 (see [27]). Consider system (13) with 𝑤(𝑘)=0 and the following parameters: 𝐴₁=1.010.100.1,𝐴_𝑑1=,𝐴−0.10−0.1−0.1₂=0.100.11.02,𝐴_𝑑2=.0.1200.110.11 (58) Let 𝛼₁=𝛼₂=0.5, the minimum delay bound 𝑑_𝑚=1, and decay rate 𝜆=1. The delay upper bound obtained by [27] is 3. However, our result obtained by the application of Theorem 6 could be infinite to guarantee the asymptotic stability of system (13) with (58) by switching signal in (12). The switching regions can be designed in the same way as in Example 1.

Example 3. Consider the discrete-time switched system (13) with 𝑤(𝑘)=0 and the following parameters: 𝐴₁=1.210.100.1,𝐴_𝑑1=,𝐴−0.10−0.1−0.1₂=0.100.11.02,𝐴_𝑑2=.0.1200.110.11 (59) One can easily find that both the above subsystems are unstable. The state responses of the two subsystems are shown in Figures 3 and 4. Let 𝑑_𝑚=1. By Theorem 6, via solving LMIs (14) and (15) with 𝜆=1 and 𝛼₁=𝛼₂=0.5, we have 𝑑_𝑀=5 and ,.𝑃=1.0𝑒+003∗3.54860.1556∗0.5634𝑈=1.0𝑒+003∗2.62040.2602∗0.3184 (60) From (9) and (11), one can obtain the switching regions Ω¹_𝑖=𝑥₁𝑥₂^𝑇∈𝑅∶2.5751𝑥²₁+0.376𝑥₁𝑥₂−0.2741𝑥²₂<0,Ω₂=𝑅²⧵Ω₁. (61) Select the switching signal by 𝜎(𝑥(𝑘))=𝑖,𝑥(𝑘)∈Ω_𝑖,𝑖=1,2. (62)
With time-varying delay 𝑑(𝑘)=5 and the initial condition 𝜙(𝜃)=1−1^𝑇, 𝜃=−5,−4,…,0. The state responses of system are shown in Figure 5. It can be seen from Figure 5 that the designed switching law is effective although all subsystems are unstable. However, with the same parameters and 𝑑_𝑀=5, the results in [27] cannot find any feasible solution to guarantee the asymptotic stability of system (13) with (59).

Figure 3 

State responses of the subsystems 1.

Figure 4 

State responses of the subsystems 2.

Figure 5 

State responses of the switched systems 2.

Example 4. Consider system (45) with the following parameters: 𝐴₁=10.0100,𝐴_𝑑1=,𝐵00.10−0.1₁=0.1000.1,𝐶₁=,𝐶0.01000.08_𝑑1=0.02000.01,𝐷₁=,𝐴0.10.100.2₂=10.010.011.01,𝐴_𝑑2=,𝐵0.100.10.1₂=0.200.10.1,𝐶₂=,𝐶0.03000.05_𝑑2=0.02000.02,𝐷₂=,𝑀0.2000.1₂=0.1000.05,𝑁₂=,𝑀0.01000.02₁=𝐵₁,𝑁₁=𝐶_𝑑1,𝑁₃=0.1×𝐵₁. (63) For a given 𝑑_𝑚=1, 𝛾=0.98, 𝜅=1.53, and 𝛼₁=𝛼₂=0.5. We compute the upper delay bound 𝑑_𝑀=23 such that the uncertain system is exponential stability by applying Theorem 10. However, with the same parameters and 𝑑_𝑀=23, the results in [30] cannot find any feasible solution to guarantee the exponentially stable with 𝐻_∞ performance 𝜅=1.53 of system (45). Via solving LMIs (50) and (15), we have 𝑃=0.68360.0942∗0.6615,𝑈=0.34240.00670.00670.3387. (64) From (9) and (11), one can obtain the switching regions Ω₁=𝑥₁𝑥₂^𝑇∈𝑅∶0.3413𝑥²₁+0.0002𝑥₁𝑥₂−0.3386𝑥²₂<0,Ω₂=𝑅²⧵Ω₁. (65) Select the switching signal by 𝜎(𝑥(𝑘))=𝑖, 𝑥(𝑘)∈Ω_𝑖, 𝑖=1,2. On the other side, by setting the upper delay bound 𝑑_𝑀=3, we have the 𝐻_∞ performance 𝜅=0.275.

5. Conclusions

By using improved discrete LK function combined with LMIs technique, in this paper, we propose new criteria for the exponential stability with 𝐻_∞ performance of uncertain switched linear discrete-time systems with interval time-varying delay. If there is a feasible solution for the proposed LMIs conditions under some given upper bounds of delay, the switching law can be designed and the exponential stability of the systems can be achieved. Finally, the obtained results are shown to be less conservative than previous ones via the numerical examples.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (nos. 61104064 and 61174038).