Stabilization and Controller Design of 2D Discrete Switched Systems with State Delays under Asynchronous Switching

Abstract

This paper is concerned with the problem of robust stabilization for a class of uncertain two-dimensional (2D) discrete switched systems with state delays under asynchronous switching. The asynchronous switching here means that the switching instants of the controller experience delays with respect to those of the system. The parameter uncertainties are assumed to be norm-bounded. A state feedback controller is proposed to guarantee the exponential stability. The dwell time approach is utilized for the stability analysis and controller design. A numerical example is given to illustrate the effectiveness of the proposed method.

1. Introduction

Two-dimensional (2D) systems have attracted considerable attention for several decades due to their numerous applications in many areas, such as multidimensional digital filtering, linear image processing, signal processing and process control [1–3]. It is well known that 2D systems can be represented by different models such as Roesser model, Fornasini-Marchesini model, and Attasi model. The issues of stability analysis and control synthesis of these systems have been studied in [4–8]. Considering that time delays frequently occur in practical systems and are often the source of instability, many authors have devoted their energies to studying time-delay systems. Recently, many results on delay systems have been reported in the literature. For example, the delay-fractional approach has been utilized to deal with discrete time-delay systems in [9–11]. The stability of 2D discrete systems with state delays has been investigated in [12–16].

On the other hand, because of their wide applications in many fields, such as mechanical systems, automotive industry, aircraft and air traffic control, and switched power converters, switched systems have also received considerable attention during the past few decades. A switched system is a hybrid system which consists of a finite number of continuous-time or discrete-time subsystems and a switching signal specifying the switch between these subsystems. The stability and stabilization problems have been extensively studied in [17–25].

In many modelling problems of physical processes, a 2D switching representation is needed. One can cite a 2D physically based model for advanced power bipolar devices [26] and heat flux switching, and modulating in a thermal transistor [27]. This class of systems can correspond to 2D state space or 2D time space switched systems. Recently, there are a few reports on 2D switched systems. Benzaouia et al. firstly studied the stabilization problem of 2D discrete switched systems with arbitrary switching sequences in [28, 29]. By using the common Lyapunov function method and multiple Lyapunov functions method, two different sufficient conditions for the existence of state feedback controllers were proposed. In [30], the authors first extended the concept of average dwell time to 2D switched systems and designed a switching signal to guarantee the exponential stability of delay-free 2D switched systems. It should be pointed out that a very common assumption in [30] is that the controllers are switched synchronously with the switching of system modes, which is quite unpractical. As stated in [31, 32], there inevitably exists asynchronous switching in actual operation that is, the switching instants of the controllers exceed or lag behind those of system modes. Thus, it is necessary to consider asynchronous switching for efficient control design. Some results on the control synthesis for switched systems under asynchronous switching have been proposed in [33–36]. However, to the best of our knowledge, the stabilization problem for 2D switched systems under asynchronous switching has not been yet investigated to date, especially for 2D switched systems with state delays, which motivates our present study.

In this paper, we are interested in designing a stabilizing controller for 2D discrete switched delayed systems represented by a model of Roesser type under asynchronous switching such that the corresponding closed-loop systems are exponentially stable. The dwell time approach is utilized for the stability analysis and controller design. The main contributions of this paper can be summarized as follows: (i) the asynchronous stabilization problem is for the first time addressed in the paper; (ii) an exponential stability criterion is established for 2D switched systems with state delays; and (iii) an asynchronous switching controller design scheme is proposed to guarantee the exponential stability of the resulting closed-loop system.

This paper is organized as follows. In Section 2, problem formulation and some necessary lemmas are given. In Section 3, based on the dwell time approach, stability and stabilization for 2D discrete switched systems with state delays are addressed. Then, a sufficient condition for the existence of a stabilizing controller for such 2D discrete switched systems under asynchronous switching is derived in terms of a set of matrix inequalities. A numerical example is provided to illustrate the effectiveness of the proposed approach in Section 4. Concluding remarks are given in Section 5.

Notations. Throughout this paper, the superscript ‘‘𝑇’’ denotes the transpose, and the notation 𝑋≥𝑌(𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (positive definite, resp.). ‖⋅‖ denotes the Euclidean norm. 𝐼 represents the identity matrix with an appropriate dimension. 𝐼_ℎ is the identity matrix with 𝑛₁ dimension and 𝐼_𝑣 is the identity matrix with 𝑛₂ dimension. diag{𝑎_𝑖} denotes the diagonal matrix with the diagonal elements 𝑎_𝑖, and 𝑖=1,2,…,𝑛. 𝑋⁻¹ denotes the inverse of 𝑋. The asterisk ∗ in a matrix is used to denote the term that is induced by symmetry. The set of all nonnegative integers is represented by 𝑍₊.

2. Problem Formulation and Preliminaries

Consider the following uncertain 2D discrete switched systems with state delays: 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣=𝐴𝑖,𝑗+1_{𝜎(𝑖,𝑗)}𝑥^ℎ𝑥𝑖,𝑗_𝑣+𝐴𝑖,𝑗^{𝑑𝜎(𝑖,𝑗)}𝑥^ℎ𝑖−𝑑_ℎ𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣+𝐵^{𝜎(𝑖,𝑗)}𝑢𝑖,𝑗, (1) where 𝑥^ℎ(𝑖,𝑗)∈𝑅^𝑛1 and 𝑥^𝑣(𝑖,𝑗)∈𝑅^𝑛2 are the horizontal state and the vertical state, respectively, 𝑥(𝑖,𝑗) is the whole state in 𝑅^𝑛 with 𝑛=𝑛₁+𝑛₂, and 𝑢(𝑖,𝑗)∈𝑅^𝑚 is the control input. 𝑖 and 𝑗 are integers in 𝑍₊. 𝜎(𝑖,𝑗)∶𝑍₊×𝑍₊→𝑁={1,2,...,𝑁} is the switching signal. 𝑁 is the number of subsystems, and 𝜎(𝑖,𝑗)=𝑘 means that the 𝑘th subsystem is activated. 𝑑_ℎ and 𝑑_𝑣 are constant delays along horizontal and vertical directions, respectively. 𝐴^𝑘 and 𝐴^𝑘_𝑑 (𝑘∈𝑁) are uncertain real-valued matrices with appropriate dimensions and are assumed to be of the form 𝐴^𝑘=𝐴^𝑘+𝐻^𝑘𝐹^𝑘𝑖,𝑗𝐸^𝑘₁,𝐴^𝑘_𝑑=𝐴^𝑘_𝑑+𝐻^𝑘𝐹^𝑘𝑖,𝑗𝐸^𝑘₂,𝐵^𝑘=𝐵^𝑘+𝐻^𝑘𝐹^𝑘𝑖,𝑗𝐸^𝑘₃, (2) with 𝐴^𝑘=𝐴^𝑘₁₁𝐴^𝑘₁₂𝐴^𝑘₂₁𝐴^𝑘₂₂,𝐴^𝑘_𝑑=𝐴^𝑘_𝑑11𝐴^𝑘_𝑑12𝐴^𝑘_𝑑21𝐴^𝑘_𝑑22,𝐻^𝑘=𝐻^𝑘₁𝐻^𝑘₂,𝐸^𝑘₁=𝐸^𝑘₁₁𝐸^𝑘₁₂,𝐸^𝑘₂=𝐸^𝑘₂₁𝐸^𝑘₂₂,𝐸^𝑘₃=𝐸^𝑘₃₁𝐸^𝑘₃₂, (3) where matrices 𝐴^𝑘₁₁∈𝑅^𝑛1×𝑛1, 𝐴^𝑘₁₂∈𝑅^𝑛1×𝑛2, 𝐴^𝑘₂₁∈𝑅^𝑛2×𝑛1, 𝐴^𝑘₂₂∈𝑅^𝑛2×𝑛2, 𝐴^𝑘_𝑑11∈𝑅^𝑛1×𝑛1, 𝐴^𝑘_𝑑12∈𝑅^𝑛1×𝑛2,  𝐴^𝑘_𝑑21∈𝑅^𝑛2×𝑛1, 𝐴^𝑘_𝑑22∈𝑅^𝑛2×𝑛2, 𝐻^𝑘₁, 𝐻^𝑘₂, 𝐸^𝑘₁₁, 𝐸^𝑘₁₂, 𝐸^𝑘₂₁, 𝐸^𝑘₂₂, 𝐸^𝑘₃₁, 𝐸^𝑘₃₂ are constant matrices. 𝐹^𝑘(𝑖,𝑗) (𝑘∈𝑁) is an unknown matrix representing parameter uncertainty and satisfies 𝐹^𝑘𝑇𝑖,𝑗𝐹^𝑘𝑖,𝑗≤𝐼. (4) The boundary conditions are given by 𝑥^ℎ𝑖,𝑗=ℎ_𝑖𝑗,∀0≤𝑗≤𝑧₁,−𝑑_ℎ𝑥≤𝑖≤0,_ℎ𝑖,𝑗=0,∀𝑗>𝑧₁,−𝑑_ℎ𝑥≤𝑖≤0,_𝑣𝑖,𝑗=𝑣_𝑖𝑗,∀0≤𝑖≤𝑧₂,−𝑑_𝑣𝑥≤𝑗≤0,_𝑣𝑖,𝑗=0,∀𝑖>𝑧₂,−𝑑_𝑣≤𝑗≤0, (5) where 𝑧₁<∞ and 𝑧₂<∞ are positive integers, and ℎ_𝑖𝑗 and 𝑣_𝑖𝑗 are given vectors.

In this paper, it is assumed that (1) at each time only one subsystem is active; (2) the switching signal is not known a priori, but its value is available at each sampling period; (3) the switch occurs only at each sampling point of 𝑖 or 𝑗. The switching sequence can be described as follows: 𝑖₀,𝑗₀,𝜎𝑖₀,𝑗₀,𝑖₁,𝑗₁,𝜎𝑖₁,𝑗₁𝑖,…,_𝜅,𝑗_𝜅,𝜎𝑖_𝜅,𝑗_𝜅,…, (6) where (𝑖_𝜅,𝑗_𝜅) denotes the 𝜅th switching instant. It should be noted that the value of 𝜎(𝑖,𝑗) only depends on 𝑖+𝑗 (see [29, 30]).

However, in actual operation, there inevitably exists asynchronous switching between the controller and the system. Without loss of generality, we only consider the case where the switching instants of the controller experience delays with respect to those of the system. Let 𝜎^′(𝑖,𝑗) denote the switching signal of the controller. Denoting 𝑚_𝜅=𝑖_𝜅+𝑗_𝜅, Δ𝑚_𝜅=Δ𝑖_𝜅+Δ𝑗_𝜅, 𝜅=1,2,…, then the switching points of the controller can be described as 𝑖₀,𝑗₀,𝑖₁+Δ𝑖₁,𝑗₁+Δ𝑗₁,…,𝑖_𝜅+Δ𝑖_𝜅,𝑗_𝜅+Δ𝑗_𝜅,…, (7) where Δ𝑖_𝜅 and Δ𝑗_𝜅 represent the delayed period along horizontal and vertical directions, respectively. Δ𝑚_𝜅<inf(𝑚_𝜅+1−𝑚_𝜅) is said to be the mismatched period.

Remark 1. Similar to the one-dimensional switched system case [33–36], the mismatched period Δ𝑚_𝜅<inf(𝑚_𝜅+1−𝑚_𝜅) guarantees that there always exists a period in which the controller and the system operate synchronously. This period is said to be the matched period in the later section.

Remark 2. If there is only one subsystem in system (1), it will degenerate to the following 2D system in Roesser model with state delays [12]: 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣=𝐴𝑥𝑖,𝑗+1𝑥^ℎ𝑖,𝑗_𝑣+𝐴𝑖,𝑗^𝑑𝑥^ℎ𝑖−𝑑_ℎ𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣, (8)

Definition 3. System (1) is said to be exponentially stable under 𝜎(𝑖,𝑗) if, for a given 𝑧≥0, there exist positive constants 𝑐 and 𝜂, such that _{𝑖+𝑗=𝐷}𝑥𝑖,𝑗²≤𝜂𝑒^{−𝑐(𝐷−𝑧)}_{𝑖+𝑗=𝑧}𝑥𝑖,𝑗²_𝐶, (9) holds for all 𝐷≥𝑧, where _{𝑖+𝑗=𝑧}𝑥𝑖,𝑗²_𝐶≜sup_{−𝑑ℎ≤𝜃ℎ≤0,−𝑑𝑣≤𝜃𝑣≤0𝑖+𝑗=𝑧}𝑥^ℎ𝑖−𝜃_ℎ,𝑗²+𝑥^𝑣𝑖,𝑗−𝜃_𝑣². (10)

Remark 4. From Definition 3, it is easy to see that when 𝑧 is given, ∑_{𝑖+𝑗=𝑧}‖𝑥(𝑖,𝑗)‖²_𝐶 will be bounded and ∑_{𝑖+𝑗=𝐷}‖𝑥(𝑖,𝑗)‖² will tend to be zero exponentially as 𝐷 goes to infinity, which implies that ‖𝑥(𝑖,𝑗)‖ tends to be zero.

Definition 5. Let (𝑖_𝜅,𝑗_𝜅) denote the 𝜅th switching point and (𝑖_𝜅+1,𝑗_𝜅+1) denote the (𝜅+1)th switching point. Denote 𝑚_𝜅=𝑖_𝜅+𝑗_𝜅, 𝑚_𝜅+1=𝑖_𝜅+1+𝑗_𝜅+1, 𝜏=inf(𝑚_𝜅+1−𝑚_𝜅), then 𝜏 is called the dwell time.

Definition 6 (see [30]). For any 𝑖+𝑗=𝐷≥𝑧=𝑖_𝑧+𝑗_𝑧, let 𝑁_{𝜎(𝑖,𝑗)}(𝑧,𝐷) denote the switching number of 𝜎(𝑖,𝑗) on the interval [𝑧,𝐷). If 𝑁_{𝜎(𝑖,𝑗)}(𝑧,𝐷)≤𝑁₀+𝐷−𝑧𝜏_𝑎 (11) holds for given 𝑁₀≥0 and 𝜏_𝑎≥0, then the constant 𝜏_𝑎 is called the average dwell time and 𝑁₀ is the chatter bound.

Lemma 7 (see [37]). For a given matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}, where 𝑆₁₁ and 𝑆₂₂ are square matrices, the following conditions are equivalent:(i)𝑆<0,(ii)𝑆₁₁<0,  𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0,(iii)𝑆₂₂<0, 𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<0.

Lemma 8 (see [38]). Let 𝑈, 𝑉, 𝑊, and 𝑋 be real matrices of appropriate dimensions with 𝑋 satisfying 𝑋=𝑋^𝑇, then for all 𝑉^𝑇𝑉≤𝐼, 𝑋+𝑈𝑉𝑊+𝑊^𝑇𝑉^𝑇𝑈^𝑇<0, (12) if and only if there exists a scalar 𝜀 such that 𝑋+𝜀𝑈𝑈^𝑇+𝜀⁻¹𝑊^𝑇𝑊<0. (13)

3. Main results

3.1. Stability Analysis

In this section, we first focus on the stability analysis for system (8).

Lemma 9. Consider system (8) with the boundary conditions (5), suppose that there exists a 𝐶¹ function 𝑉∶𝑅^𝑛→𝑅. For a given positive constant 𝛼, if there exist positive definite symmetric matrices 𝑃=diag{𝑃_ℎ,𝑃_𝑣} and 𝑄=diag{𝑄_ℎ,𝑄_𝑣} with appropriate dimensions, such that the following inequality holds: 𝑄−𝛼𝑃0𝐴^𝑇𝑃∗−Λ₁𝑄𝐴^𝑇_𝑑𝑃∗∗−𝑃<0, (14) where Λ₁=diag{𝛼^𝑑ℎ𝐼_ℎ,𝛼^𝑑𝑣𝐼_𝑣}, then along the trajectory of systems (8), the following inequality holds for any 𝐷≥𝐷^′: _{𝑖+𝑗=𝐷}𝑉𝑥𝑖,𝑗<𝛼^{𝐷−𝐷′}_{𝑖+𝑗=𝐷′}𝑉𝑥𝑖,𝑗, (15) where 𝐷^′≥𝑧 and 𝑧=max{𝑧₁,𝑧₂}.

Proof. See appendix for the detailed proof, it is omitted here.

Remark 10. Lemma 9 provides a method for the estimation of the 𝐶¹ function 𝑉 which will be used to design the controller for system (1) under asynchronous switching. It is worth pointing out that when 0<𝛼<1, (15) presents the decay estimation of the 𝐶¹ function 𝑉, and when 𝛼>1, (15) shows the growth estimation of the 𝐶¹ function 𝑉.

Remark 11. It is noted that the block diagonal matrices 𝑃 and 𝑄 are often chosen as the matrices for Lyapunov functional analysis of 2D systems by the Roesser model in the existing literature (see, e.g., [12, 13, 15]). This is because 2D systems in Roesser model may be unstable when the block diagonal matrices are not chosen, which has been shown in the literature [4, 5].

3.2. Controller Design

Consider system (1), under the following asynchronous switching controller: 𝑢𝑖,𝑗=𝐾^{𝜎′(𝑖,𝑗)}𝑥𝑖,𝑗,𝐾^{𝜎′(𝑖,𝑗)}=𝐾^𝜎′₁^(𝑖,𝑗)𝐾^𝜎′₂^(𝑖,𝑗), (16) where 𝐾^𝜎′₁^(𝑖,𝑗)∈𝑅^𝑚×𝑛1 and 𝐾^𝜎′₂^(𝑖,𝑗)∈𝑅^𝑚×𝑛2, the corresponding closed-loop system is given by 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣=𝐴𝑖,𝑗+1_{𝜎(𝑖,𝑗)}+𝐵^{𝜎(𝑖,𝑗)}𝐾^{𝜎′(𝑖,𝑗)}𝑥^ℎ𝑥𝑖,𝑗_𝑣+𝐴𝑖,𝑗_{𝑑𝜎(𝑖,𝑗)}𝑥^ℎ𝑖−𝑑_ℎ𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣. (17)

Without loss of generality, we denote 𝜎(𝑖_𝜅,𝑗_𝜅)=𝑘∈𝑁 and (𝑖_𝜅+1,𝑗_𝜅+1)=𝑙∈𝑁, then due to the existence of asynchronous switching, we can obtain from (7) that 𝜎^′𝑖_𝜅+Δ𝑖_𝜅,𝑗_𝜅+Δ𝑗_𝜅𝜎=𝑘,_′𝑖_𝜅+1+Δ𝑖_𝜅+1,𝑗_𝜅+1+Δ𝑗_𝜅+1=𝑙. (18)

In many actual applications, it is always difficult to verify each asynchronous period in advance, but the maximal asynchronous period can be easily predicted offline. Let Δ=max_𝜅=1,2,…(Δ𝑚_𝜅) denote the maximal asynchronous period, then we can get the following result.

Theorem 12. Consider system (1), for given positive constants 𝛼<1 and 𝛽>1, if there exist positive definite symmetric matrices 𝑋^𝑘=diag{𝑋^𝑘_ℎ,𝑋^𝑘_𝑣}, 𝑌^𝑘=diag{𝑌^𝑘_ℎ,𝑌^𝑘_𝑣}, 𝑋^𝑘𝑙=diag{𝑋_ℎ^𝑘𝑙,𝑋_𝑣^𝑘𝑙}, 𝑌^𝑘𝑙=diag{𝑌_ℎ^𝑘𝑙,𝑌_𝑣^𝑘𝑙} and 𝑊^𝑘 with appropriate dimensions, and positive scalars 𝜀_𝑘 and 𝜀_𝑘𝑙 such that, for 𝑘,𝑙∈𝑁, 𝑘≠𝑙, the following inequalities hold, −𝛼𝑋^𝑘0𝐴^𝑘𝑋^𝑘+𝐵^𝑘𝑊^𝑘𝑇𝑋^𝑘𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇∗−Λ₁𝑌^𝑘𝐴^𝑘_𝑑𝑌^𝑘𝑇0𝐸^𝑘₂𝑌^𝑘𝑇∗∗−𝑋^𝑘+𝜀_𝑘𝐻^𝑘𝐻^𝑘𝑇00∗∗∗−𝑌^𝑘0∗∗∗∗−𝜀_𝑘𝐼<0, (19)−𝛽𝑋^𝑘𝑙0𝐴^𝑙𝑋^𝑘𝑙+𝐵^𝑙𝐾^𝑘𝑋^𝑘𝑙𝑇𝑋^𝑘𝑙𝐸^𝑙₁𝑋^𝑘𝑙+𝐸^𝑙₃𝐾^𝑘𝑋^𝑘𝑙𝑇∗−Λ₂𝑌^𝑘𝑙𝐴^𝑙_𝑑𝑌^𝑘𝑙𝑇0𝐸^𝑙₂𝑌^𝑘𝑙𝑇∗∗−𝑋^𝑘𝑙+𝜀_𝑘𝑙𝐻^𝑙𝐻^𝑙𝑇00∗∗∗−𝑌^𝑘𝑙0∗∗∗∗−𝜀_𝑘𝑙𝐼<0, (20)where Λ₂=diag{𝛽^𝑑ℎ𝐼_ℎ,𝛽^𝑑𝑣𝐼_𝑣}, then under the following switching controller: 𝑢𝑖,𝑗=𝐾^{𝜎′(𝑖,𝑗)}𝑥𝑖,𝑗,𝐾^𝑘=𝑊^𝑘𝑋^𝑘−1, (21) and the following average dwell time scheme: 𝜏_𝑎>𝜏^∗_𝑎=Δln𝛽−ln𝛼+ln𝜇₁𝜇₂,−ln𝛼 (22) the resulting closed-loop system (17) is exponentially stable, where 𝜇=(𝛼/𝛽)^𝑑, 𝑑=max{𝑑_ℎ,𝑑_𝑣}, and 𝜇₁𝜇₂𝜇≥1 satisfies 𝑋_𝑙⁻¹≤𝜇₁𝑋⁻¹_𝑘𝑙,𝑋⁻¹_𝑘𝑙≤𝜇₂𝑋_𝑘⁻¹,𝑌_𝑙⁻¹≤𝜇₁𝑌⁻¹_𝑘𝑙,𝑌⁻¹_𝑘𝑙≤𝜇₂𝜇𝑌_𝑘⁻¹. (23)

Proof. See the appendix.

Remark 13. In Theorem 12, we propose a sufficient condition for the existence of a state feedback controller such that the resulting closed-loop system (17) is exponentially stable. It is worth noting that this condition is obtained by using the average dwell time approach. Here, 𝛼 plays a key role in controlling the rate of decaying of the system in the matched period, and 𝛽 plays a key role in controlling the rate of increasing of the system in the mismatched period. From (22), we know that the value of 𝜏^∗_𝑎 is proportional to the value of Δ, which also means that it needs much larger dwell time to guarantee the stability of the considered system when there exists much larger asynchronous period.

Remark 14. It is noticed that (19) and (20) are mutually dependent. Therefore, we can firstly solve (19) to obtain the solution of matrices 𝑋^𝑘, 𝑌^𝑘, and 𝑊^𝑘. Then, (20) can be transformed into the LMI by substituting 𝐾^𝑘=𝑊^𝑘(𝑋^𝑘)⁻¹ into it. By adjusting the parameters 𝛼 and 𝛽, we can find a feasible solution of 𝑋^𝑘, 𝑌^𝑘, 𝑊^𝑘, 𝑋^𝑘𝑙, and 𝑌^𝑘𝑙 such that (19) and (20) hold.

The procedure of the controller design for system (1) can be given as follows.

Step 1. Given system matrices and positive scalar 0<𝛼<1, and by solving LMI (19), we can get the feasible solution of matrices 𝑋^𝑘, 𝑌^𝑘, and 𝑊^𝑘 and positive scalar 𝜀_𝑘 then, the controller can be obtained from (21).

Step 2. Substituting 𝐾^𝑘 into (20), we can find the feasible solution of 𝑋^𝑘𝑙, 𝑌^𝑘𝑙, and 𝜀_𝑘𝑙 such that (20) holds by adjusting the parameter 𝛽.

Step 3. From (23), we can obtain 𝜇₁ and 𝜇₂ satisfying 𝜇₁𝜇₂𝜇≥1.

Step 4. Taking the value of Δ, we can compute the value of 𝜏^∗_𝑎 by (22).

Remark 15. From the procedure above, it can be seen that the proposed method is feasible. We can find the desire controller and switching signal according to the procedure. However, we would like to point out that there still exists the conservatism to some extent for this method because (19) and (20) are mutually dependent, which brings about the increase of the complex computation. The result can be improved by adopting the method presented in [31, 32].
When 𝐴_𝑑^{𝜎(𝑖,𝑗)}=0, system (17) degenerates to the following delay-free system: 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣=𝐴𝑖,𝑗+1^{𝜎(𝑖,𝑗)}+𝐵^{𝜎(𝑖,𝑗)}𝐾^{𝜎′(𝑖,𝑗)}𝑥^ℎ𝑥𝑖,𝑗_𝑣𝑖,𝑗. (24) Then, we can get the following result.

Corollary 16. Consider system (1) with 𝐴_𝑑^{𝜎(𝑖,𝑗)}=0, for given positive constants 𝛼<1 and 𝛽>1, if there exist positive definite symmetric matrices 𝑋^𝑘=diag{𝑋^𝑘_ℎ,𝑋^𝑘_𝑣} and 𝑋^𝑘𝑙=diag{𝑋_ℎ^𝑘𝑙,𝑋_𝑣^𝑘𝑙} with appropriate dimensions, and positive scalars 𝜀_𝑘 and 𝜀_𝑘𝑙 such that, for 𝑘,𝑙∈𝑁, 𝑘≠𝑙, the following inequalities hold: −𝛼𝑋^𝑘𝐴^𝑘𝑋^𝑘+𝐵^𝑘𝑊^𝑘𝑇𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇∗−𝑋^𝑘+𝜀_𝑘𝐻^𝑘𝐻^𝑘𝑇0∗∗−𝜀_𝑘𝐼<0,−𝛽𝑋^𝑘𝑙𝐴^𝑙𝑋^𝑘𝑙+𝐵^𝑙𝐾^𝑘𝑋^𝑘𝑙𝑇𝐸^𝑙₁𝑋^𝑘𝑙+𝐸^𝑙₃𝐾^𝑘𝑋^𝑘𝑙𝑇∗−𝑋^𝑘𝑙+𝜀_𝑘𝑙𝐻^𝑙𝐻^𝑙𝑇0∗∗−𝜀_𝑘𝑙𝐼<0, (25) then under the following switching controller: 𝑢𝑖,𝑗=𝐾^{𝜎′(𝑖,𝑗)}𝑥𝑖,𝑗,𝐾^𝑘=𝑊^𝑘𝑋^𝑘−1, (26) and the following average dwell time scheme: 𝜏_𝑎>𝜏^∗_𝑎=Δln𝛽−ln𝛼+ln𝜇₁𝜇₂,−ln𝛼 (27) the resulting closed-loop system (24) is exponentially stable, where 𝜇₁𝜇₂≥1 satisfies 𝑋_𝑙⁻¹≤𝜇₁𝑋⁻¹_𝑘𝑙,𝑋⁻¹_𝑘𝑙≤𝜇₂𝑋_𝑘⁻¹. (28)

Furthermore, it should also be noted that if the criterion in Theorem 12 is satisfied when Δ=0, which means that the controller and the subsystem are synchronous, in other words, the results presented in Theorem 12 can be reduced to the synchronous case, then we can obtain the following corollary.

Corollary 17. Consider system (1) under synchronous switching, for a given positive scalar 𝛼<1, if there exist positive definite symmetric matrices 𝑋^𝑘=diag{𝑋^𝑘_ℎ,𝑋^𝑘_𝑣}, 𝑌^𝑘=diag{𝑌^𝑘_ℎ,𝑌^𝑘_𝑣}, and 𝑊^𝑘, with appropriate dimensions, and a positive scalar 𝜀_𝑘, such that, for 𝑘∈𝑁, the following inequality holds: −𝛼𝑋^𝑘0𝐴^𝑘𝑋^𝑘+𝐵^𝑘𝑊^𝑘𝑇𝑋^𝑘𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇∗−Λ₁𝑌^𝑘𝐴^𝑘_𝑑𝑌^𝑘𝑇0𝐸^𝑘₂𝑌^𝑘𝑇∗∗−𝑋^𝑘+𝜀_𝑘𝐻^𝑘𝐻^𝑘𝑇00∗∗∗−𝑌^𝑘0∗∗∗∗−𝜀_𝑘𝐼<0, (29) then under the following controller: 𝑢𝑖,𝑗=𝐾^𝑘𝑥𝑖,𝑗,𝐾^𝑘=𝑊^𝑘𝑋^𝑘−1, (30) and the following average dwell time scheme: 𝜏_𝑎>𝜏^∗_𝑎=ln𝜇₁,−ln𝛼 (31) the resulting closed-loop system is exponentially stable, where 𝜇₁≥1 satisfies 𝑋_𝑙⁻¹≤𝜇₁𝑋_𝑘⁻¹,𝑌_𝑙⁻¹≤𝜇₁𝑌_𝑘⁻¹. (32)

Remark 18. In [30], by using the average dwell time approach, a criterion of exponential stability for a class of 2D discrete delay-free switched systems is developed. However, the focus of our work is on stability analysis and controller design under asynchronous switching, which is different from [30], and this is also the major contribution of our work. In fact, if we let Δ=0 and do not consider the uncertainties and state delays, then the closed-loop system (24) is the same as (36) in [30]. In this case, Corollary 17 can be reduced to Theorem 2 in [30].

4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the proposed approach.

Consider system (1) with the following parameters: 𝐴¹=11.510.5,𝐴¹_𝑑=𝐵−0.150−0.1−0.12₁,=−4.501−3,𝐻¹=𝐸0.20.150.10.2₁₁,=−0.20−0.2−0.2,𝐸¹₂=,𝐹0.100.10.2₁,𝐸=diagsin0.5𝜋𝑖+𝑗,sin0.5𝜋𝑖+𝑗₁₃=0.1500.130.12,𝐴²=,𝐴1211_2𝑑=−0.10.20−0.2,𝐵²=,𝐻−51−1−3₂=0.20.250.20.3,𝐸²₁=𝐸0.10.20.20.1₂₂=0.20.10.20.1,𝐸²₃=,𝐹0.120.150.120.1₂,𝑑=diagcos0.5𝜋𝑖+𝑗,cos0.5𝜋𝑖+𝑗_ℎ=2,𝑑_𝑣=3. (33)(34)(35) The boundary conditions are given as follows: 𝑥^ℎ=𝑥0,𝑗5,0≤𝑗≤20,0,𝑗>20,_𝑣(𝑖,0)=3,0≤𝑖≤20,0,𝑖>20, (36) where the state dimensions are 𝑛₁=1 and 𝑛₂=1.

Take 𝛼=0.6 and 𝛽=1.2, then solving (19) in Theorem 12 gives rise to 𝑋¹=36.39031.02471.024738.3909,𝑋²=,𝑌40.0833−6.6640−6.664034.4516₁=96.6213−4.5896−4.589696.9472,𝑌²=,𝑊99.6213−8.8472−8.847279.5315₁=8.586613.060415.556811.4091,𝑊²=,𝜀6.984413.29308.37904.4234₁=74.2107,𝜀₂=73.4149. (37) By (21), 𝐾¹ and 𝐾² can be obtained as follows: 𝐾¹=0.22650.33410.41940.2860,𝐾²=0.24630.43350.23800.1744. (38) Substituting 𝐾¹ and 𝐾² into (20), and solving it, we get the following solution: 𝑋¹²=0.5400−0.0934−0.09340.4811,𝑋²¹=,𝑌0.5805−0.0926−0.09260.5677₁₂=0.9398−0.0418−0.04180.8451,𝑌²¹=,𝜀0.9436−0.0340−0.03400.8867₁₂=0.9011,𝜀₂₁=0.7725. (39) Then, from (23), we can get that 𝜇₁=0.0171 and 𝜇₂=917.7224. Taking Δ=2, it is easy to obtain from (22) that 𝜏^∗_𝑎=7.9. Choosing 𝜏_𝑎=8, the trajectories of the states 𝑥^ℎ(𝑖,𝑗) and 𝑥^𝑣(𝑖,𝑗) are shown in Figures 1 and 2, respectively. The system switching signal 𝜎(𝑖,𝑗) and the controller switching signal 𝜎^′(𝑖,𝑗) are shown in Figure 3. One can see that the states of the closed-loop system converge to zero under the asynchronous switching. This demonstrates the effectiveness of the proposed approach.

Figure 1 

The trajectory of the state 𝑥ℎ(𝑖,𝑗).

Figure 2 

The trajectory of the state 𝑥𝑣(𝑖,𝑗).

Figure 3 

Switching signal.

5. Conclusions

This paper has investigated the problem of stabilization for a class of 2D discrete switched systems with constant state delays under asynchronous switching. A state feedback controller is proposed to stabilize such system, and the dwell time approach is utilized for the stability analysis and controller design. A sufficient condition for the existence of such controller is formulated in terms of a set of LMIs. An example is also given to illustrate the applicability of the proposed approach. Our future work will focus on extending the proposed design method to other problems such as robust 𝐻_∞ control for 2D discrete switched systems with time-varying delays and fractional uncertainties under asynchronous switching.

Appendix

Proof of Lemma 9. Consider the following Lyapunov-Krasovskii functional candidate: 𝑉𝑥𝑖,𝑗=𝑉^ℎ𝑥^ℎ𝑖,𝑗+𝑉^𝑣𝑥^𝑣𝑖,𝑗, (A.1) where 𝑉^ℎ𝑥^ℎ𝑖,𝑗=𝑥^ℎ𝑖,𝑗^𝑇𝑃_ℎ𝑥^ℎ+𝑖,𝑗_{𝑖−1𝑟=𝑖−𝑑ℎ}𝑥^ℎ𝑟,𝑗^𝑇𝑄_ℎ𝑥^ℎ𝑟,𝑗𝛼^{𝑖−𝑟−1},𝑉^𝑣𝑥^𝑣𝑖,𝑗=𝑥^𝑣𝑖,𝑗^𝑇𝑃_𝑣𝑥^𝑣+𝑖,𝑗_{𝑗−1𝑡=𝑗−𝑑𝑣}𝑥^𝑣(𝑖,𝑡)^𝑇𝑄_𝑣𝑥^𝑣(𝑖,𝑡)𝛼^{𝑗−𝑡−1}. (A.2) Along the trajectory of system (8), we have 𝑉^ℎ𝑥^ℎ𝑖+1,𝑗−𝛼𝑉^ℎ𝑥^ℎ𝑖,𝑗=𝑥^ℎ𝑖+1,𝑗^𝑇𝑃_ℎ𝑥^ℎ𝑖+1,𝑗−𝛼𝑥^ℎ𝑖,𝑗^𝑇𝑃_ℎ𝑥^ℎ𝑖,𝑗+𝑥^ℎ𝑖,𝑗^𝑇𝑄_ℎ𝑥^ℎ𝑖,𝑗−𝛼^𝑑ℎ𝑥^ℎ𝑖−𝑑_ℎ,𝑗^𝑇𝑄_ℎ𝑥^ℎ𝑖−𝑑_ℎ,𝑉,𝑗_𝑣𝑥^𝑣𝑖,𝑗+1−𝛼𝑉^𝑣𝑥^𝑣𝑖,𝑗=𝑥^𝑣𝑖,𝑗+1^𝑇𝑃_𝑣𝑥^𝑣𝑖,𝑗+1−𝛼𝑥^𝑣𝑖,𝑗^𝑇𝑃_𝑣𝑥^𝑣𝑖,𝑗+𝑥^𝑣𝑖,𝑗^𝑇𝑄_𝑣𝑥^𝑣𝑖,𝑗−𝛼^𝑑𝑣𝑥^𝑣𝑖,𝑗−𝑑_𝑣^𝑇𝑄_𝑣𝑥^𝑣𝑖,𝑗−𝑑_𝑣. (A.3)(A.4) It follows that 𝑉^ℎ𝑥^ℎ𝑖+1,𝑗−𝛼𝑉^ℎ𝑥^ℎ𝑖,𝑗+𝑉^𝑣𝑥^𝑣𝑖,𝑗+1−𝛼𝑉^𝑣𝑥^𝑣=𝑥𝑖,𝑗_ℎ𝑥𝑖,𝑗_𝑣𝑥𝑖,𝑗_ℎ𝑖−𝑑_ℎ𝑥,𝑗_ℎ𝑖,𝑗−𝑑_𝑣^𝑇Φ₁₁Φ₁₂Φ^𝑇₁₂Φ₂₂𝑥^ℎ𝑥𝑖,𝑗_𝑣𝑥𝑖,𝑗_ℎ𝑖−𝑑_ℎ𝑥,𝑗_ℎ𝑖,𝑗−𝑑_𝑣, (A.5) where Φ₁₁=𝑄−𝛼𝑃+𝐴^𝑇𝑃𝐴,Φ₁₂=𝐴^𝑇𝑃𝐴_𝑑,Φ₂₂=𝐴^𝑇_𝑑𝑃𝐴_𝑑−Λ₁𝑄,Λ₁=diag𝛼^𝑑ℎ𝐼_ℎ,𝛼^𝑑𝑣𝐼_𝑣. (A.6) Applying Lemma 7, it can be obtained from (14) that Φ₁₁Φ₁₂Φ^𝑇₁₂Φ₂₂<0. (A.7) For simplicity, we denote 𝑉^ℎ𝑖,𝑗=𝑉^ℎ𝑥^ℎ𝑖,𝑗,𝑉^𝑣𝑖,𝑗=𝑉^𝑣𝑥^𝑣,𝑉𝑥𝑖,𝑗𝑖,𝑗=𝑉𝑖,𝑗,𝑉^ℎ𝑖+1,𝑗=𝑉^ℎ𝑥,𝑉𝑖+1,𝑗_𝑣𝑖,𝑗+1=𝑉^𝑣𝑥𝑖,𝑗+1. (A.8) Thus, it is easy to get that 𝑉^ℎ𝑖+1,𝑗+𝑉^𝑣𝑖,𝑗+1<𝛼𝑉^ℎ𝑖,𝑗+𝑉^𝑣𝑖,𝑗. (A.9) Notice that for any nonnegative integer 𝐷>𝑧=max(𝑧₁,𝑧₂), it holds that 𝑉^ℎ(0,𝐷)=𝑉^𝑣(𝐷,0)=0; then summing up both sides of (A.9) from 𝐷−1 to 0 with respect to 𝑗 and 0 to 𝐷−1 with respect to 𝑖, for any nonnegative integer 𝐷>𝐷^′≥𝑧=max(𝑧₁,𝑧₂), one gets _{𝑖+𝑗=𝐷}𝑉𝑖,𝑗=𝑉^ℎ(0,𝐷)+𝑉^ℎ(1,𝐷−1)+𝑉^ℎ(2,𝐷−2)+⋯+𝑉^ℎ(𝐷−1,1)+𝑉^ℎ(𝐷,0)+𝑉^𝑣(0,𝐷)+𝑉^𝑣(1,𝐷−1)+𝑉^𝑣(2,𝐷−2)+⋯+𝑉^𝑣(𝐷−1,1)+𝑉^𝑣()𝑉𝐷,0<𝛼^ℎ(0,𝐷−1)+𝑉^𝑣(0,𝐷−1)+𝑉^ℎ(1,𝐷−2)+𝑉^𝑣(1,𝐷−2)+⋯+𝑉^ℎ(𝐷−1,0)+𝑉^𝑣(𝐷−1,0)=𝛼_{𝑖+𝑗=𝐷−1}𝑉𝑖,𝑗<⋯<𝛼^{𝐷−𝐷′}_{𝑖+𝑗=𝐷′}𝑉𝑖,𝑗. (A.10)
The proof is completed.

Proof of Theorem 12. When 𝐷∈[𝑚_𝜅+Δ𝑚_𝜅,𝑚_𝜅+1), the closed-loop system (17) can be written as 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣=𝐴𝑖,𝑗+1_𝑘+𝐵^𝑘𝐾^𝑘𝑥^ℎ𝑥𝑖,𝑗_𝑣+𝐴𝑖,𝑗^𝑘𝑑𝑥^ℎ𝑖−𝑑_ℎ𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣. (A.11) For the system, we consider the following Lyapunov function candidate: 𝑉_𝑘𝑥𝑖,𝑗=𝑉^ℎ_𝑘𝑥^ℎ𝑖,𝑗+𝑉^𝑣_𝑘𝑥^𝑣𝑖,𝑗, (A.12) where 𝑉^ℎ_𝑘𝑥^ℎ𝑖,𝑗=𝑥^ℎ𝑖,𝑗^𝑇𝑃^𝑘_ℎ𝑥^ℎ𝑖,𝑗+^𝑖−1_{𝑟=𝑖−𝑑ℎ}𝑥^ℎ𝑟,𝑗^𝑇𝑄^𝑘_ℎ𝑥^ℎ𝑟,𝑗𝛼^{𝑖−𝑟−1},𝑉^𝑣_𝑘𝑥^𝑣𝑖,𝑗=𝑥^𝑣𝑖,𝑗^𝑇𝑃^𝑘_𝑣𝑥^𝑣𝑖,𝑗+^𝑗−1_{𝑡=𝑗−𝑑𝑣}𝑥^𝑣(𝑖,𝑡)^𝑇𝑄^𝑘_𝑣𝑥^𝑣(𝑖,𝑡)𝛼^{𝑗−𝑡−1}. (A.13) By Lemma 9, one gets that if there exist positive definite symmetric matrices 𝑃^𝑘=diag{𝑃^𝑘_ℎ,𝑃^𝑘_𝑣} and 𝑄^𝑘=diag{𝑄^𝑘_ℎ,𝑄^𝑘_𝑣} with appropriate dimensions, such that 𝑄^𝑘−𝛼𝑃^𝑘0𝐴^𝑘+𝐵^𝑘𝐾^𝑘𝑇𝑃^𝑘∗−Λ₁𝑄^𝑘𝐴_𝑑^𝑘𝑇𝑃^𝑘∗∗−𝑃^𝑘<0 (A.14) holds, then the following inequality holds for any 𝐷≥𝑚_𝜅+Δ𝑚_𝜅≥𝑧: _{𝑖+𝑗=𝐷}𝑉_𝑘𝑖,𝑗<𝛼^{𝐷−𝑚𝑘−Δ𝑚𝑘}_{𝑖+𝑗=𝑚𝜅+Δ𝑚𝜅}𝑉_𝑘𝑖,𝑗. (A.15)
When 𝐷∈[𝑚_𝜅+1,𝑚_𝜅+1+Δ𝑚_𝜅+1), the closed-loop system (17) can be written as 𝑥^ℎ𝑥𝑖+1,𝑗_𝑣=𝐴𝑖,𝑗+1_𝑙+𝐵^𝑙𝐾^𝑘𝑥^ℎ𝑥𝑖,𝑗_𝑣+𝐴𝑖,𝑗^𝑙𝑑𝑥^ℎ𝑖−𝑑_ℎ𝑥,𝑗_𝑣𝑖,𝑗−𝑑_𝑣. (A.16) Consider the following Lyapunov function candidate: 𝑉_𝑘𝑙𝑥𝑖,𝑗=𝑉^ℎ_𝑘𝑙𝑥^ℎ𝑖,𝑗+𝑉^𝑣_𝑘𝑙𝑥^𝑣𝑖,𝑗, (A.17) where 𝑉^ℎ_𝑘𝑙𝑥^ℎ𝑖,𝑗=𝑥^ℎ𝑖,𝑗^𝑇𝑃_ℎ^𝑘𝑙𝑥^ℎ+𝑖,𝑗_{𝑖−1𝑟=𝑖−𝑑ℎ}𝑥^ℎ𝑟,𝑗^𝑇𝑄_ℎ^𝑘𝑙𝑥^ℎ𝑟,𝑗𝛼^{𝑖−𝑟−1},𝑉^𝑣_𝑘𝑙𝑥^𝑣𝑖,𝑗=𝑥^𝑣𝑖,𝑗^𝑇𝑃_𝑣^𝑘𝑙𝑥^𝑣+𝑖,𝑗_{𝑗−1𝑡=𝑗−𝑑𝑣}𝑥^𝑣(𝑖,𝑡)^𝑇𝑄_𝑣^𝑘𝑙𝑥^𝑣(𝑖,𝑡)𝛼^{𝑗−𝑡−1}. (A.18) Similarly, by Lemma 9, we get that if there exist positive definite symmetric matrices 𝑃^𝑘𝑙=diag{𝑃_ℎ^𝑘𝑙,𝑃_𝑣^𝑘𝑙} and 𝑄^𝑘𝑙=diag{𝑄_ℎ^𝑘𝑙,𝑄_𝑣^𝑘𝑙} with appropriate dimensions, such that 𝑄^𝑘𝑙−𝛽𝑃^𝑘𝑙0𝐴^𝑙+𝐵^𝑙𝐾^𝑘𝑇𝑃^𝑘𝑙∗−Λ₂𝑄^𝑘𝑙𝐴_𝑑^𝑙𝑇𝑃^𝑘𝑙∗∗−𝑃^𝑘𝑙<0 (A.19) holds, then the following inequality holds for any 𝐷≥𝑚_𝜅+1≥𝑧: _{𝑖+𝑗=𝐷}𝑉_𝑘𝑙𝑖,𝑗<𝛽^{𝐷−𝑚𝑘+1}_{𝑖+𝑗=𝑚𝜅+1}𝑉_𝑘𝑙𝑖,𝑗. (A.20) Consider the following piecewise Lyapunov functional candidate for system (17): 𝑉=𝑥𝑖,𝑗^ℎ𝑖,𝑗^𝑇𝑃_ℎ^{𝜎(𝑖,𝑗)}𝑥^ℎ𝑖,𝑗+𝑥^𝑣𝑖,𝑗^𝑇𝑃_𝑣^{𝜎(𝑖,𝑗)}𝑥^𝑣+𝑖,𝑗_{𝑖−1𝑟=}∑_{𝑖−𝑑ℎ}𝑥^ℎ𝑟,𝑗^𝑇𝑄_ℎ^{𝜎(𝑖,𝑗)}𝑥^ℎ𝑟,𝑗𝛼^{𝑖−𝑟−1}+∑^𝑗−1_{𝑡=𝑗−𝑑𝑣}𝑥^𝑣(𝑖,𝑡)^𝑇𝑄_𝑣^{𝜎(𝑖,𝑗)}𝑥^𝑣(𝑖,𝑡)𝛼^{𝑗−𝑡−1},𝐷∈𝑚₀,𝑚₁∪𝑚_𝜋+Δ𝑚_𝜋,𝑚_𝜋+1,𝑥𝜋=1,2,…,𝜅…,_ℎ𝑖,𝑗^𝑇𝑃^𝜎′_ℎ^{(𝑖,𝑗)𝜎(𝑖,𝑗)}𝑥^ℎ𝑖,𝑗+𝑥^𝑣𝑖,𝑗^𝑇𝑃^𝜎′_𝑣^{(𝑖,𝑗)𝜎(𝑖,𝑗)}𝑥^𝑣+𝑖,𝑗_{𝑖−1𝑟=}∑_{𝑖−𝑑ℎ}𝑥^ℎ𝑟,𝑗^𝑇𝑄^𝜎′_ℎ^{(𝑖,𝑗)𝜎(𝑖,𝑗)}𝑥^ℎ𝑟,𝑗𝛼^{𝑖−𝑟−1}+∑^𝑗−1_{𝑡=𝑗−𝑑𝑣}𝑥^𝑣(𝑖,𝑡)^𝑇𝑄^𝜎′_𝑣^{(𝑖,𝑗)𝜎(𝑖,𝑗)}𝑥^𝑣(𝑖,𝑡)𝛼^{𝑗−𝑡−1},𝐷∈𝑚_𝜋,𝑚_𝜋+Δ𝑚_𝜋,𝜋=1,2,…,𝜅…. (A.21)
Now, let 𝜐=𝑁_{𝜎(𝑖,𝑗)}(𝑧,𝐷) denote the switch number of 𝜎(𝑖,𝑗) on the interval [𝑧,𝐷), and let (𝑖_{𝜅−𝜐+1},𝑗_{𝜅−𝜐+1}),(𝑖_{𝜅−𝜐+2},𝑗_{𝜅−𝜐+2}),…,(𝑖_𝜅,𝑗_𝜅) denote the switching points of 𝜎(𝑖,𝑗) over the interval [𝑧,𝐷); then, the switching points of 𝜎^′(𝑖,𝑗) can be denoted as follows: 𝑖_{𝜅−𝜐+1}+Δ𝑖_{𝜅−𝜐+1},𝑗_{𝜅−𝜐+1}+Δ𝑗_{𝜅−𝜐+1},𝑖_{𝜅−𝜐+2}+Δ𝑖_{𝜅−𝜐+2},𝑗_{𝜅−𝜐+2}+Δ𝑗_{𝜅−𝜐+2},…,𝑖_𝜅+Δ𝑖_𝜅,𝑗_𝜅+Δ𝑗_𝜅. (A.22) Denoting 𝑚_𝑔=𝑖_𝑔+𝑗_𝑔, 𝑔=𝜅−𝜐+1,…,𝜅, then we can get from (23) and (A.21) that _{𝑖+𝑗=𝑚𝑔+Δ𝑚𝑔}𝑉𝑖,𝑗≤𝜇_{1𝑚𝑖+𝑗=𝑔+Δ𝑚𝑔−}𝑉𝑖,𝑗_{𝑖+𝑗=𝑚𝑔},𝑉𝑖,𝑗≤𝜇_{2𝑚𝑖+𝑗=𝑔−}𝑉𝑖,𝑗, (A.23) where (𝑚_𝑔)⁻ and (𝑚_𝑔+Δ𝑚_𝑔)⁻ satisfy the following conditions: 0<𝑚_𝑔−𝑚_𝑔⁻<𝜆,0<𝑚_𝑔+Δ𝑚_𝑔−𝑚_𝑔+Δ𝑚_𝑔⁻<𝜆, (A.24) where 𝜆 is a sufficient small positive constant.
When 𝑚_{𝜅−𝜐+1}>𝑧≥𝑚_𝜅−𝜐+Δ𝑚_𝜅−𝜐, we have, for 𝐷>𝑚_𝜅+Δ𝑚_𝜅, _{𝑖+𝑗=𝐷}𝑉𝑖,𝑗<𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}_{𝑖+𝑗=𝑚𝜅+Δ𝑚𝜅}𝑉𝑖,𝑗≤𝜇₁𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}_{𝑖+𝑗=(𝑚𝜅+Δ𝑚𝜅)−}𝑉𝑖,𝑗<𝜇₁𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}𝛽^Δ𝑚𝜅_{𝑖+𝑗=𝑚𝜅}𝑉𝑖,𝑗≤𝜇₁𝜇₂𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}𝛽^Δ𝑚𝜅_{𝑚𝑖+𝑗=𝜅−}𝑉<𝜇𝑖,𝑗<⋯₁𝜇₂^𝜐𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅+𝑚𝜅−𝑚𝜅−1−Δ𝑚𝜅−1+⋯+𝑚𝜅−𝜐+2−𝑚𝜅−𝜐+1−Δ𝑚𝜅−𝜐+1}𝛽^{Δ𝑚𝜅+Δ𝑚𝜅−1+⋯+Δ𝑚𝜅−𝜐+1}_{𝑖+𝑗=(𝑚𝑘−𝜐+1)−}𝑉<𝜇𝑖,𝑗₁𝜇₂^𝜐𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅+𝑚𝜅−𝑚𝜅−1−Δ𝑚𝜅−1+⋯+𝑚𝜅−𝜐+2−𝑚𝜅−𝜐+1−Δ𝑚𝜅−𝜐+1+𝑚𝜅−𝜐+1−𝑧}𝛽^{Δ𝑚𝜅+Δ𝑚𝜅−1+⋯+Δ𝑚𝜅−𝜐+1}_{𝑖+𝑗=𝑧}𝑉≤𝜇𝑖,𝑗₁𝜇₂^𝜐𝛼^{𝐷−𝑧−𝜐Δ}𝛽^𝜐Δ_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗=𝛼^𝐷−𝑧𝑒^{𝜐ln(𝜇1𝜇2)+𝜐Δ(ln𝛽−ln𝛼)}_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗. (A.25) When 𝑚_𝜅−𝜐<𝑧<𝑚_𝜅−𝜐+Δ𝑚_𝜅−𝜐, we can also get that _𝑖+∑_𝑗=𝐷𝑉𝑖,𝑗<𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}∑_{𝑖+𝑗=𝑚𝜅+Δ𝑚𝜅}𝑉𝑖,𝑗<𝜇₁𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}∑_{𝑖+𝑗=(𝑚𝜅+Δ𝑚𝜅)−}𝑉𝑖,𝑗<𝜇₁𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}𝛽^Δ𝑚𝜅∑_{𝑖+𝑗=𝑚𝜅}𝑉𝑖,𝑗<𝜇₁𝜇₂𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅}𝛽^Δ𝑚𝜅∑_{𝑚𝑖+𝑗=𝜅−}𝑉<𝜇𝑖,𝑗<⋯₁𝜇₂^𝜐𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅+𝑚𝜅−𝑚𝜅−1−Δ𝑚𝜅−1+⋯+𝑚𝜅−𝜐+2−𝑚𝜅−𝜐+1−Δ𝑚𝜅−𝜐+1}𝛽^{Δ𝑚𝜅+Δ𝑚𝜅−1+...+Δ𝑚𝜅−𝜐+1}∑_{𝑖+𝑗=(𝑚𝑘−𝜐+1)−}𝑉<𝜇𝑖,𝑗₁𝜇₂^𝜐𝛼^{𝐷−𝑚𝜅−Δ𝑚𝜅+𝑚𝜅−𝑚𝜅−1−Δ𝑚𝜅−1+⋯+𝑚𝜅−𝜐+2−𝑚𝜅−𝜐+1−Δ𝑚𝜅−𝜐+1+𝑚𝜅−𝜐+1−𝑚𝜅−𝜐−Δ𝑚𝜅−𝜐}𝛽^{Δ𝑚𝜅+Δ𝑚𝜅−1+⋯+Δ𝑚𝜅−𝜐+1+Δ𝑚𝜅−𝜐+𝑚𝜅−𝜐−𝑧}∑_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗≤𝜇₁𝜇₁𝜇₂^𝜐𝛼^{𝐷−𝑧−𝜐Δ}𝛽^𝜐Δ∑_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗=𝜇₁𝛼^𝐷−𝑧𝑒^{𝜐ln(𝜇1𝜇2)+𝜐Δ(ln𝛽−ln𝛼)}∑_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗. (A.26) From (A.25) and (A.26), we can obtain _{𝑖+𝑗=𝐷}𝑉𝜇𝑖,𝑗<max₁,1𝛼^𝐷−𝑧𝑒^{𝜐ln(𝜇1𝜇2)+Δ𝜐(ln𝛽−ln𝛼)}_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗. (A.27) According to Definition 6, one has 𝜐=𝑁_{𝜎(𝑖,𝑗)}(𝑧,𝐷)≤𝑁₀+𝐷−𝑧𝜏_𝑎. (A.28) From condition (22), the following inequality can be obtained: Δ𝜇−ln𝛼>ln𝛽−ln𝛼+ln₁𝜇₂𝜏_𝑎. (A.29) Thus, it is easy to get the following inequality: _{𝑖+𝑗=𝐷}𝑉𝜇𝑖,𝑗<max₁,1𝜇₁𝜇₂^𝑁0×𝛽𝛼^𝑁0Δ𝑒^{((ln(𝜇1𝜇2)+Δ(ln𝛽−ln𝛼))/𝜏𝑎+ln𝛼)(𝐷−𝑧)}_{𝑖+𝑗=𝑧}𝑉𝑖,𝑗. (A.30) It follows that _{𝑖+𝑗=𝐷}𝑥𝑖,𝑗²<𝜁₁𝜁₂max𝜇₁,1𝜇₁𝜇₂^𝑁0×𝛽𝛼^𝑁0Δ𝑒^{((ln(𝜇1𝜇2)+Δ(ln𝛽−ln𝛼))/𝜏𝑎+ln𝛼)(𝐷−𝑧)}_{𝑖+𝑗=𝑧}𝑥𝑖,𝑗², (A.31) where 𝜁₁=max_{𝑘,𝑙∈𝑁,𝑘≠𝑙}𝜆_max𝑃^𝑘+max𝑑_ℎ,𝑑_𝑣𝜆_max𝑄^𝑘,𝜆_max𝑃^𝑘𝑙+max𝑑_ℎ,𝑑_𝑣𝜆_max𝑄^𝑘𝑙,𝜁₂=min_{𝑘,𝑙∈𝑁,𝑘≠𝑙}𝜆_min𝑃^𝑘,𝜆_min𝑃^𝑘𝑙,_{𝑖+𝑗=𝑧}𝑥𝑖,𝑗²_𝐶≜sup_{−𝑑ℎ≤𝜃ℎ≤0,−𝑑𝑣≤𝜃𝑣≤0𝑖+𝑗=𝑧}𝑥^ℎ𝑖−𝜃_ℎ,𝑗²+𝑥^𝑣𝑖,𝑗−𝜃_𝑣², (A.32) Thus, it can be obtained from (22) that the closed-loop system (17) is exponentially stable.
Denote 𝑋^𝑘=(𝑃^𝑘)⁻¹ and 𝑌^𝑘=(𝑄^𝑘)⁻¹, then it is easy to get (𝑋^𝑘)^𝑇=𝑋^𝑘 and (𝑌^𝑘)^𝑇=𝑌^𝑘. Using diag{𝑋^𝑘,𝑌^𝑘,𝑋^𝑘} to pre- and postmultiply the left of (A.14), respectively, and applying Lemma 7, it follows that (A.33) and (A.14) are equivalent: −𝛼𝑋^𝑘0𝐴^𝑘𝑋^𝑘+𝐵^𝑘𝑊^𝑘𝑇𝑋^𝑘∗−Λ₁𝑌^𝑘𝐴^𝑘_𝑑𝑌^𝑘𝑇0∗∗−𝑋^𝑘0∗∗∗−𝑌^𝑘<0, (A.33) where 𝑊^𝑘=𝐾^𝑘𝑋^𝑘.
By substituting (2) into (A.33), we can get the following inequality 𝑇=𝑇₀+𝑇₁<0, (A.34) where 𝑇₀=−𝛼𝑋^𝑘0𝐴^𝑘𝑋^𝑘+𝐵^𝑘𝑊^𝑘𝑇𝑋^𝑘∗−Λ₁𝑌^𝑘𝐴^𝑘_𝑑𝑌^𝑘𝑇0∗∗−𝑋^𝑘0∗∗∗−𝑌^𝑘,𝑇₁=𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇𝐸^𝑘₂𝑌^𝑘𝑇00𝐹^𝑘𝑇00𝐻^𝑘0^𝑇+00𝐻^𝑘0𝐹^𝑘𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇𝐸^𝑘₂𝑌^𝑘𝑇00^𝑇. (A.35) By Lemma 8, we get 𝑇₀+𝜀_𝑘⁻¹𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇𝐸^𝑘₂𝑌^𝑘𝑇00𝐸^𝑘₁𝑋^𝑘+𝐸^𝑘₃𝑊^𝑘𝑇𝐸^𝑘₂𝑌^𝑘𝑇00^𝑇+𝜀_𝑘00𝐻^𝑘000𝐻^𝑘0^𝑇<0. (A.36) Applying Lemma 7 again, we obtain that (A.33) holds if (19) is satisfied.
Similarly, substitute (2) into (A.19) and denote 𝑋^𝑘𝑙=(𝑃^𝑘𝑙)⁻¹ and 𝑌^𝑘𝑙=(𝑄^𝑘𝑙)⁻¹, then using diag{𝑋^𝑘𝑙,𝑌^𝑘𝑙,𝑋^𝑘𝑙} to pre- and postmultiply the left of (A.19), respectively, and applying Lemmas 7 and 8, it is easy to get that (A.19) holds if (20) is satisfied.
The proof is completed.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grant no. 61273120.