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

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.

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. Section 2 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:𝐑^𝑛×𝑛: set of 𝑛×𝑛 real matrices,𝐑^𝑛: 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,𝐆^𝑇: 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,…,𝑎_𝑛}: diagonal matrix with diagonal elements 𝑎₁,𝑎₂,…,𝑎_𝑛,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 by𝐱(𝑖==+𝐟1𝑓(,𝐲𝑗(+𝑖,𝑗1)))₁𝐲(𝑖,𝑗)=𝐀₁𝑦₁(𝑖,𝑗)𝑓₂𝑦₂(𝑖,𝑗)⋯𝑓_𝑛𝑦_𝑛(𝑖,𝑗)^𝑇,(1a)+𝐀+Δ𝐀₁𝐱(𝑖+1,𝑗)₂+𝐀+Δ𝐀₂𝐱(𝑖,𝑗+1)_𝑑1+𝐀_𝑑2+Δ𝐀_𝑑1𝐱𝑖+1,𝑗−𝑑₁=𝑦+Δ𝐀_𝑑2𝐱𝑖−𝑑₂,𝑗+1₁(𝑖,𝑗)𝑦₂(𝑖,𝑗)⋯𝑦_𝑛(𝑖,𝑗)^𝑇,(1b)where 0≤𝑖,𝑗∈𝐙are horizontal and vertical coordinates, respectively; 𝐱(𝑖,𝑗)∈𝐑^𝑛 is the local state vector; 𝐀₁, 𝐀₂, 𝐀_𝑑1, 𝐀_𝑑2 are the known real constant 𝑛×𝑛 matrices; Δ𝐀₁,Δ𝐀₂,Δ𝐀_𝑑1,Δ𝐀_𝑑2 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 𝑓𝐿≤𝑓_𝑘𝑦_𝑘(𝑖,𝑗)≤𝐿₁,𝑦_𝑘(𝑖,𝑗)>1,𝑘=1,2,…,𝑛_𝑘−𝐿₂𝑦_𝑘(𝑖,𝑗)=𝑦_𝑘(𝑖,𝑗),−1≤𝑦_𝑘(𝑖,𝑗)≤1,𝑘=1,2,…,𝑛≤𝑓_𝑘𝑦_𝑘(𝑖,𝑗)≤−𝐿,𝑦_𝑘(𝑖,𝑗)<−1,𝑘=1,2,…,𝑛(2a) where −1≤𝐿≤1𝐿≤𝐿₁≤1𝐿≤𝐿₂≤1.(2b)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 Δ𝐀==Δ𝐇𝐀𝐅𝐄₁₁Δ𝐀₂Δ𝐀_𝑑1Δ𝐀_𝑑2𝐇𝐅𝐄₂𝐇𝐅𝐄_𝑑1𝐇𝐅𝐄_𝑑2,(3a) where 𝐇,𝐄₁,𝐄₂,𝐄_𝑑1,𝐄_𝑑2 are known real constant matrices with appropriate dimensions and 𝐅 is an unknown real matrix satisfying𝐅^𝑇𝐅≤𝐈.(3b)

In the uncertainty structure given by (3a)-(3b), the matrices 𝐇,𝐄₁,𝐄₂,𝐄_𝑑1,𝐄_𝑑2 characterize how the uncertain parameters in 𝐅enter the state matrices. The matrix𝐅 can always be restricted as (3b) by appropriately selecting 𝐇,𝐄₁,𝐄₂,𝐄_𝑑1,𝐄_𝑑2. 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 that𝐱𝐱((𝑖𝑖,,𝑗𝑗))==𝟎𝟎,,∀∀𝑗𝑖≥≥𝐿𝐾,,𝑗𝑖==−−𝑑𝑑₁₂,−𝑑₁+1,…,0;,−𝑑₂+1,…,0.(4)

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 for 𝐿−01≤≤𝐿𝐿≤<10𝐿𝐿≤≤𝐿𝐿₁₁≤1𝐿≤𝐿₂≤1,(5)≤1𝐿≤𝐿₂≤1.(6)

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=𝐌^𝑇;then 𝐌+𝚺𝐅𝚪+𝚪^𝑇𝐅^𝑇𝚺^𝑇<𝟎(7) for all 𝐅^𝑇𝐅≤𝐈 if and only if there exists a positive scalar 𝜀 such that 𝐌+𝜀⁻¹𝚺𝚺^𝑇+𝜀𝚪^𝑇𝚪<𝟎.(8)

Next, suppose 𝐂=[𝑐_𝑘𝑙]∈𝐑^𝑛×𝑛 is a matrix characterized by𝑐_𝑘𝑘=^𝑛_𝑙=1_,𝑙≠𝑘𝛼_𝑘𝑙𝑐+𝛽_𝑘𝑙,𝑘=1,2,…,𝑛,(9a)_𝑘𝑙𝛼=𝐿𝛼_𝑘𝑙−𝛽_𝑘𝑙,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙),(9b)_𝑘𝑙>0,𝛽_𝑘𝑙>0,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙),(9c)where it is understood that, for 𝑛=1, 𝐂 corresponds to a scalar𝛾>0. Thus, corresponding to 𝑛=3, the matrix 𝐂 takes the form𝐂=⎡⎢⎢⎢⎢⎣𝛼¹²+L𝛽𝛼₁₂₂₁+𝛼₁₃+𝛽₁₃L𝛼₁₂−𝛽₁₂L𝛼₁₃−𝛽₁₃L𝛼₃₁−𝛽₂₁𝛼₂₁+𝛽₂₁+𝛼₂₃+𝛽₂₃L𝛼₂₃−𝛽₂₃−𝛽₃₁L𝛼₃₂−𝛽₃₂𝛼₃₁+𝛽₃₁+𝛼₃₂+𝛽₃₂⎤⎥⎥⎥⎥⎦,(10)where 𝛼_𝑘𝑙>0,𝛽_𝑘𝑙>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 scalars𝛼_𝑘𝑙,𝛽_𝑘𝑙(𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙)), 𝜀, and𝑛×𝑛 positive definite symmetric matrices 𝐏,𝐏_𝟏,𝐐_𝟏,𝐐_𝟐 such that⎡⎢⎢⎢⎢⎣−𝐉𝐂+𝜀𝐄^𝑇𝐄𝐀^𝑇𝐂𝟎_𝑇𝟎𝐀𝐏−𝐇𝐂_𝑇−𝐂^𝑇𝐂^𝑇𝐇𝐂−𝜀𝐈⎤⎥⎥⎥⎥⎦<𝟎,(11a) where 𝐉=⎡⎢⎢⎢⎢⎢⎢⎣𝐏−𝐏¹𝟎−𝐐₁𝐏₁𝟎𝟎𝟎𝟎−𝟎𝐐₂𝐐𝟎₁𝟎𝟎𝟎𝟎𝐐𝟎₂𝐀=𝐀₁⎤⎥⎥⎥⎥⎥⎥⎦,(11b)𝐄=𝐄₁𝐀₂𝐀_𝑑1𝐀_𝑑2,(11c)𝐄₂𝐄_𝑑1𝐄_𝑑2,(11d)and 𝐂=[𝑐_𝑘𝑙]∈𝐑^𝑛×𝑛 is defined by (9a)–(9c).

Proof. Consider a 2D quadratic Lyapunov-Krasovskii function [24]𝑉(𝐱(𝑖,𝑗))=𝑉_ℎ(𝐱(𝑖,𝑗))+𝑉_𝑣(𝐱(𝑖,𝑗)),(12a) where 𝑉_ℎ(𝐱(𝑖,𝑗))=+𝐱^𝑇(𝑖,𝑗)𝐏₁𝐱(𝑖,𝑗)_−1𝑙=_−𝑑2𝑉𝐱^𝑇(𝑖+𝑙,𝑗)𝐐₂𝐱(𝑖+𝑙,𝑗),(12b)_𝑣(𝐱(𝑖,𝑗))=+𝐱^𝑇(𝑖,𝑗)𝐏−𝐏₁𝐱(𝑖,𝑗)_−1𝑙=_−𝑑1𝐱^𝑇(𝑖,𝑗+𝑙)𝐐₁𝐱(𝑖,𝑗+𝑙).(12c)Application of (12a)–(12c) to system (1a)-(1b) yields Δ𝑉(𝐱(𝑖,𝑗))=−𝑉_ℎ𝑉(_ℎ𝐱(𝑖+1,𝑗+1))+𝑉_𝑣(𝐱(𝑖+1,𝑗+1))(𝐱(𝑖,𝑗+1))−𝑉_𝑣(𝐱(𝑖+1,𝑗)).(13) It follows from (1a), (12a)–(12c), and (13) Δ𝑉(𝐱(𝑖,𝑗))=−𝐟^𝑇𝐱(_𝑇𝐲(𝑖,𝑗))𝐏𝐟(𝐲(𝑖,𝑗))−𝐱_𝑇(𝑖,𝑗+1)𝐏₁−𝐐₂𝐱(𝑖,𝑗+1)−𝐱_𝑇𝑖−𝑑₂,𝑗+1𝐐₂𝐱𝑖−𝑑₂,𝑗+1−𝐱_𝑇(𝑖+1,𝑗)𝐏−𝐏₁−𝐐₁𝐱(𝑖+1,𝑗)𝑖+1,𝑗−𝑑₁𝐐₁𝐱𝑖+1,𝑗−𝑑₁.(14)
Now consider the quantity “𝛿” given by 𝛿=^𝑛_𝑘=1×2𝑦_𝑘(𝑖,𝑗)−𝑓_𝑘𝑦_𝑘(𝑖,𝑗)_𝑛𝑙=_{1,𝑙≠𝑘}𝛼_𝑘𝑙+𝐿𝛼+_𝑘𝑙𝛽_𝑘𝑙𝑓_𝑘𝑦_𝑘(𝑖,𝑗)=𝐲_𝑇−𝛽_𝑘𝑙𝑓_𝑙𝑦_𝑙(𝑖,𝑗)−𝐟_𝑇(𝑖,𝑗)𝐂𝐟(𝐲(𝑖,𝑗))+𝐟^𝑇(𝐲(𝑖,𝑗))𝐂^𝑇𝐲(𝑖,𝑗)(𝐲(𝑖,𝑗))𝐂+𝐂^𝑇𝐟(𝐲(𝑖,𝑗)).(15) It is clear that, owing to (2a), (5), and (9c), 𝛿 is nonnegative. (14) can be rearranged as Δ𝑉(𝐱(𝑖,𝑗))=−𝐱̂^𝑇(𝑖,𝑗)𝐙𝐱̂(𝑖,𝑗)−𝛿,(16) where𝐱̂^𝑇(𝑖,𝑗)=𝐱^𝑇(𝑖+1,𝑗)𝐱^𝑇(𝑖,𝑗+1)𝐱^𝑇𝑖+1,𝑗−𝑑₁𝐱^𝑇𝑖−𝑑₂,𝑗+1𝐟^𝑇(𝐲(𝑖,𝑗)),(17)𝐙=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐏−𝐏¹𝟎−𝐐₁𝐏₁𝟎𝟎𝟎−𝐀₁+Δ𝐀₁^𝑇𝐂𝟎−𝟎𝐐₂𝐐𝟎₁𝟎−𝐀₂+Δ𝐀₂^𝑇𝐂𝟎𝟎𝟎⋅𝐐𝟎₂−𝐀_𝑑1+Δ𝐀_𝑑1^𝑇𝐂−𝐂_𝑇−𝐀_𝑑2+Δ𝐀_𝑑2^𝑇𝐂𝐀₁+Δ𝐀₁−𝐂^𝑇𝐀₂+ΔA₂−𝐂^𝑇𝐀_𝑑1+Δ𝐀_𝑑1−𝐂^𝑇𝐀_𝑑2+Δ𝐀_𝑑2𝐂+𝐂^𝑇−𝐏⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦(1.8)Observe that, if𝐙>𝟎, thenΔ𝑉(𝐱(𝑖,𝑗))≤0. From (13) and condition Δ𝑉(𝐱(𝑖,𝑗))≤0, it follows that 𝑉_ℎ(𝐱≤(𝑖𝑉+_ℎ1,𝑗+1))+𝑉_𝑣(𝐱(𝑖+1,𝑗+1))(𝐱(𝑖,𝑗+1))+𝑉_𝑣(𝐱(𝑖+1,𝑗)),(19) 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 𝐷(𝑟)≜{(𝑖,𝑗)∶𝑖+𝑗=𝑟,𝑖≥0,𝑗≥0}.(20) For any nonnegative integer 𝑟≥ max {𝐾,𝐿}, it follows from (19) and the initial condition (4) that _(𝑖+𝑗)_{∈𝐷(𝑟+1)}=𝑉(𝐱(𝑖,𝑗))_{(𝑖+𝑗)∈𝐷(𝑟+1)}=𝑉_ℎ𝑉_ℎ(𝐱(𝑖,𝑗))+𝑉_𝑣(𝐱(𝑖,𝑗))+𝑉(_ℎ𝐱(𝑟+1,0))+𝑉_ℎ(𝐱(𝑟,1))+𝑉_ℎ(𝐱(𝑟−1,2))+⋯+𝑉_ℎ(𝐱(2,𝑟−1))+𝑉_𝑣(𝐱(1,𝑟))+𝑉_ℎ(𝐱(0,𝑟+1))+𝑉_𝑣(𝐱(𝑟+1,0))+𝑉_𝑣(𝐱(𝑟,1))+𝑉_𝑣(𝐱(𝑟−1,2))+⋯+𝑉_𝑣(𝐱(2,𝑟−1))≤𝑉_ℎ(𝐱(1,𝑟))+𝑉_𝑣(𝐱(0,𝑟+1))+𝑉(_ℎ𝐱(𝑟+1,0))+𝑉_ℎ(𝐱(𝑟−1,1))+𝑉_ℎ(𝐱(𝑟−2,2))+⋯+𝑉_ℎ(𝐱(1,𝑟−1))+𝑉_𝑣(𝐱(0,𝑟))+𝑉_ℎ(𝐱(0,𝑟+1))+𝑉_𝑣(𝐱(𝑟+1,0))+𝑉_𝑣(𝐱(𝑟,0))+𝑉_𝑣(𝐱(𝑟−1,1))+⋯+𝑉_𝑣(𝐱(2,𝑟−2))+𝑉_ℎ(𝐱(1,𝑟−1))+𝑉_𝑣(𝐱(0,𝑟+1))−𝑉_ℎ(𝐱(𝑟,0))+𝑉_𝑣(𝐱(0,𝑟))=(𝐱(𝑟,0))−𝑉_𝑣(𝐱(0,𝑟))_{(𝑖+𝑗)∈𝐷(𝑟)}𝑉(𝐱(𝑖,𝑗)),(21) where the equality sign holds only when _(𝑖+𝑗_{)∈𝐷(𝑟)}𝑉(𝐱(𝑖,𝑗))=0.(22) 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 lim_{𝑟→∞(𝑖+𝑗}_{)∈𝐷(𝑟)}𝑉(𝐱(𝑖,𝑗))=0.(23) Consequently, lim_{𝑖+𝑗→∞}‖𝐱(𝑖,𝑗)‖=𝟎.(24) 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:𝐌+𝐇𝐅𝐄+𝐄^𝑇𝐅^𝑇𝐇^𝑇<𝟎,(25a) where 𝐌=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝐏+𝐏¹𝟎+𝐐₁−𝐏₁𝟎𝟎𝟎𝐀^𝑇1𝐂𝟎𝟎+𝐐₂−𝟎𝐐₁𝟎𝐀^𝑇2𝐂𝟎𝟎𝟎−𝟎𝐐₂𝐀^𝑇𝑑₁𝐂𝐂_𝑇𝐀^𝑇𝑑₂𝐂𝐀₁𝐂^𝑇𝐀₂𝐂^𝑇𝐀_𝑑1𝐂^𝑇𝐀_𝑑2𝐏−𝐂−𝐂^𝑇⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,(25b)𝐇^𝑇=𝟎𝟎𝟎𝟎𝐇^𝑇𝐂,(25c)𝐄=𝐄₁𝐄₂𝐄_𝑑1𝐄_𝑑2𝟎.(25d)By Lemma 1, (25a)–(25d) is equivalent to 𝐌+𝜀⁻¹𝐇𝐇^𝑇+𝜀𝐄^𝑇𝐄<𝟎,(26) 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,𝛼_𝑘𝑙,𝛽_𝑘𝑙(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,…,𝑑_𝑛), and a positive scalar 𝜀, satisfying⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝐒−2𝟎𝜀𝐿𝐄^𝑇𝐄(1−𝐿)𝟎₂√−2𝐿𝐀^𝑇𝐃𝟎√2𝐿𝐃+𝐏(√1−+2𝐿𝐿)𝐃𝟎−2𝐿𝐃𝐀(1+𝐿)𝟎√−𝟎2𝐿𝐃𝐇−_T𝐃𝐃𝐇𝐃−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦<𝟎,(27a) where 𝐒=⎡⎢⎢⎢⎢⎢⎢⎣𝐏−𝐏¹𝟎−𝐐₁𝐏₁𝟎𝟎𝟎𝟎−𝟎𝐐₂𝐐𝟎₁𝟎𝟎𝟎𝟎𝐐𝟎₂⎤⎥⎥⎥⎥⎥⎥⎦.(27b)

Proof. Consider the quantity 𝜙 given by 𝜙=^𝑛_𝑘=1×−2𝑑𝐿_𝑘𝑦_𝑘𝑦_𝑘(𝑖,𝑗)−𝑓_𝑘𝑦_𝑘(𝑖,𝑗)=(1+𝐿)𝐲(_𝑇𝑖,𝑗)+𝑓_𝑘𝑦_𝑘(𝑖,𝑗)+(1+𝐿)𝐟(_𝑇𝑖,𝑗)𝐃𝐟(𝐲(𝑖,𝑗))−2𝐟_𝑇(𝐲(𝑖,𝑗))𝐃𝐲(𝑖,𝑗)−2𝐿𝐲(𝐲_𝑇(𝑖,𝑗))𝐃𝐟(𝐲(𝑖,𝑗))(𝑖,𝑗)𝐃𝐲(𝑖,𝑗).(28) Observe that, for 𝐿∈[−1,0), 𝜙 is nonnegative [10]. Equation (14) takes the form Δ𝑉(𝐱(𝑖,𝑗))=−𝐱̂^𝑇(𝑖,𝑗)𝐆𝐱̂(𝑖,𝑗)−𝜙,where ^{⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝐆=𝐏−𝐏1−𝐐1𝐀+2𝐿1+Δ𝐀1𝑇𝐃𝐀1+Δ𝐀1𝐀2𝐿1+Δ𝐀1𝑇𝐃𝐀2+Δ𝐀2𝐀2𝐿1+Δ𝐀1𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀2𝐿2+Δ𝐀2𝑇𝐃𝐀1+Δ𝐀1𝐏1−𝐐2𝐀+2𝐿2+Δ𝐀2𝑇𝐃𝐀𝟐+Δ𝐀2𝐀2𝐿2+Δ𝐀2𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀2𝐿d1+Δ𝐀𝑑1𝑇𝐃𝐀1+Δ𝐀1𝐀2𝐿𝑑1+Δ𝐀𝑑1𝑇𝐃𝐀2+Δ𝐀2𝐐1𝐀+2𝐿𝑑1+Δ𝐀𝑑1𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀1+Δ𝐀1𝐀2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀2+Δ𝐀2𝐀2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀−(𝟏+𝐿)𝐃1+Δ𝐀1𝐀−(𝟏+𝐿)𝐃2+Δ𝐀2𝐀−(1+𝐿)𝐃𝑑1+Δ𝐀𝑑12𝐿}_{A1+Δ𝐀1𝑇𝐃𝐀d2+Δ𝐀𝑑2𝐀−(1+𝐿)1+Δ𝐀1𝑇𝐃𝐀2𝐿2+Δ𝐀2𝑇𝐃𝐀𝑑2+Δ𝐀𝑑2𝐀−(1+𝐿)2+Δ𝐀2𝑇𝐃𝐀2𝐿𝑑1+Δ𝐀𝑑1𝑇𝐃𝐀𝑑2+Δ𝐀𝑑2𝐀−(1+𝐿)𝑑1+Δ𝐀𝑑1𝑇𝐃𝐐2𝐀+2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀𝑑2+Δ𝐀𝑑2𝐀−(1+𝐿)𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀−(1+𝐿)𝐃𝑑2+Δ𝐀𝑑2⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.2𝐃−𝐏}(29) 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 ⎡⎢⎢⎢⎢⎢⎢⎢⎣^{𝐏−𝐏1−𝐐1𝐀𝟎𝟎𝟎𝟎−𝜻1+Δ𝐀1𝑇𝐃𝟎𝐏}_{1−𝐐2𝐀𝟎𝟎𝟎−𝜻2+Δ𝐀2𝑇𝐃𝟎𝟎𝐐1𝐀𝟎𝟎−𝜻𝑑1+Δ𝐀𝑑1𝑇𝐃𝟎𝟎𝟎𝐐2𝐀𝟎−𝜻𝑑2+Δ𝐀𝑑2𝑇𝐃𝟎𝟎𝟎𝟎−(1−𝐿)22𝐿𝐃−𝐏−(1+𝐿)𝜻𝐃𝐀−𝜻𝐃1+Δ𝐀1𝐀−𝜻𝐃2+Δ𝐀2𝐀−𝜻𝐃𝑑1+Δ𝐀𝑑1𝐀−𝜻𝐃𝑑2+Δ𝐀𝑑2−(1+𝐿)𝜻𝐃𝐃}⎤⎥⎥⎥⎥⎥⎥⎥⎦>𝟎.(30) where 𝜻 denotes √−2𝐿.
Applying (3a), (30) can be rewritten in the following form:𝐖+𝐇𝐅𝐄+𝐄^𝑇𝐅^𝑇𝐇^𝑇<𝟎,(31a) where𝐖=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−𝐏−𝐏¹𝟎−𝐐₁−𝐏₁𝟎𝟎𝟎𝟎√−2𝐿𝐀^𝑇1𝐃𝟎𝟎−𝐐₂−𝟎𝐐₁𝟎𝟎√−2𝐿𝐀^𝑇2𝐃𝟎𝟎𝟎−𝟎𝐐₂𝟎√−2𝐿𝐀^𝑇𝑑₁𝐃𝟎𝟎𝟎𝟎(1−𝐿)𝟎₂√−2𝐿𝐀^𝑇𝑑₂𝐃√2L𝐃+𝐏(√1−+2𝐿𝐿)𝐃−2𝐿𝐃𝐀₁√−2𝐿𝐃𝐀₂√−2𝐿𝐃𝐀_𝑑1√−2𝐿𝐃𝐀_𝑑2(1+𝐿)√−2𝐿𝐃−𝐃⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(31b)𝐇^𝑇=𝟎𝟎𝟎𝟎𝟎𝐇^𝑇𝐃,(31c)𝐄=√−2𝐿𝐄₁√−2𝐿𝐄₂√−2𝐿𝐄_𝑑1√−2𝐿𝐄_𝑑2𝟎𝟎.(31d)By Lemma 1, (31a)–(31d) is equivalent to 𝐖+𝜀⁻¹𝐇𝐇^𝑇+𝜀𝐄^𝑇𝐄<𝟎,(32) 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 by𝐿𝑦_2𝑘𝑓_𝑘(0)=0,(𝑖,𝑗)≤𝑓_𝑘𝑦_𝑘(𝑖,𝑗)𝑘𝑦_𝑘=(𝑖1,,𝑗2),≤…𝑦,^2𝑘𝑛(,𝑖,𝑗),(33a)−1≤𝐿<0.(33b)

Clearly, the quantity 𝜙 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, Theorem 3 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 becomes𝐱(𝑖+=1𝐟,𝑗+𝐀1₁)+Δ𝐀₁𝐀𝐱(=𝑖+𝐀1,₁𝑗)+𝐀₂+Δ𝐀₂𝐱(𝑖,𝑗+1),(34a)Δ𝐀₁𝐀₂,(34b)Δ𝐀₂=𝐅𝐇𝐅𝐄,𝐄=𝐄₁𝐄₂,(34c)_𝑇𝐅≤𝐈.(34d)

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 𝛼_𝑘𝑙,𝛽_𝑘𝑙(𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙)),𝜀, and 𝑛×𝑛 positive definite symmetric matrices 𝐏and𝐏₁such that ⎡⎢⎢⎢⎢⎢⎢⎣−⎡⎢⎢⎣𝐏−𝟎𝐏₁𝐏𝟎₁⎤⎥⎥⎦+𝜀𝐄_𝑇𝐂𝐄𝐀^𝑇𝐂𝟎_𝑇𝟎𝐀𝐏−𝐇𝐂_𝑇−𝐂^𝑇𝐂^𝑇𝐇𝐂−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎦<𝟎,(35) where 𝐂=[𝑐_𝑘𝑙]∈𝐑^𝑛×𝑛 is defined by (9a)–(9c).

Proof. Pertaining to the system described by (34a)–(34d), (2a), and (5), choosing 𝐀_𝑑1=𝐀_𝑑2=𝟎 and 𝐄_𝑑1=𝐄_𝑑2=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 Theorem 2.

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 ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−⎡⎢⎢⎣𝐏−𝟎𝐏₁𝐏𝟎₁⎤⎥⎥⎦−2𝜀𝐿𝐄_𝑇𝟎𝐄(1−𝐿)𝟎₂√−2𝐿𝐀^𝑇𝐃𝟎√2𝐿𝐃+𝐏(√1−+2𝐿𝐿)𝐃𝟎−2𝐿𝐃𝐀(1+𝐿)𝟎√−𝟎2𝐿𝐃𝐇−_𝑇𝐃𝐃𝐇𝐃−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<𝟎.(36)

Proof. For the system described by (34a)–(34d), (2a), and (6), substituting 𝐀_𝑑1=𝐀_𝑑2=𝟎 and 𝐄_𝑑1=𝐄_𝑑2=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,𝛼_𝑘𝑙,𝛽_𝑘𝑙(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 ⎡⎢⎢⎢⎢⎢⎢⎣−⎡⎢⎢⎣𝐏−𝟎𝐏₁𝐏𝟎₁⎤⎥⎥⎦+𝜀𝐄_𝑇𝐃𝟎𝐀𝐄𝐏𝐇𝐀−^𝑇_𝑇2𝐃𝐃𝐃𝟎𝐇𝐃−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎦<𝟎.(37)

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 matrices𝐏_𝟏, 𝐏=[𝑝_𝑘𝑙] such that𝑝_𝑘𝑘≥^𝑛_𝑙=1_,𝑙≠𝑘⎡⎢⎢⎢⎢⎢⎢⎣−𝐏𝐏𝐀₁||𝑝_𝑘𝑙||,𝑘=1,2,…,𝑛,(38a)𝐀𝐏𝐀₂𝐏𝐇_𝑇1𝐀_𝑇2𝐏−𝐏+𝐏₁+𝜀𝐄^𝑇1𝐄₁𝜀𝐄^𝑇1𝐄₂𝟎𝐇_𝑇𝐏𝜀𝐄^𝑇2𝐄₁−𝐏₁+𝜀𝐄^𝑇2𝐄₂𝟎𝐏𝟎𝟎−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎦<𝟎.(38b)

Theorem 6 (Singh [14]). Suppose there exists a matrix 𝐖=[𝑤_𝑖𝑗]∈𝐑^𝑛×𝑛, positive definite diagonal matrices 𝐔=diag(𝑢₁,𝑢₂,…,𝑢_𝑛), 𝐕=diag(𝑣₁,𝑣₂,…,𝑣_𝑛), 𝑛×𝑛 positive definite symmetric matrices 𝐏 and 𝐏₁, a scalar 𝜀 > 0 satisfying⎡⎢⎢⎢⎢⎢⎢⎣−⎡⎢⎢⎣𝐏−𝟎𝐏₁𝐏𝟎₁⎤⎥⎥⎦+𝜀𝐄_𝑇𝐔+𝐖_𝑇𝐄𝐀^𝑇(𝐔+𝐕𝐖)𝟎𝟎𝐕𝐀𝐏−−2𝐇𝐔_𝑇−𝐕𝐖−𝐖^𝑇𝐕−𝐔+𝐖^𝑇𝐕𝐇(𝐔+𝐕𝐖)−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎦<𝟎,(39a)where 𝑤𝑤_𝑘𝑘=𝑛−1,𝑘=1,2,…,𝑛,(39b)_𝑘𝑙𝜆=𝜆_𝑘,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙),(39c)_𝑘∈(1,−1,0),𝑘=1,2,…,𝑛.(39d)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 𝐱(𝑖+1,𝑗+1)=+𝐀𝐀₁𝐱_d1(𝑖+1,𝑗)+𝐀₂𝐱(𝑖,𝑗+1)𝐱𝑖+1,𝑗−𝑑₁+𝐀_d2𝐱𝑖−𝑑₂,𝑗+1.(40) 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 𝐘=⎡⎢⎢⎢⎢⎢⎢⎣𝐏−𝐏¹𝟎−𝐐₁𝐏₁𝟎𝟎𝟎𝟎−𝟎𝐐₂𝐐𝟎₁𝟎𝟎𝟎𝟎𝐐𝟎₂−⎡⎢⎢⎢⎢⎢⎢⎣𝐀_𝑇1⎤⎥⎥⎥⎥⎥⎥⎦𝐀^𝑇2𝐀^𝑇𝑑₁𝐀^𝑇𝑑₂⎤⎥⎥⎥⎥⎥⎥⎦𝐏𝐀₁𝐀₂𝐀_𝑑1𝐀_𝑑2>𝟎.(41)

Now, we have the following propositions.

Proposition 1. Corollary 1 implies Theorem 4.

Proof. With 𝛼_𝑘𝑙=𝛽_𝑘𝑙(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. Corollary 1 implies Theorem 5.

Proof. It is not difficult to show that, with 𝐂=𝐂^𝑇=𝐏,(42)(35) reduces to (38b). Condition (42) along with (9a)–(9c) and 𝐿=𝐿₁=𝐿₂=1 yields𝑝_𝑘𝑘=𝑐_𝑘𝑘=^𝑛_𝑙=1_,𝑙≠𝑘𝛼_𝑘𝑙𝑝+𝛽_𝑘𝑙,𝑘=1,2,…,𝑛,(43a)_𝑘𝑙=𝑐_𝑘𝑙=𝑐_𝑙𝑘=𝛼_𝑘𝑙−𝛽_𝑘𝑙,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙),(43b)where 𝛼_𝑘𝑙>0,𝛽_𝑘𝑙>0,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙). Using (43a)-(43b), we obtain ^𝑛_𝑙_{=1,𝑙≠𝑘}||𝑝_𝑘𝑙||=^𝑛_𝑙=_{1,𝑙≠𝑘}<_𝑛||𝛼_𝑘𝑙−𝛽_𝑘𝑙||_𝑙=_{1,𝑙≠𝑘}||𝛼_𝑘𝑙+𝛽_𝑘𝑙||=𝑝_𝑘𝑘,𝑘=1,2,…,𝑛.(44) Thus, condition (42) leads to (38a).

Proposition 3. Corollary 1 implies Theorem 6.

Proof. Using Schur complement, (39a) can equivalently be expressed as⎡⎢⎢⎢⎢⎢⎢⎣−⎡⎢⎢⎣𝐏−𝟎𝐏₁𝐏𝟎₁⎤⎥⎥⎦+𝜀𝐄_𝑇𝐔+𝐖_𝑇𝐄𝐀^𝑇(𝐔+𝐕𝐖)𝟎𝟎𝐕𝐀𝐏−2𝐇𝐔_𝑇−𝐕𝐖−𝐖^𝑇𝐕𝐔+𝐖^𝑇𝐕𝐇(𝐔+𝐕𝐖)−𝜀𝐈⎤⎥⎥⎥⎥⎥⎥⎦<𝟎.(45)
Now, consider the case where 𝑛=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 where 𝑛≥2. In this case, let us choose the positive scalars 𝛼_𝑘𝑙 and 𝛽_𝑘𝑙 given by (9c) as𝛼_𝑘𝑙=𝑢^𝑘+(𝑛−1)𝑣_𝑘2(𝑛−1)+𝑣^𝑘𝜆_𝑘2,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙),(46a)𝛽_𝑘𝑙=𝑢^𝑘+(𝑛−1)𝑣_𝑘2(𝑛−1)−𝑣^𝑘𝜆_𝑘2,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙),(46b)where 𝑢_𝑘>0,𝑣_𝑘>0,𝜆_𝑘∈(1,−1,0),𝑘=1,2,…,𝑛. In view of (46a)-(46b) and 𝐿=𝐿₁=𝐿₂=1, one obtains from (9a)–(9c) 𝑐𝑐_𝑘𝑘=𝑢_𝑘+(𝑛−1)𝑣_𝑘,𝑘=1,2,…,𝑛,(47a)_𝑘𝑙=𝑣_𝑘𝜆_𝑘,𝑘,𝑙=1,2,…,𝑛(𝑘≠𝑙).(47b)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 matrix 𝐂 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 Theorem 6, one is required to test (39a) (possibly) for all possible combinations of 𝜆_𝑘’s (𝑘=1,2,…,𝑛) belonging to (1, −1, 0). For an 𝑛th-order system, the number of such possible combinations of 𝜆_𝑘’s (𝑘=1,2,…,𝑛) is 3^𝑛. Therefore, (39a)–(39d) is computationally more complex than (35).

Proposition 4. Theorem 1 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=Δ𝐀_𝑑1=Δ𝐀_𝑑2=𝟎. In this case, (14) can be rearranged as Δ𝑉(𝐱(𝑖,𝑗))=−⎡⎢⎢⎢⎢⎢⎢⎣𝐱𝐱𝑖𝐱+((𝑖𝑖,1+𝑗,𝑗+1,−1𝑗))𝑑₁𝐱𝑖−𝑑₂,𝑗+1⎤⎥⎥⎥⎥⎥⎥⎦^𝑇𝐘⎡⎢⎢⎢⎢⎢⎢⎣𝐱𝐱𝑖𝐱+((𝑖𝑖,1+𝑗,𝑗+1,−1𝑗))𝑑₁𝐱𝑖−𝑑₂,𝑗+1⎤⎥⎥⎥⎥⎥⎥⎦,(48) 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) with𝐀𝐀₁=⎡⎢⎢⎣10..015−02⎤⎥⎥⎦,𝐀₂=⎡⎢⎢⎣0.00010.0001⎤⎥⎥⎦,(49a)_𝑑1𝐄=𝐀_𝑑2=⎡⎢⎢⎣0.00010.0001⎤⎥⎥⎦,𝐇=⎡⎢⎢⎣0.001⎤⎥⎥⎦,(49b)₁=𝐄₂=𝐄_𝑑1=𝐄_𝑑2=0.010,𝐿=1.(49c)Using the Matlab LMI Toolbox [38, 39], it has been found that LMI (11a)–(11d) is feasible for the following values of unknown parameters. 𝛽𝛼₁₂=62.4706,𝛼₂₁=298.8216,₁₂𝐏𝐏==1⎡⎢⎢⎣4−62.41345785.7.968,27343𝛽−4₂₁146=81.066.299₁333.5⎤⎥⎥⎦6,76,𝐐=⎡⎢⎢⎣−3107.5.7341997−30187..6744395₁7⎤⎥⎥⎦,𝐐=⎡⎢⎢⎣−1150.7.6143056−18120..5694907₂6⎤⎥⎥⎦,=⎡⎢⎢⎣−1𝜀190=.8.1824785797.0−81171180...91441547⎤⎥⎥⎦,(50)
Thus, according to Theorem 1, the system under consideration is globally asymptotically stable.

Figure 1 shows the trajectories of the two state variables for the present example with 𝑑₁=𝑑₂=1,𝐅=sin𝜋(𝑖+𝑗)𝐱𝐱(𝐱𝐱𝑖(,𝑖(,(𝑗0𝑖𝑗),,)0=𝑗=))𝟎==𝟎,,⎡⎢⎢⎣⎡⎢⎢⎣1111∀⎤⎥⎥⎦⎤⎥⎥⎦∀𝑖𝑗,,≥≥30030≤≤0,,𝑗𝑖𝑗𝑖<==<13−−3800110,.,,00,,,(((555111bac)))

(a)

(b)

Figure 1 

Trajectories of the two state variables (Example 1).

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 Theorem 1) 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 𝐄₁=𝐄₂=0.010,𝐇=⎡⎢⎢⎣0.001⎤⎥⎥⎦,𝐿=1.(52) 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. 𝛽𝛼₁₂=61.6039,𝛼₂₁=237.6665,₁₂𝐏=𝐏=1⎡⎢⎢⎣0−41.31830119.5.679,34897𝛽−3₂₁131=17.975.335₁867.3⎤⎥⎥⎦2,85,=⎡⎢⎢⎣𝜀−25=8..80344392348.022−24894...09452486⎤⎥⎥⎦,(53) 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 Example 1 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].

Acknowledgments

The authors wish to thank Professor Derong Liu and the anonymous reviewers for their constructive comments and suggestions.