# LMI-Based Criterion for the Robust Stability of 2D Discrete State-Delayed Systems Using Generalized Overflow Nonlinearities

Authors: Anurita Dey; Haranath Kar
Journal: Journal of Control Science and Engineering
 (2011)

---

## Abstract

This paper addresses the problem of global asymptotic stability of a class of discrete uncertain state-delayed systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model using generalized overflow nonlinearities. The uncertainties are assumed to be norm bounded. A computationally tractable, that is, linear-matrix-inequality-(LMI-) based new criterion for the global asymptotic stability of such system is proposed. It is demonstrated that several previously reported stability criteria for two-dimensional (2D) systems are recovered from the presented approach as special cases. Numerical examples are given to illustrate the usefulness of the presented approach.


---

## Body

## 1. Introduction

Two-dimensional (2D) systems play an important role in filtering, image data processing and transmission, water stream heating, seismographic data processing, thermal processes, biomedical imaging, gas absorption [1, 2], river pollution modeling [3], process of gas filtration [4], grid-based wireless sensor networks [5, 6], and many other areas. The study of such systems has received considerable attention in the last two decades.When designing a linear discrete system with fixed point arithmetic using a digital computer or special-purpose hardware, several kinds of nonlinearities, such as overflow and quantization, are produced. These nonlinearities may lead to instability in the designed system. The study of nonlinear systems has received much attention in the literature [6–21].Physical systems usually suffer from uncertainties that arise because of variations in system parameters, modeling errors, or some ignored factors. The subject of uncertain systems has received a considerable amount of interest [13–16, 21–29].Delays are frequently encountered in many dynamical systems due to finite speed of information propagation and computational lags. Time delays are another source of instability in the designed system. A significant amount of work concerning the stability of time-delay systems has been done [18, 19, 24–32].The design of a 2D system so as to ensure the stability of the designed system is an interesting and challenging problem. During the past few decades, the stability properties of 2D discrete systems described by the Fornasini-Marchesini second local state-space (FMSLSS) model [33] have been investigated extensively [6–10, 12–16, 18, 19, 21–26, 28, 30–32, 34–36]. Several publications [7–10, 12, 18, 19] relating to the issue of the global asymptotic stability of the FMSLSS model with overflow nonlinearities have appeared. The stability analysis of uncertain FMSLSS model with overflow nonlinearities has been carried out in [13–16, 21]. In [24], the problem of robust stability and stabilization of 2D state-delayed systems has been addressed. Reference [37] deals with the problem of stabilization of nonlinear 2D discrete Takagi-Sugeno fuzzy systems. The problem of global asymptotic stability of 2D state-delayed FMSLSS model with saturation nonlinearities has been studied in [18, 19]. A 2D filtering approach with H2/H∞ performance measure for state-delayed FMSLSS model has been developed in [25]. The guaranteed cost control problem for 2D state-delayed systems has been studied in [28].The stability analysis of 2D discrete systems described by the FMSLSS model in the simultaneous presence of nonlinearity, state delay, and parameter uncertainty in their physical models is an important and realistic problem. Since the characterization of the evolution of nonlinear uncertain dynamical state-delayed systems as a deterministic set of state equations is a formidable task, the stability analysis of such systems is generally difficult. To the best of authors’ knowledge, such problem has not been addressed so far in the literature.This paper, therefore, deals with the problem of global asymptotic stability of a class of 2D discrete uncertain state-delayed systems described by the FMSLSS model employing generalized overflow nonlinearities. Parametric uncertainties involved in the system are assumed to be norm bounded. The paper is organized as follows. Section2 presents a description of the system under consideration. In Section 3, the main result and its corollaries are presented in the form of linear matrix inequality (LMI) which can be effectively solved using Matlab LMI Toolbox [38, 39]. It is demonstrated in Section 4 that several previously reported results are recovered from the presented approach as special cases. Numerical examples highlighting the usefulness of the presented criteria are discussed in Section 5.


## 2. System Description

The following notations are used throughout the paper:Rn×n: set of n×n real matrices,Rn: set of n×1 real vectors,Z: set of integers,I: identity matrix of appropriate dimension,0: null matrix or null vector of appropriate dimension,GT: transpose of the matrix (or vector) G,G>0: G is positive definite symmetric matrix,G<0: G is negative definite symmetric matrix,∥·∥: any vector or matrix norm,diag{a1,a2,…,an}: diagonal matrix with diagonal elements a1,a2,…,an,max{v,w}:  maximum value of scalars v and w.The 2D discrete uncertain state-delayed system to be studied presently is described by the FMSLSS model [33] employing generalized overflow arithmetic. The system under consideration is given by(1a)x(i+1,j+1)=f(y(i,j))=[f1(y1(i,j))f2(y2(i,j))⋯fn(yn(i,j))]T,(1b)y(i,j)=(A1+ΔA1)x(i+1,j)+(A2+ΔA2)x(i,j+1)+(Ad1+ΔAd1)x(i+1,j-d1)+(Ad2+ΔAd2)x(i-d2,j+1)=[y1(i,j)y2(i,j)⋯yn(i,j)]T,where 0≤i,j∈Zare horizontal and vertical coordinates, respectively; x(i,j)∈Rn is the local state vector; A1, A2, Ad1, Ad2 are the known real constant n×n matrices; ΔA1,ΔA2,ΔAd1,ΔAd2 are the unknown real n×n matrices representing parametric uncertainties in the state matrices; d1, d2 are constant positive integers representing delays along vertical direction and horizontal direction, respectively. The generalized overflow characteristic is given by (2a)L≤fk(yk(i,j))≤L1,yk(i,j)>1,k=1,2,…,nfk(yk(i,j))=yk(i,j),-1≤yk(i,j)≤1,k=1,2,…,n-L2≤fk(yk(i,j))≤-L,yk(i,j)<-1,k=1,2,…,n
where
(2b)-1≤L≤1L≤L1≤1L≤L2≤1.With the appropriate selection of L,L1, andL2, (2a) characterizes the common types of overflow arithmetics employed in applications such as saturation (L=L1=L2=1), zeroing (L=L1=L2=0), triangular (L=-1,L1=L2=1), and two’s complement (L=-1,L1=L2=1).The uncertain matrixΔA is defined in the norm-bounded form [13–16, 21–24, 27–29] as (3a)ΔA=[ΔA1ΔA2ΔAd1ΔAd2]=[HFE1HFE2HFEd1HFEd2],
where H,E1,E2,Ed1,Ed2 are known real constant matrices with appropriate dimensions and F is an unknown real matrix satisfying(3b)FTF≤I.In the uncertainty structure given by (3a)-(3b), the matrices H,E1,E2,Ed1,Ed2 characterize how the uncertain parameters in Fenter the state matrices. The matrixF can always be restricted as (3b) by appropriately selecting H,E1,E2,Ed1,Ed2. In other words, there is no loss of generality in choosing F as in (3b).It is assumed [32] that the system has a finite set of initial conditions, that is, there exist two positive integers K and L such that(4)x(i,j)=0,∀i≥K,j=-d1,-d1+1,…,0;x(i,j)=0,∀j≥L,i=-d2,-d2+1,…,0.Equations (1a)–(4) represent a class of 2D discrete dynamical systems which include 2D discrete systems implemented in a finite register length, 2D digital control systems with overflow nonlinearities, models of various physical phenomena (e.g., compartmental systems, single carriageway traffic flow [40], grid-based wireless sensor networks modeling [6], etc.), and various dynamical processes represented by the Darboux equation [41–43].


## 3. Main Result and Its Corollaries

Consider two ranges forL(5)0≤L≤1L≤L1≤1L≤L2≤1,(6)-1≤L<0L≤L1≤1L≤L2≤1.These two ranges together constitute (2b). In what follows, a criterion applicable to (5) and a different criterion applicable to (6) for the global asymptotic stability of the system described by (1a)-(1b), (2a), (3a)-(3b), and (4) are presented.Before presenting our main result, we recall the following lemma.Lemma 1 (see Xie et al. [44], Boyd et al. [38]).
LetΣ,Γ,F,andM be real matrices of appropriate dimensions with  M satisfying M=MT;then
(7)M+𝚺FΓ+ΓTFT𝚺T<0
for all FTF≤I if and only if there exists a positive scalar  ε  such that
(8)M+ε-1𝚺𝚺T+εΓTΓ<0.Next, supposeC=[ckl]∈Rn×n is a matrix characterized by(9a)ckk=∑l=1,l≠kn(αkl+βkl),k=1,2,…,n,(9b)ckl=L(αkl-βkl),k,l=1,2,…,n(k≠l),(9c)αkl>0,βkl>0,k,l=1,2,…,n(k≠l),where it is understood that, for  n=1, C corresponds to a scalarγ>0. Thus, corresponding to n=3, the matrix C takes the form(10)C=[α12+β12+α13+β13L(α12-β12)L(α13-β13)L(α21-β21)α21+β21+α23+β23L(α23-β23)L(α31-β31)L(α32-β32)α31+β31+α32+β32],where αkl>0,βkl>0,k,l=1,2,3(k≠l).Now, we have the following result.Theorem 1.
The null solution of the system characterized by (1a)-(1b), (2a), (3a)-(3b)–(5) is globally asymptotically stable if there exist positive scalarsαkl,βkl(k,l=1,2,…,n(k≠l)), ε, andn×n positive definite symmetric matrices P,P1,Q1,Q2 such that(11a)[-J+εETEATC0CTAP-C-CTCTH0HTC-εI]<0,
where
(11b)J=[P-P1-Q10000P1-Q20000Q10000Q2],(11c)A=[A1A2Ad1Ad2],(11d)E=[E1E2Ed1Ed2],and C=[ckl]∈Rn×n is defined by (9a)–(9c).Proof.
Consider a 2D quadratic Lyapunov-Krasovskii function [24](12a)V(x(i,j))=Vh(x(i,j))+Vv(x(i,j)),
where
(12b)Vh(x(i,j))=xT(i,j)P1x(i,j)+∑l=-d2-1xT(i+l,j)Q2x(i+l,j),(12c)Vv(x(i,j))=xT(i,j)(P-P1)x(i,j)+∑l=-d1-1xT(i,j+l)Q1x(i,j+l).Application of (12a)–(12c) to system (1a)-(1b) yields
(13)ΔV(x(i,j))=Vh(x(i+1,j+1))+Vv(x(i+1,j+1))-Vh(x(i,j+1))-Vv(x(i+1,j)).
It follows from (1a), (12a)–(12c), and (13)
(14)ΔV(x(i,j))=fT(y(i,j))Pf(y(i,j))-xT(i,j+1)(P1-Q2)x(i,j+1)-xT(i-d2,j+1)Q2x(i-d2,j+1)-xT(i+1,j)(P-P1-Q1)x(i+1,j)-xT(i+1,j-d1)Q1x(i+1,j-d1).
Now consider the quantity “δ” given by
(15)δ=∑k=1n2[yk(i,j)-fk(yk(i,j))]×[∑l=1,l≠kn{(αkl+βkl)fk(yk(i,j))+L(αkl-βkl)fl(yl(i,j))}∑l=1,l≠kn{(αkl+βkl)fk(yk(i,j))]=yT(i,j)Cf(y(i,j))+fT(y(i,j))CTy(i,j)-fT(y(i,j))(C+CT)f(y(i,j)).
It is clear that, owing to (2a), (5), and (9c), δ is nonnegative. (14) can be rearranged as
(16)ΔV(x(i,j))=-x̂T(i,j)Zx̂(i,j)-δ,
where(17)x̂T(i,j)=[xT(i+1,j)xT(i,j+1)xT(i+1,j-d1)xT(i-d2,j+1)fT(y(i,j))],(18)Z=[P-P1-Q1000-(A1+ΔA1)TC0P1-Q200-(A2+ΔA2)TC00Q1⋅0-(Ad1+ΔAd1)TC000Q2-(Ad2+ΔAd2)TC-CT(A1+ΔA1)-CT(A2+ΔA2)-CT(Ad1+ΔAd1)-CT(Ad2+ΔAd2)C+CT-P].Observe that, ifZ>0, thenΔV(x(i,j))≤0. From (13) and condition ΔV(x(i,j))≤0, it follows that
(19)Vh(x(i+1,j+1))+Vv(x(i+1,j+1))≤Vh(x(i,j+1))+Vv(x(i+1,j)),
where the equality sign holds only when x(i+1,j)=x(i,j+1)=x(i+1,j-d1)=x(i-d2,j+1)=0. Let D(r) denote the set defined by
(20)D(r)≜{(i,j):i+j=r,i≥0,j≥0}.
For any nonnegative integer r≥ max {K,L}, it follows from (19) and the initial condition (4) that
(21)∑(i+j)∈D(r+1)V(x(i,j))=∑(i+j)∈D(r+1)[Vh(x(i,j))+Vv(x(i,j))]=Vh(x(r+1,0))+Vh(x(r,1))+Vh(x(r-1,2))+⋯+Vh(x(2,r-1))+Vh(x(1,r))+Vh(x(0,r+1))+Vv(x(r+1,0))+Vv(x(r,1))+Vv(x(r-1,2))+⋯+Vv(x(2,r-1))+Vv(x(1,r))+Vv(x(0,r+1))≤Vh(x(r+1,0))+Vh(x(r-1,1))+Vh(x(r-2,2))+⋯+Vh(x(1,r-1))+Vh(x(0,r))+Vh(x(0,r+1))+Vv(x(r+1,0))+Vv(x(r,0))+Vv(x(r-1,1))+⋯+Vv(x(2,r-2))+Vv(x(1,r-1))+Vv(x(0,r+1))+Vh(x(r,0))+Vv(x(0,r))-Vh(x(r,0))-Vv(x(0,r))=∑(i+j)∈D(r)V(x(i,j)),
where the equality sign holds only when
(22)∑(i+j)∈D(r)V(x(i,j))=0.
In the above derivation, the fact that  x(r+1,j)=0,  j=-d1,  -d1+1,…,0,x(i,r+1)=0,  i=-d2,-d2+1,…,0,x(r,0)=0,x(0,r)=0, and the positive definiteness of the function  V(x(i,j))  have been used.
From (21), we obtain
(23)lim⁡r→∞∑(i+j)∈D(r)V(x(i,j))=0.
Consequently,
(24)lim⁡i+j→∞‖x(i,j)‖=0.
Thus condition Z>0 is a sufficient condition for the global asymptotic stability of the system described by (1a)-(1b), (2a), (3a)-(3b)–(5). Using (3a) the condition Z>0 can be rewritten in the following form:(25a)M+H̅FE̅+E̅TFTH̅T<0,
where
(25b)M=[-P+P1+Q1000A1TC0-P1+Q200A2TC00-Q10Ad1TC000-Q2Ad2TCCTA1CTA2CTAd1CTAd2P-C-CT],(25c)H̅T=[0000HTC],(25d)E̅=[E1E2Ed1Ed20].By Lemma 1, (25a)–(25d) is equivalent to
(26)M+ε-1H̅H̅T+εE̅TE̅<0,
where ε>0. The equivalence of (26) and (11a)–(11d) follows trivially from Schur complement. This completes the proof of Theorem 1.Remark 1.
The matrix inequality (11a)–(11d) is linear in the unknown parametersP,P1,Q1,Q2,αkl,βkl(k,l=1,2,…,n(k≠l)), and  ε. Therefore, it can be solved using the Matlab LMI Toolbox [38, 39].Next, we prove the following result.Theorem 2.
The null solution of the system characterized by (1a)-(1b), (2a), (3a)-(3b), (4), and (6) is globally asymptotically stable if there exist  n×n  positive definite symmetric matrices P,P1,Q1,Q2, an n×n positive definite diagonal matrix D=diag(d1,d2,…,dn), and a positive scalar ε, satisfying(27a)[-S-2εLETE0-2LATD00(1-L)22LD+P(1+L)-2LD0-2LDA(1+L)-2LD-DDH00HTD-εI]<0,
where
(27b)S=[P-P1-Q10000P1-Q20000Q10000Q2].Proof.
Consider the quantityϕ given by
(28)ϕ=∑k=1n2dk[yk(i,j)-fk(yk(i,j))]×[-Lyk(i,j)+fk(yk(i,j))]=(1+L)yT(i,j)Df(y(i,j))+(1+L)fT(y(i,j))Dy(i,j)-2fT(y(i,j))Df(y(i,j))-2LyT(i,j)Dy(i,j).
Observe that, for  L∈[-1,0), ϕ is nonnegative [10]. Equation (14) takes the form ΔV(x(i,j))=-x̂T(i,j)Gx̂(i,j)-ϕ,where
(29)G=[P-P1-Q1+2L(A1+ΔA1)TD(A1+ΔA1)2L(A1+ΔA1)TD(A2+ΔA2)2L(A1+ΔA1)TD(Ad1+ΔAd1)2L(A2+ΔA2)TD(A1+ΔA1)P1-Q2+2L(A2+ΔA2)TD(A2+ΔA2)2L(A2+ΔA2)TD(Ad1+ΔAd1)2L(Ad1+ΔAd1)TD(A1+ΔA1)2L(Ad1+ΔAd1)TD(A2+ΔA2)Q1+2L(Ad1+ΔAd1)TD(Ad1+ΔAd1)2L(Ad2+ΔAd2)TD(A1+ΔA1)2L(Ad2+ΔAd2)TD(A2+ΔA2)2L(Ad2+ΔAd2)TD(Ad1+ΔAd1)-(1+L)D(A1+ΔA1)-(1+L)D(A2+ΔA2)-(1+L)D(Ad1+ΔAd1)2L(A1+ΔA1)TD(Ad2+ΔAd2)-(1+L)(A1+ΔA1)TD2L(A2+ΔA2)TD(Ad2+ΔAd2)-(1+L)(A2+ΔA2)TD2L(Ad1+ΔAd1)TD(Ad2+ΔAd2)-(1+L)(Ad1+ΔAd1)TDQ2+2L(Ad2+ΔAd2)TD(Ad2+ΔAd2)-(1+L)(Ad2+ΔAd2)TD-(1+L)D(Ad2+ΔAd2)2D-P].
Note that, ifG>0, then ΔV(x(i,j))<0 for  x̂(i,j)≠0. Thus condition G>0 is a sufficient condition for the global asymptotic stability of system (1a)-(1b), (2a), (3a)-(3b), (4), and (6). By the Schur complement, the condition G>0 is equivalent to
(30)[P-P1-Q10000-ζ(A1+ΔA1)TD0P1-Q2000-ζ(A2+ΔA2)TD00Q100-ζ(Ad1+ΔAd1)TD000Q20-ζ(Ad2+ΔAd2)TD0000-(1-L)22LD-P-(1+L)ζD-ζD(A1+ΔA1)-ζD(A2+ΔA2)-ζD(Ad1+ΔAd1)-ζD(Ad2+ΔAd2)-(1+L)ζDD]>0.
where  ζ  denotes  -2L.
Applying (3a), (30) can be rewritten in the following form:(31a)W+H̃FẼ+ẼTFTH̃T<0,
where(31b)W=[-(P-P1-Q1)0000-2LA1TD0-(P1-Q2)000-2LA2TD00-Q100-2LAd1TD000-Q20-2LAd2TD0000(1-L)22LD+P(1+L)-2LD-2LDA1-2LDA2-2LDAd1-2LDAd2(1+L)-2LD-D],(31c)H̃T=[00000HTD],(31d)Ẽ=[-2LE1-2LE2-2LEd1-2LEd200].By Lemma 1, (31a)–(31d) is equivalent to
(32)W+ε-1H̃H̃T+εẼTẼ<0,
where ε>0. Using Schur complement, (32) can equivalently be expressed as (27a)-(27b). The rest of the proof is similar to that of Theorem 1.Next, consider a class of nonlinearities characterized by(33a)fk(0)=0,Lyk2(i,j)≤fk(yk(i,j))yk(i,j)≤yk2(i,j),k=1,2,…,n,(33b)-1≤L<0.Clearly, the quantityϕ defined by (28) is nonnegative under the restriction (33a)-(33b). Therefore, using the steps of the proof of Theorem 2, one arrives at the following.Theorem 3.
The null solution of the system described by (1a)-(1b), (3a)-(3b), (4), and (33a)-(33b) is globally asymptotically stable provided there exist n×n positive definite symmetric matrices P, P1, Q1, Q2, an n×n positive definite diagonal matrix D, and a positive scalar ε such that (27a)-(27b) holds true.Remark 2.
Observe that, for triangular overflow arithmetic, Theorem3 will always lead to less conservative conditions than Theorem 2. As far as triangular overflow arithmetic is concerned, one is required to choose L=-1/3 in (33a)-(33b), rather than L=-1. By contrast, (2a)-(2b) is overly restrictive for the characterization of triangular overflow arithmetic. As a matter of fact, (2a)-(2b) fails to make any distinction between two’s complement and triangular nonlinearities.Next, consider the system (1a)-(1b), (3a)-(3b), and (4) in absence of state delay; therefore, the system becomes(34a)x(i+1,j+1)=f[(A1+ΔA1)x(i+1,j)+(A2+ΔA2)x(i,j+1)],(34b)Ã=[A1A2],(34c)[ΔA1ΔA2]=HFÊ,Ê=[E1E2],(34d)FTF≤I.Now, as a consequence of the presented approach, we have the following corollaries.Corollary 1.
The null solution of the system described by (34a)–(34d), (2a), and (5) is globally asymptotically stable if there exist positive scalars  αkl,βkl(k,l=1,2,…,n(k≠l)),ε, and n×n positive definite symmetric matrices PandP1such that
(35)[-[P-P100P1]+εÊTÊÃTC0CTÃP-C-CTCTH0HTC-εI]<0,
where  C=[ckl]∈Rn×n  is defined by (9a)–(9c).Proof.
Pertaining to the system described by (34a)–(34d), (2a), and (5), choosing Ad1=Ad2=0 and Ed1=Ed2=0, and  Q1=Q2=0 in (11a)–(11d), we obtain (35). This completes the proof of Corollary 1.The following result can easily be obtained as a direct consequence of Theorem2.Corollary 2.
The null solution of the system described by (34a)–(34d), (2a), and (6) is globally asymptotically stable if there exist scalarε > 0, n×n positive definite symmetric matrices P, P1, and an n×n positive definite diagonal matrix D satisfying (36)[-[P-P100P1]-2εLÊTÊ0-2LÃTD00(1-L)22LD+P(1+L)-2LD0-2LDÃ(1+L)-2LD-DDH00HTD-εI]<0.Proof.
For the system described by (34a)–(34d), (2a), and (6), substituting Ad1=Ad2=0 and Ed1=Ed2=0 and Q1=Q2=0 in (27a)-(27b), we obtain (36). This completes the proof of Corollary 2.Remark 3.
In [13], a criterion (see [13, Theorem  5]) for the global asymptotic stability of 2D discrete uncertain systems described by the FMSLSS model employing (2a) and (5) has been established. However, as indicated in [13, Remark  4], [13, Theorem  5] is computationally demanding. On the other hand, (35) is linear in the unknown parameters ε, P, P1,αkl,βkl(k,l=1,2,…,n(k≠l)), and, hence, is computationally tractable [38, 39]. Further, pertaining to the 2D system without state-delay, Corollary 2 is identical to [13, Theorem  4]. Therefore, the present work may be treated as an extension of [13] from the case of nonlinear uncertain discrete systems to a general class of systems in the simultaneous presence of nonlinearity, uncertainty, and state delays.


## 4. Comparison

In this section, we will compare the main results of this paper with the results stated in [14, 15, 21, 24].Theorem 4 (Du and Xie [15]).
The null solution of the 2D system described by (34a)–(34d), employing saturation nonlinearities (i.e., (2a)-(2b) with L=L1=L2=1) is globally asymptotically stable if there exist n×n positive definite symmetric matrices P,P1 and a positive definite diagonal matrix D, and a positive scalar ε such that
(37)[-[P-P100P1]+εÊTÊÃTD0DÃP-2DDH0HTD-εI]<0.Pertaining to the saturation nonlinearity [21, Theorem  2] leads to the following result.Theorem 5 (Dey and Kar [21]).
The null solution of the system described by (34a)–(34d) employing saturation nonlinearities (i.e., (2a)-(2b) with L=L1=L2=1) is globally asymptotically stable if there exist a positive scalar ε and n×n positive definite symmetric matricesP1, P=[pkl] such that(38a)pkk≥∑l=1,l≠kn|pkl|,k=1,2,…,n,(38b)[-PPA1PA2PHA1TP-P+P1+εE1TE1εE1TE20A2TPεE2TE1-P1+εE2TE20HTP00-εI]<0.Theorem 6 (Singh [14]).
Suppose there exists a matrixW=[wij]∈Rn×n, positive definite diagonal matrices U=diag(u1,u2,…,un), V=diag(v1,v2,…,vn), n×n positive definite symmetric matrices  P and P1, a scalar  ε > 0 satisfying(39a)[-[P-P100P1]+εÊTÊÃT(U+VW)0(U+WTV)ÃP-2U-VW-WTV-(U+WTV)H0-HT(U+VW)-εI]<0,where
(39b)wkk=n-1,k=1,2,…,n,(39c)wkl=λk,k,l=1,2,…,n(k≠l),(39d)λk∈(1,-1,0),k=1,2,…,n.Then the null solution of the system (34a)–(34d) using saturation nonlinearities (i.e., (2a)-(2b) with L=L1=L2=1) is globally asymptotically stable.Theorem 7 (Paszke et al. [24]).
Consider the system (1a)–(4) with no nonlinearity and no uncertainty; then system becomes
(40)x(i+1,j+1)=A1x(i+1,j)+A2x(i,j+1)+Ad1x(i+1,j-d1)+Ad2x(i-d2,j+1).
The null solution of the system described by (40) is globally asymptotically stable if there exist n×n positive definite symmetric matrices P,P1,Q1, and Q2 such that
(41)Y=[P-P1-Q10000P1-Q20000Q10000Q2]-[A1TA2TAd1TAd2T]P[A1A2Ad1Ad2]>0.Now, we have the following propositions.Proposition 1.
Corollary1 implies Theorem 4.Proof.
Withαkl=βkl(k,l=1,2,…,n(k≠l)) and L=L1=L2=1 (saturation nonlinearities), matrix C in Corollary 1 reduces to a positive definite diagonal matrix D and, consequently, Corollary 1 becomes Theorem 4. Thus, Theorem 4 is recovered from Corollary 1 as a special case.Proposition 2.
Corollary1 implies Theorem 5.Proof.
It is not difficult to show that, with(42)C=CT=P,(35) reduces to (38b). Condition (42) along with (9a)–(9c) and L=L1=L2=1 yields(43a)pkk=ckk=∑l=1,l≠kn(αkl+βkl),k=1,2,…,n,(43b)pkl=ckl=clk=αkl-βkl,k,l=1,2,…,n(k≠l),where  αkl>0,βkl>0,k,l=1,2,…,n(k≠l). Using (43a)-(43b), we obtain
(44)∑l=1,l≠kn|pkl|=∑l=1,l≠kn|αkl-βkl|<∑l=1,l≠kn|αkl+βkl|=pkk,k=1,2,…,n.
Thus, condition (42) leads to (38a).Proposition 3.
Corollary1 implies Theorem 6.Proof.
Using Schur complement, (39a) can equivalently be expressed as(45)[-[P-P100P1]+εÊTÊÃT(U+VW)0(U+WTV)ÃP-2U-VW-WTV(U+WTV)H0HT(U+VW)-εI]<0.
Now, consider the case wheren=1. In this case, the matrix  C  in Corollary 1 corresponds to a positive scalar. On the other hand, for  n=1, the matrices  W  and  U  in Theorem 6 reduce to a scalar equal to zero and a positive scalar, respectively. Now, from (35) and (45), it is clear that, for  n=1,  Corollary 1 is equivalent to Theorem 6.
Next, consider the case wheren≥2. In this case, let us choose the positive scalars αkl and βkl given by (9c) as(46a)αkl=uk+(n-1)vk2(n-1)+vkλk2,k,l=1,2,…,n(k≠l),(46b)βkl=uk+(n-1)vk2(n-1)-vkλk2,k,l=1,2,…,n(k≠l),where uk>0,vk>0,λk∈(1,-1,0),k=1,2,…,n. In view of (46a)-(46b) and L=L1=L2=1, one obtains from (9a)–(9c) (47a)ckk=uk+(n-1)vk,k=1,2,…,n,(47b)ckl=vkλk,k,l=1,2,…,n(k≠l).With (47a)-(47b), the matrix C in Corollary 1 reduces to (U+VW) and, consequently, (35) reduces to (45). Thus, Theorem 6 is recovered from Corollary 1 as a special case.Remark 4.
Pertaining to saturation nonlinearities, the matrixC  in Corollary 1 is more general than the matrix (U+VW) in Theorem 6. For a given row, the off-diagonal elements of the matrix (U+VW) are equal. However, for  L=L1=L2=1, such restrictions are not required for the matrix  C  in Corollary 1. In other words, Corollary 1 offers a greater flexibility than Theorem 6 for testing the saturation overflow stability of system (34a)–(34d).Remark 5.
To establish global asymptotic stability via Theorem6, one is required to test (39a) (possibly) for all possible combinations of  λk’s (k=1,2,…,n) belonging to (1, −1, 0). For an nth-order system, the number of such possible combinations of  λk’s (k=1,2,…,n) is 3n. Therefore, (39a)–(39d) is computationally more complex than (35).Proposition 4.
Theorem1 implies Theorem 7.Proof.
The system described by (1a)–(4) employing “no nonlinearity” and “no uncertainty” implies f(y(i,j))=y(i,j) and ΔA1=ΔA2=ΔAd1=ΔAd2=0. In this case, (14) can be rearranged as
(48)ΔV(x(i,j))=-[x(i+1,j)x(i,j+1)x(i+1,j-d1)x(i-d2,j+1)]TY[x(i+1,j)x(i,j+1)x(i+1,j-d1)x(i-d2,j+1)],
which satisfies  ΔV(x(i,j))≤0  if (41) holds true and  ΔV(x(i,j))=0 only when x(i+1,j)=x(i,j+1)=x(i+1,j-d1)=x(i-d2,j+1)=0. Thus, Theorem 7 is recovered from Theorem 1 as a special case.


## 5. Illustrative Examples

To illustrate the effectiveness of the presented results, we now consider the following examples.Example 1.
Consider the 2D discrete system represented by (1a)-(1b), (2a), (3a)-(3b), (4) with(49a)A1=[1.05-20.10],A2=[0.001000.001],(49b)Ad1=Ad2=[0.001000.001],H=[00.01],(49c)E1=E2=Ed1=Ed2=[0.010],L=1.Using the Matlab LMI Toolbox [38, 39], it has been found that LMI (11a)–(11d) is feasible for the following values of unknown parameters.
(50)α12=62.4706,α21=298.8216,β12=146.1559,β21=693.5676,P=[237.7874-448.6233-448.62331610.9],P1=[30.5319-17.7497-17.7497308.6435],Q1=[15.7135-10.6406-10.6406182.5997],Q2=[19.8877-10.1457-10.1457178.9414],ε=289.0811.
Thus, according to Theorem1, the system under consideration is globally asymptotically stable.Figure1 shows the trajectories of the two state variables for the present example with (51a)d1=d2=1,F=sin⁡π(i+j)180,(51b)x(i,j)=0,∀i≥30,j=-1,0,x(i,j)=0,∀j≥30,i=-1,0,(51c)x(i,0)=[11],0≤i<30,x(0,j)=[11],0≤j<30.Trajectories of the two state variables (Example1).
(a)(b)Likewise, the trajectory trace of the present system has been carried out for a number of arbitrary (randomly generated) initial conditions and it supports the fact (which has been arrived at via Theorem1) that the system is asymptotically stable.Example 2.
Consider the 2D discrete system (34a)–(34d), (2a) where A1 and A2 are given by (49a) and
(52)E1=E2=[0.010],H=[00.01],L=1.
Using the Matlab LMI Toolbox [38, 39], it turns out that [15, 21] fail to determine the global asymptotic stability of the present system. On the other hand, LMI (35) is feasible for the following values of unknown parameters.
(53)α12=61.6039,α21=237.6665,β12=104.1096,β21=536.3285,P=[181.5949-331.7387-331.73871179.5],P1=[25.8494-8.0428-8.0428244.9546],ε=333.0229.
Therefore, Corollary 1 affirms the global asymptotic stability for the present system.It may be noted that, in the absence of state-delay, the system of Example1 reduces to that of Example 2. Whereas [15, 21] fail to determine the global asymptotic stability of the system of Example 2, the present approach succeeds to establish the global asymptotic stability of the same system even in the presence of state-delay.


## 6. Conclusion

Sufficient conditions in terms of LMIs are established for the global asymptotic stability of a class of 2D discrete uncertain state-delayed systems using generalized overflow nonlinearities. The presented approach turns out to be a generalization over the results reported in [14, 15, 21, 24]. Pertaining to the saturation nonlinearities, the presented criteria turn out to be less restrictive than previously reported criteria [14, 15, 21].



---

*Source: 271515-2011-10-23.xml*