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

Authors: Shipei Huang; Zhengrong Xiang; Hamid Reza Karimi
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 Section2, 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. Ih is the identity matrix with n1 dimension and Iv is the identity matrix with n2 dimension. diag{ai} denotes the diagonal matrix with the diagonal elements ai, 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:(1)[xh(i+1,j)xv(i,j+1)]=A^σ(i,j)[xh(i,j)xv(i,j)]+A^dσ(i,j)[xh(i-dh,j)xv(i,j-dv)]+B^σ(i,j)u(i,j),
where xh(i,j)∈Rn1 and xv(i,j)∈Rn2 are the horizontal state and the vertical state, respectively, x(i,j) is the whole state in Rn with n=n1+n2, and u(i,j)∈Rm 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. dh and dv are constant delays along horizontal and vertical directions, respectively. A^k and A^dk (k∈N_) are uncertain real-valued matrices with appropriate dimensions and are assumed to be of the form
(2)A^k=Ak+HkFk(i,j)E1k,A^dk=Adk+HkFk(i,j)E2k,B^k=Bk+HkFk(i,j)E3k,
with
(3)Ak=[A11kA12kA21kA22k],Adk=[Ad11kAd12kAd21kAd22k],Hk=[H1kH2k],E1k=[E11kE12k],E2k=[E21kE22k],E3k=[E31kE32k],
where matrices A11k∈Rn1×n1, A12k∈Rn1×n2, A21k∈Rn2×n1, A22k∈Rn2×n2, Ad11k∈Rn1×n1, Ad12k∈Rn1×n2,  Ad21k∈Rn2×n1, Ad22k∈Rn2×n2, H1k, H2k, E11k, E12k, E21k, E22k, E31k, E32k are constant matrices. Fk(i,j) (k∈N_) is an unknown matrix representing parameter uncertainty and satisfies
(4)FkT(i,j)Fk(i,j)≤I.
The boundary conditions are given by
(5)xh(i,j)=hij,∀0≤j≤z1,-dh≤i≤0,xh(i,j)=0,∀j>z1,-dh≤i≤0,xv(i,j)=vij,∀0≤i≤z2,-dv≤j≤0,xv(i,j)=0,∀i>z2,-dv≤j≤0,
where z1<∞ and z2<∞ are positive integers, and hij and vij 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 ofi or j. The switching sequence can be described as follows:
(6)((i0,j0),σ(i0,j0)),((i1,j1),σ(i1,j1)),…,((iκ,jκ),σ(iκ,jκ)),…,
where (iκ,jκ) 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. Letσ′(i,j) denote the switching signal of the controller. Denoting mκ=iκ+jκ, Δmκ=Δiκ+Δjκ, κ=1,2,…, then the switching points of the controller can be described as
(7)(i0,j0),(i1+Δi1,j1+Δj1),…,(iκ+Δiκ,jκ+Δjκ),…,
where Δiκ and Δjκ represent the delayed period along horizontal and vertical directions, respectively. Δmκ<inf(mκ+1-mκ) is said to be the mismatched period.Remark 1.
Similar to the one-dimensional switched system case [33–36], the mismatched period Δmκ<inf(mκ+1-mκ) 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]:
(8)[xh(i+1,j)xv(i,j+1)]=A^[xh(i,j)xv(i,j)]+A^d[xh(i-dh,j)xv(i,j-dv)],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
(9)∑i+j=D∥x(i,j)∥2≤ηe-c(D-z)∑i+j=z∥x(i,j)∥C2,
holds for all D≥z, where
(10)∑i+j=z∥x(i,j)∥C2≜sup-dh≤θh≤0,-dv≤θv≤0∑i+j=z(∥xh(i-θh,j)∥2+∥xv(i,j-θv)∥2).Remark 4.
From Definition3, it is easy to see that when z is given, ∑i+j=z∥x(i,j)∥C2 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(iκ,jκ) denote the κth switching point and (iκ+1,jκ+1) denote the (κ+1)th switching point. Denote mκ=iκ+jκ, mκ+1=iκ+1+jκ+1, τ=inf(mκ+1-mκ), then τ is called the dwell time.Definition 6 (see [30]).
For anyi+j=D≥z=iz+jz, let Nσ(i,j)(z,D) denote the switching number of σ(i,j) on the interval [z,D). If
(11)Nσ(i,j)(z,D)≤N0+D-zτa
holds for given N0≥0 and τa≥0, then the constant τa is called the average dwell time and N0 is the chatter bound.Lemma 7 (see [37]).
For a given matrixS=[S11S12S12TS22], where S11 and S22 are square matrices, the following conditions are equivalent:(i)
S<0,(ii)
S11<0,  S22-S12TS11-1S12<0,(iii)
S22<0, S11-S12S22-1S12T<0.Lemma 8 (see [38]).
LetU, V, W, and X be real matrices of appropriate dimensions with X satisfying X=XT, then for all VTV≤I,
(12)X+UVW+WTVTUT<0,
if and only if there exists a scalar ε such that
(13)X+εUUT+ε-1WTW<0.


## 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:Rn→R. For a given positive constant α, if there exist positive definite symmetric matrices P=diag{Ph,Pv} and Q=diag{Qh,Qv} with appropriate dimensions, such that the following inequality holds:
(14)[Q-αP0A^TP*-Λ1QA^dTP**-P]<0,
where Λ1=diag{αdhIh,αdvIv}, then along the trajectory of systems (8), the following inequality holds for any D≥D′:
(15)∑i+j=DV(x(i,j))<αD-D′∑i+j=D′V(x(i,j)),
where D′≥z and z=max{z1,z2}.Proof.
See appendix for the detailed proof, it is omitted here.Remark 10.
Lemma9 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 matricesP 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:
(16)u(i,j)=Kσ′(i,j)x(i,j),Kσ′(i,j)=[K1σ′(i,j)K2σ′(i,j)],
where K1σ′(i,j)∈Rm×n1 and K2σ′(i,j)∈Rm×n2, the corresponding closed-loop system is given by
(17)[xh(i+1,j)xv(i,j+1)]=(A^σ(i,j)+B^σ(i,j)Kσ′(i,j))[xh(i,j)xv(i,j)]+A^dσ(i,j)[xh(i-dh,j)xv(i,j-dv)].Without loss of generality, we denoteσ(iκ,jκ)=k∈N_ and (iκ+1,jκ+1)=l∈N_, then due to the existence of asynchronous switching, we can obtain from (7) that
(18)σ′(iκ+Δiκ,jκ+Δjκ)=k,σ′(iκ+1+Δiκ+1,jκ+1+Δjκ+1)=l.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. LetΔ=maxκ=1,2,…(Δmκ) 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 Xk=diag{Xhk,Xvk}, Yk=diag{Yhk,Yvk}, Xkl=diag{Xhkl,Xvkl}, Ykl=diag{Yhkl,Yvkl} and Wk with appropriate dimensions, and positive scalars εk and εkl such that, for k,l∈N_, k≠l, the following inequalities hold,
(19)[-αXk0(AkXk+BkWk)TXk(E1kXk+E3kWk)T*-Λ1Yk(AdkYk)T0(E2kYk)T**-Xk+εkHkHkT00***-Yk0****-εkI]<0,(20)[-βXkl0(AlXkl+BlKkXkl)TXkl(E1lXkl+E3lKkXkl)T*-Λ2Ykl(AdlYkl)T0(E2lYkl)T**-Xkl+εklHlHlT00***-Ykl0****-εklI]<0,where Λ2=diag{βdhIh,βdvIv}, then under the following switching controller:
(21)u(i,j)=Kσ′(i,j)x(i,j),Kk=Wk(Xk)-1,
and the following average dwell time scheme:
(22)τa>τa*=Δ(lnβ-lnα)+ln(μ1μ2)-lnα,
the resulting closed-loop system (17) is exponentially stable, where μ=(α/β)d-, d-=max{dh,dv}, and μ1μ2μ≥1 satisfies
(23)Xl-1≤μ1Xkl-1,Xkl-1≤μ2Xk-1,Yl-1≤μ1Ykl-1,Ykl-1≤μ2μYk-1.Proof.
See the appendix.Remark 13.
In Theorem12, 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 τa* 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 Xk, Yk, and Wk. Then, (20) can be transformed into the LMI by substituting Kk=Wk(Xk)-1 into it. By adjusting the parameters α and β, we can find a feasible solution of Xk, Yk, Wk, Xkl, and Ykl 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 scalar0<α<1, and by solving LMI (19), we can get the feasible solution of matrices Xk, Yk, and Wk and positive scalar εk then, the controller can be obtained from (21).Step 2.
SubstitutingKk into (20), we can find the feasible solution of Xkl, Ykl, and εkl 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 ofΔ, we can compute the value of τa* 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].
WhenA^dσ(i,j)=0, system (17) degenerates to the following delay-free system:
(24)[xh(i+1,j)xv(i,j+1)]=(A^σ(i,j)+B^σ(i,j)Kσ′(i,j))[xh(i,j)xv(i,j)].
Then, we can get the following result.Corollary 16.
Consider system (1) with A^dσ(i,j)=0, for given positive constants α<1 and β>1, if there exist positive definite symmetric matrices Xk=diag{Xhk,Xvk} and Xkl=diag{Xhkl,Xvkl} with appropriate dimensions, and positive scalars εk and εkl such that, for k,l∈N_, k≠l, the following inequalities hold:
(25)[-αXk(AkXk+BkWk)T(E1kXk+E3kWk)T*-Xk+εkHkHkT0**-εkI]<0,[-βXkl(AlXkl+BlKkXkl)T(E1lXkl+E3lKkXkl)T*-Xkl+εklHlHlT0**-εklI]<0,
then under the following switching controller:
(26)u(i,j)=Kσ′(i,j)x(i,j),Kk=Wk(Xk)-1,
and the following average dwell time scheme:
(27)τa>τa*=Δ(lnβ-lnα)+ln(μ1μ2)-lnα,
the resulting closed-loop system (24) is exponentially stable, where μ1μ2≥1 satisfies
(28)Xl-1≤μ1Xkl-1,Xkl-1≤μ2Xk-1.Furthermore, it should also be noted that if the criterion in Theorem12 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 Xk=diag{Xhk,Xvk}, Yk=diag{Yhk,Yvk},  and Wk, with appropriate dimensions, and a positive scalar εk,  such that, for k∈N_, the following inequality holds:
(29)[-αXk0(AkXk+BkWk)TXk(E1kXk+E3kWk)T*-Λ1Yk(AdkYk)T0(E2kYk)T**-Xk+εkHkHkT00***-Yk0****-εkI]<0,
then under the following controller:
(30)u(i,j)=Kkx(i,j),Kk=Wk(Xk)-1,
and the following average dwell time scheme:
(31)τa>τa*=lnμ1-lnα,
the resulting closed-loop system is exponentially stable, where μ1≥1 satisfies
(32)Xl-1≤μ1Xk-1,Yl-1≤μ1Yk-1.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].


## 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:Rn→R. For a given positive constant α, if there exist positive definite symmetric matrices P=diag{Ph,Pv} and Q=diag{Qh,Qv} with appropriate dimensions, such that the following inequality holds:
(14)[Q-αP0A^TP*-Λ1QA^dTP**-P]<0,
where Λ1=diag{αdhIh,αdvIv}, then along the trajectory of systems (8), the following inequality holds for any D≥D′:
(15)∑i+j=DV(x(i,j))<αD-D′∑i+j=D′V(x(i,j)),
where D′≥z and z=max{z1,z2}.Proof.
See appendix for the detailed proof, it is omitted here.Remark 10.
Lemma9 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 matricesP 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:
(16)u(i,j)=Kσ′(i,j)x(i,j),Kσ′(i,j)=[K1σ′(i,j)K2σ′(i,j)],
where K1σ′(i,j)∈Rm×n1 and K2σ′(i,j)∈Rm×n2, the corresponding closed-loop system is given by
(17)[xh(i+1,j)xv(i,j+1)]=(A^σ(i,j)+B^σ(i,j)Kσ′(i,j))[xh(i,j)xv(i,j)]+A^dσ(i,j)[xh(i-dh,j)xv(i,j-dv)].Without loss of generality, we denoteσ(iκ,jκ)=k∈N_ and (iκ+1,jκ+1)=l∈N_, then due to the existence of asynchronous switching, we can obtain from (7) that
(18)σ′(iκ+Δiκ,jκ+Δjκ)=k,σ′(iκ+1+Δiκ+1,jκ+1+Δjκ+1)=l.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. LetΔ=maxκ=1,2,…(Δmκ) 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 Xk=diag{Xhk,Xvk}, Yk=diag{Yhk,Yvk}, Xkl=diag{Xhkl,Xvkl}, Ykl=diag{Yhkl,Yvkl} and Wk with appropriate dimensions, and positive scalars εk and εkl such that, for k,l∈N_, k≠l, the following inequalities hold,
(19)[-αXk0(AkXk+BkWk)TXk(E1kXk+E3kWk)T*-Λ1Yk(AdkYk)T0(E2kYk)T**-Xk+εkHkHkT00***-Yk0****-εkI]<0,(20)[-βXkl0(AlXkl+BlKkXkl)TXkl(E1lXkl+E3lKkXkl)T*-Λ2Ykl(AdlYkl)T0(E2lYkl)T**-Xkl+εklHlHlT00***-Ykl0****-εklI]<0,where Λ2=diag{βdhIh,βdvIv}, then under the following switching controller:
(21)u(i,j)=Kσ′(i,j)x(i,j),Kk=Wk(Xk)-1,
and the following average dwell time scheme:
(22)τa>τa*=Δ(lnβ-lnα)+ln(μ1μ2)-lnα,
the resulting closed-loop system (17) is exponentially stable, where μ=(α/β)d-, d-=max{dh,dv}, and μ1μ2μ≥1 satisfies
(23)Xl-1≤μ1Xkl-1,Xkl-1≤μ2Xk-1,Yl-1≤μ1Ykl-1,Ykl-1≤μ2μYk-1.Proof.
See the appendix.Remark 13.
In Theorem12, 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 τa* 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 Xk, Yk, and Wk. Then, (20) can be transformed into the LMI by substituting Kk=Wk(Xk)-1 into it. By adjusting the parameters α and β, we can find a feasible solution of Xk, Yk, Wk, Xkl, and Ykl 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 scalar0<α<1, and by solving LMI (19), we can get the feasible solution of matrices Xk, Yk, and Wk and positive scalar εk then, the controller can be obtained from (21).Step 2.
SubstitutingKk into (20), we can find the feasible solution of Xkl, Ykl, and εkl 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 ofΔ, we can compute the value of τa* 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].
WhenA^dσ(i,j)=0, system (17) degenerates to the following delay-free system:
(24)[xh(i+1,j)xv(i,j+1)]=(A^σ(i,j)+B^σ(i,j)Kσ′(i,j))[xh(i,j)xv(i,j)].
Then, we can get the following result.Corollary 16.
Consider system (1) with A^dσ(i,j)=0, for given positive constants α<1 and β>1, if there exist positive definite symmetric matrices Xk=diag{Xhk,Xvk} and Xkl=diag{Xhkl,Xvkl} with appropriate dimensions, and positive scalars εk and εkl such that, for k,l∈N_, k≠l, the following inequalities hold:
(25)[-αXk(AkXk+BkWk)T(E1kXk+E3kWk)T*-Xk+εkHkHkT0**-εkI]<0,[-βXkl(AlXkl+BlKkXkl)T(E1lXkl+E3lKkXkl)T*-Xkl+εklHlHlT0**-εklI]<0,
then under the following switching controller:
(26)u(i,j)=Kσ′(i,j)x(i,j),Kk=Wk(Xk)-1,
and the following average dwell time scheme:
(27)τa>τa*=Δ(lnβ-lnα)+ln(μ1μ2)-lnα,
the resulting closed-loop system (24) is exponentially stable, where μ1μ2≥1 satisfies
(28)Xl-1≤μ1Xkl-1,Xkl-1≤μ2Xk-1.Furthermore, it should also be noted that if the criterion in Theorem12 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 Xk=diag{Xhk,Xvk}, Yk=diag{Yhk,Yvk},  and Wk, with appropriate dimensions, and a positive scalar εk,  such that, for k∈N_, the following inequality holds:
(29)[-αXk0(AkXk+BkWk)TXk(E1kXk+E3kWk)T*-Λ1Yk(AdkYk)T0(E2kYk)T**-Xk+εkHkHkT00***-Yk0****-εkI]<0,
then under the following controller:
(30)u(i,j)=Kkx(i,j),Kk=Wk(Xk)-1,
and the following average dwell time scheme:
(31)τa>τa*=lnμ1-lnα,
the resulting closed-loop system is exponentially stable, where μ1≥1 satisfies
(32)Xl-1≤μ1Xk-1,Yl-1≤μ1Yk-1.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:
(33)A1=[11.510.5],Ad1=[-0.150-0.1-0.12],(34)B1=[-4.501-3],H1=[0.20.150.10.2],(35)E11=[-0.20-0.2-0.2],E21=[0.100.10.2],F1=diag{sin(0.5π(i+j)),sin(0.5π(i+j))},E31=[0.1500.130.12],A2=[1211],Ad2=[-0.10.20-0.2],B2=[-51-1-3],H2=[0.20.250.20.3],E12=[0.10.20.20.1]E22=[0.20.10.20.1],E32=[0.120.150.120.1],F2=diag{cos(0.5π(i+j)),cos(0.5π(i+j))},dh=2,dv=3.
The boundary conditions are given as follows:
(36)xh(0,j)={5,0≤j≤20,0,j>20,xv(i,0)={3,0≤i≤20,0,i>20,
where the state dimensions are n1=1 and n2=1.Takeα=0.6 and β=1.2, then solving (19) in Theorem 12 gives rise to
(37)X1=[36.39031.02471.024738.3909],X2=[40.0833-6.6640-6.664034.4516],Y1=[96.6213-4.5896-4.589696.9472],Y2=[99.6213-8.8472-8.847279.5315],W1=[8.586613.060415.556811.4091],W2=[6.984413.29308.37904.4234],ε1=74.2107,ε2=73.4149.
By (21), K1 and K2 can be obtained as follows:
(38)K1=[0.22650.33410.41940.2860],K2=[0.24630.43350.23800.1744].
Substituting K1 and K2 into (20), and solving it, we get the following solution:
(39)X12=[0.5400-0.0934-0.09340.4811],X21=[0.5805-0.0926-0.09260.5677],Y12=[0.9398-0.0418-0.04180.8451],Y21=[0.9436-0.0340-0.03400.8867],ε12=0.9011,ε21=0.7725.
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 τa*=7.9. Choosing τa=8, the trajectories of the states xh(i,j) and xv(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 statexh(i,j).Figure 2
The trajectory of the statexv(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 robustH∞ control for 2D discrete switched systems with time-varying delays and fractional uncertainties under asynchronous switching.



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*Source: 961870-2013-05-16.xml*