Journal of Control Science and Engineering

Journal of Control Science and Engineering / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 271515 | https://doi.org/10.1155/2011/271515

Anurita Dey, Haranath Kar, "LMI-Based Criterion for the Robust Stability of 2D Discrete State-Delayed Systems Using Generalized Overflow Nonlinearities", Journal of Control Science and Engineering, vol. 2011, Article ID 271515, 12 pages, 2011. https://doi.org/10.1155/2011/271515

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

Academic Editor: Derong Liu
Received25 Apr 2011
Accepted18 Jul 2011
Published23 Oct 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.

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 [621].

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 [1316, 2129].

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, 2432].

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 [610, 1216, 18, 19, 2126, 28, 3032, 3436]. Several publications [710, 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 [1316, 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: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=𝑓𝐱(𝑖+1,𝑗+1)=𝐟(𝐲(𝑖,𝑗))1𝑦1𝑓(𝑖,𝑗)2𝑦2(𝑖,𝑗)𝑓𝑛𝑦𝑛(𝑖,𝑗)𝑇,𝐀(1a)𝐲(𝑖,𝑗)=1+Δ𝐀1+𝐀𝐱(𝑖+1,𝑗)2+Δ𝐀2+𝐀𝐱(𝑖,𝑗+1)𝑑1+Δ𝐀𝑑1𝐱𝑖+1,𝑗𝑑1+𝐀𝑑2+Δ𝐀𝑑2𝐱𝑖𝑑2=𝑦,𝑗+11(𝑖,𝑗)𝑦2(𝑖,𝑗)𝑦𝑛(𝑖,𝑗)𝑇,(1b)where 0𝑖,𝑗𝐙are 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 𝐿𝑓𝑘𝑦𝑘(𝑖,𝑗)𝐿1,𝑦𝑘𝑓(𝑖,𝑗)>1,𝑘=1,2,,𝑛𝑘𝑦𝑘(𝑖,𝑗)=𝑦𝑘(𝑖,𝑗),1𝑦𝑘(𝑖,𝑗)1,𝑘=1,2,,𝑛𝐿2𝑓𝑘𝑦𝑘(𝑖,𝑗)𝐿,𝑦𝑘(𝑖,𝑗)<1,𝑘=1,2,,𝑛(2a) where 1𝐿1𝐿𝐿11𝐿𝐿21.(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 [1316, 2124, 2729] as Δ𝐀=Δ𝐀1Δ𝐀2Δ𝐀𝑑1Δ𝐀𝑑2=𝐇𝐅𝐄1𝐇𝐅𝐄2𝐇𝐅𝐄𝑑1𝐇𝐅𝐄𝑑2,(3a) where 𝐇,𝐄1,𝐄2,𝐄𝑑1,𝐄𝑑2 are known real constant matrices with appropriate dimensions and 𝐅 is an unknown real matrix satisfying𝐅𝑇𝐅𝐈.(3b)

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 that𝐱(𝑖,𝑗)=𝟎,𝑖𝐾,𝑗=𝑑1,𝑑1𝐱+1,,0;(𝑖,𝑗)=𝟎,𝑗𝐿,𝑖=𝑑2,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 [4143].

3. Main Result and Its Corollaries

Consider two ranges for 𝐿0𝐿1𝐿𝐿11𝐿𝐿21,(5)1𝐿<0𝐿𝐿11𝐿𝐿21.(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=𝐌𝑇;then 𝐌+𝚺𝐅𝚪+𝚪𝑇𝐅𝑇𝚺𝑇<𝟎(7) for all 𝐅𝑇𝐅𝐈 if and only if there exists a positive scalar 𝜀 such that 𝐌+𝜀1𝚺𝚺𝑇+𝜀𝚪𝑇𝚪<𝟎.(8)

Next, suppose 𝐂=[𝑐𝑘𝑙]𝐑𝑛×𝑛 is a matrix characterized by𝑐𝑘𝑘=𝑛𝑙=1,𝑙𝑘𝛼𝑘𝑙+𝛽𝑘𝑙𝑐,𝑘=1,2,,𝑛,(9a)𝑘𝑙𝛼=𝐿𝑘𝑙𝛽𝑘𝑙𝛼,𝑘,𝑙=1,2,,𝑛(𝑘𝑙),(9b)𝑘𝑙>0,𝛽𝑘𝑙>0,𝑘,𝑙=1,2,,𝑛(𝑘𝑙),(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 form𝛼𝐂=12+𝛽12+𝛼13+𝛽13L𝛼12𝛽12L𝛼13𝛽13L𝛼21𝛽21𝛼21+𝛽21+𝛼23+𝛽23L𝛼23𝛽23L𝛼31𝛽31L𝛼32𝛽32𝛼31+𝛽31+𝛼32+𝛽32,(10)where 𝛼𝑘𝑙>0,𝛽𝑘𝑙>0,𝑘,𝑙=1,2,3(𝑘𝑙).

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(𝑘𝑙)), 𝜀, and𝑛×𝑛 positive definite symmetric matrices 𝐏,𝐏𝟏,𝐐𝟏,𝐐𝟐 such that𝐉+𝜀𝐄𝑇𝐄𝐀𝑇𝐂𝐂𝟎𝑇𝐀𝐏𝐂𝐂𝑇𝐂𝑇𝐇𝟎𝐇𝑇𝐂𝜀𝐈<𝟎,(11a) where 𝐉=𝐏𝐏1𝐐1𝟎𝟎𝟎𝟎𝐏1𝐐2𝟎𝟎𝟎𝟎𝐐1𝟎𝟎𝟎𝟎𝐐2𝐀,(11b)𝐀=1𝐀2𝐀𝑑1𝐀𝑑2𝐄,(11c)𝐄=1𝐄2𝐄𝑑1𝐄𝑑2,(11d)and 𝐂=[𝑐𝑘𝑙]𝐑𝑛×𝑛 is defined by (9a)–(9c).

Proof. Consider a 2D quadratic Lyapunov-Krasovskii function [24]𝑉(𝐱(𝑖,𝑗))=𝑉(𝐱(𝑖,𝑗))+𝑉𝑣(𝐱(𝑖,𝑗)),(12a) where 𝑉(𝐱(𝑖,𝑗))=𝐱𝑇(𝑖,𝑗)𝐏1+𝐱(𝑖,𝑗)1𝑙=𝑑2𝐱𝑇(𝑖+𝑙,𝑗)𝐐2𝑉𝐱(𝑖+𝑙,𝑗),(12b)𝑣(𝐱(𝑖,𝑗))=𝐱𝑇(𝑖,𝑗)𝐏𝐏1+𝐱(𝑖,𝑗)1𝑙=𝑑1𝐱𝑇(𝑖,𝑗+𝑙)𝐐1𝐱(𝑖,𝑗+𝑙).(12c)Application of (12a)–(12c) to system (1a)-(1b) yields Δ𝑉(𝐱(𝑖,𝑗))=𝑉(𝐱(𝑖+1,𝑗+1))+𝑉𝑣(𝐱(𝑖+1,𝑗+1))𝑉(𝐱(𝑖,𝑗+1))𝑉𝑣(𝐱(𝑖+1,𝑗)).(13) It follows from (1a), (12a)–(12c), and (13) Δ𝑉(𝐱(𝑖,𝑗))=𝐟𝑇(𝐲(𝑖,𝑗))𝐏𝐟(𝐲(𝑖,𝑗))𝐱𝑇𝐏(𝑖,𝑗+1)1𝐐2𝐱(𝑖,𝑗+1)𝐱𝑇𝑖𝑑2𝐐,𝑗+12𝐱𝑖𝑑2,𝑗+1𝐱𝑇(𝑖+1,𝑗)𝐏𝐏1𝐐1𝐱(𝑖+1,𝑗)𝐱𝑇𝑖+1,𝑗𝑑1𝐐1𝐱𝑖+1,𝑗𝑑1.(14)
Now consider the quantity “𝛿” given by 𝛿=𝑛𝑘=12𝑦𝑘(𝑖,𝑗)𝑓𝑘𝑦𝑘×(𝑖,𝑗)𝑛𝑙=1,𝑙𝑘𝛼𝑘𝑙+𝛽𝑘𝑙𝑓𝑘𝑦𝑘𝛼(𝑖,𝑗)+𝐿𝑘𝑙𝛽𝑘𝑙𝑓𝑙𝑦𝑙(𝑖,𝑗)=𝐲𝑇(𝑖,𝑗)𝐂𝐟(𝐲(𝑖,𝑗))+𝐟𝑇(𝐲(𝑖,𝑗))𝐂𝑇𝐲(𝑖,𝑗)𝐟𝑇(𝐲(𝑖,𝑗))𝐂+𝐂𝑇𝐟(𝐲(𝑖,𝑗)).(15) It is clear that, owing to (2a), (5), and (9c), 𝛿 is nonnegative. (14) can be rearranged as ̂𝐱Δ𝑉(𝐱(𝑖,𝑗))=𝑇̂(𝑖,𝑗)𝐙𝐱(𝑖,𝑗)𝛿,(16) wherê𝐱𝑇𝐱(𝑖,𝑗)=𝑇(𝑖+1,𝑗)𝐱𝑇(𝑖,𝑗+1)𝐱𝑇𝑖+1,𝑗𝑑1𝐱𝑇𝑖𝑑2𝐟,𝑗+1𝑇(𝐲(𝑖,𝑗)),(17)𝐙=𝐏𝐏1𝐐1𝐀𝟎𝟎𝟎1+Δ𝐀1𝑇𝐂𝟎𝐏1𝐐2𝐀𝟎𝟎2+Δ𝐀2𝑇𝐂𝟎𝟎𝐐1𝐀𝟎𝑑1+Δ𝐀𝑑1𝑇𝐂𝟎𝟎𝟎𝐐2𝐀𝑑2+Δ𝐀𝑑2𝑇𝐂𝐂𝑇𝐀1+Δ𝐀1𝐂𝑇𝐀2+ΔA2𝐂𝑇𝐀𝑑1+Δ𝐀𝑑1𝐂𝑇𝐀𝑑2+Δ𝐀𝑑2𝐂+𝐂𝑇.𝐏(18)Observe that, if𝐙>𝟎, thenΔ𝑉(𝐱(𝑖,𝑗))0. From (13) and condition Δ𝑉(𝐱(𝑖,𝑗))0, it follows that 𝑉(𝐱(𝑖+1,𝑗+1))+𝑉𝑣(𝐱(𝑖+1,𝑗+1))𝑉(𝐱(𝑖,𝑗+1))+𝑉𝑣(𝐱(𝑖+1,𝑗)),(19) where the equality sign holds only when 𝐱(𝑖+1,𝑗)=𝐱(𝑖,𝑗+1)=𝐱(𝑖+1,𝑗𝑑1)=𝐱(𝑖𝑑2,𝑗+1)=𝟎. Let 𝐷(𝑟) denote the set defined by 𝐷(𝑟){(𝑖,𝑗)𝑖+𝑗=𝑟,𝑖0,𝑗0}.(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(𝐱(𝑖,𝑗))+𝑉𝑣(𝐱(𝑖,𝑗))=𝑉(𝐱(𝑟+1,0))+𝑉(𝐱(𝑟,1))+𝑉(𝐱(𝑟1,2))++𝑉(𝐱(2,𝑟1))+𝑉(𝐱(1,𝑟))+𝑉(𝐱(0,𝑟+1))+𝑉𝑣(𝐱(𝑟+1,0))+𝑉𝑣(𝐱(𝑟,1))+𝑉𝑣(𝐱(𝑟1,2))++𝑉𝑣(𝐱(2,𝑟1))+𝑉𝑣(𝐱(1,𝑟))+𝑉𝑣(𝐱(0,𝑟+1))𝑉(𝐱(𝑟+1,0))+𝑉(𝐱(𝑟1,1))+𝑉(𝐱(𝑟2,2))++𝑉(𝐱(1,𝑟1))+𝑉(𝐱(0,𝑟))+𝑉(𝐱(0,𝑟+1))+𝑉𝑣(𝐱(𝑟+1,0))+𝑉𝑣(𝐱(𝑟,0))+𝑉𝑣(𝐱(𝑟1,1))++𝑉𝑣(𝐱(2,𝑟2))+𝑉𝑣(𝐱(1,𝑟1))+𝑉𝑣(𝐱(0,𝑟+1))+𝑉(𝐱(𝑟,0))+𝑉𝑣(𝐱(0,𝑟))𝑉(𝐱(𝑟,0))𝑉𝑣=(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 lim𝑟(𝑖+𝑗)𝐷(𝑟)𝑉(𝐱(𝑖,𝑗))=0.(23) Consequently, lim𝑖+𝑗𝐱(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+𝐄𝑇𝐅𝑇𝐇𝑇<𝟎,(25a) where 𝐌=𝐏+𝐏1+𝐐1𝟎𝟎𝟎𝐀𝑇1𝐂𝟎𝐏1+𝐐2𝟎𝟎𝐀𝑇2𝐂𝟎𝟎𝐐1𝟎𝐀𝑇𝑑1𝐂𝟎𝟎𝟎𝐐2𝐀𝑇𝑑2𝐂𝐂𝑇𝐀1𝐂𝑇𝐀2𝐂𝑇𝐀𝑑1𝐂𝑇𝐀𝑑2𝐏𝐂𝐂𝑇,(25b)𝐇𝑇=𝟎𝟎𝟎𝟎𝐇𝑇𝐂,(25c)𝐄𝐄=1𝐄2𝐄𝑑1𝐄𝑑2𝟎.(25d)By Lemma 1, (25a)–(25d) is equivalent to 𝐌+𝜀1𝐇𝐇𝑇+𝜀𝐄𝑇𝐄<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,𝛼𝑘𝑙,𝛽𝑘𝑙(𝑘,𝑙=1,2,,𝑛(𝑘𝑙)), 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, satisfying𝐒2𝜀𝐿𝐄𝑇𝐄𝟎2𝐿𝐀𝑇𝟎𝐃𝟎(1𝐿)22𝐿𝐃+𝐏(1+𝐿)2𝐿𝐃𝟎2𝐿𝐃𝐀(1+𝐿)2𝐿𝐃𝐃𝐃𝐇𝟎𝟎𝐇T𝐃𝜀𝐈<𝟎,(27a) where 𝐒=𝐏𝐏1𝐐1𝟎𝟎𝟎𝟎𝐏1𝐐2𝟎𝟎𝟎𝟎𝐐1𝟎𝟎𝟎𝟎𝐐2.(27b)

Proof. Consider the quantity 𝜙 given by 𝜙=𝑛𝑘=12𝑑𝑘𝑦𝑘(𝑖,𝑗)𝑓𝑘𝑦𝑘×(𝑖,𝑗)𝐿𝑦𝑘(𝑖,𝑗)+𝑓𝑘𝑦𝑘(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).(28) Observe that, for 𝐿[1,0), 𝜙 is nonnegative [10]. Equation (14) takes the form ̂𝐱Δ𝑉(𝐱(𝑖,𝑗))=𝑇̂(𝑖,𝑗)𝐆𝐱(𝑖,𝑗)𝜙,where 𝐆=𝐏𝐏1𝐐1𝐀+2𝐿1+Δ𝐀1𝑇𝐃𝐀1+Δ𝐀1𝐀2𝐿1+Δ𝐀1𝑇𝐃𝐀2+Δ𝐀2𝐀2𝐿1+Δ𝐀1𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀2𝐿2+Δ𝐀2𝑇𝐃𝐀1+Δ𝐀1𝐏1𝐐2𝐀+2𝐿2+Δ𝐀2𝑇𝐃𝐀𝟐+Δ𝐀2𝐀2𝐿2+Δ𝐀2𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀2𝐿d1+Δ𝐀𝑑1𝑇𝐃𝐀1+Δ𝐀1𝐀2𝐿𝑑1+Δ𝐀𝑑1𝑇𝐃𝐀2+Δ𝐀2𝐐1𝐀+2𝐿𝑑1+Δ𝐀𝑑1𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀1+Δ𝐀1𝐀2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀2+Δ𝐀2𝐀2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀𝑑1+Δ𝐀𝑑1𝐀(𝟏+𝐿)𝐃1+Δ𝐀1𝐀(𝟏+𝐿)𝐃2+Δ𝐀2𝐀(1+𝐿)𝐃𝑑1+Δ𝐀𝑑12𝐿A1+Δ𝐀1𝑇𝐃𝐀d2+Δ𝐀𝑑2𝐀(1+𝐿)1+Δ𝐀1𝑇𝐃𝐀2𝐿2+Δ𝐀2𝑇𝐃𝐀𝑑2+Δ𝐀𝑑2𝐀(1+𝐿)2+Δ𝐀2𝑇𝐃𝐀2𝐿𝑑1+Δ𝐀𝑑1𝑇𝐃𝐀𝑑2+Δ𝐀𝑑2𝐀(1+𝐿)𝑑1+Δ𝐀𝑑1𝑇𝐃𝐐2𝐀+2𝐿𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀𝑑2+Δ𝐀𝑑2𝐀(1+𝐿)𝑑2+Δ𝐀𝑑2𝑇𝐃𝐀(1+𝐿)𝐃𝑑2+Δ𝐀𝑑2.2𝐃𝐏(29) Note that, if𝐆>𝟎, then Δ𝑉(𝐱(𝑖,𝑗))<0 for ̂𝐱(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 𝐏𝐏1𝐐1𝐀𝟎𝟎𝟎𝟎𝜻1+Δ𝐀1𝑇𝐃𝟎𝐏1𝐐2𝐀𝟎𝟎𝟎𝜻2+Δ𝐀2𝑇𝐃𝟎𝟎𝐐1𝐀𝟎𝟎𝜻𝑑1+Δ𝐀𝑑1𝑇𝐃𝟎𝟎𝟎𝐐2𝐀𝟎𝜻𝑑2+Δ𝐀𝑑2𝑇𝐃𝟎𝟎𝟎𝟎(1𝐿)22𝐿𝐃𝐏(1+𝐿)𝜻𝐃𝐀𝜻𝐃1+Δ𝐀1𝐀𝜻𝐃2+Δ𝐀2𝐀𝜻𝐃𝑑1+Δ𝐀𝑑1𝐀𝜻𝐃𝑑2+Δ𝐀𝑑2(1+𝐿)𝜻𝐃𝐃>𝟎.(30) where 𝜻 denotes 2𝐿.
Applying (3a), (30) can be rewritten in the following form:𝐄𝐖+𝐇𝐅𝐄+𝑇𝐅𝑇𝐇𝑇<𝟎,(31a) where𝐖=𝐏𝐏1𝐐1𝟎𝟎𝟎𝟎2𝐿𝐀𝑇1𝐃𝐏𝟎1𝐐2𝟎𝟎𝟎2𝐿𝐀𝑇2𝐃𝟎𝟎𝐐1𝟎𝟎2𝐿𝐀𝑇𝑑1𝐃𝟎𝟎𝟎𝐐2𝟎2𝐿𝐀𝑇𝑑2𝐃𝟎𝟎𝟎𝟎(1𝐿)22L𝐃+𝐏(1+𝐿)𝐃2𝐿2𝐿𝐃𝐀12𝐿𝐃𝐀22𝐿𝐃𝐀𝑑12𝐿𝐃𝐀𝑑2(1+𝐿)2𝐿𝐃𝐃,(31b)𝐇𝑇=𝟎𝟎𝟎𝟎𝟎𝐇𝑇𝐃,(31c)𝐄=2𝐿𝐄12𝐿𝐄22𝐿𝐄𝑑12𝐿𝐄𝑑2𝟎𝟎.(31d)By Lemma 1, (31a)–(31d) is equivalent to 𝐖+𝜀1𝐇𝐇𝑇𝐄+𝜀𝑇𝐄<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 by𝑓𝑘(0)=0,𝐿𝑦2𝑘(𝑖,𝑗)𝑓𝑘𝑦𝑘𝑦(𝑖,𝑗)𝑘(𝑖,𝑗)𝑦2𝑘(𝑖,𝑗),𝑘=1,2,,𝑛,(33a)1𝐿<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 becomes𝐀𝐱(𝑖+1,𝑗+1)=𝐟1+Δ𝐀1𝐀𝐱(𝑖+1,𝑗)+2+Δ𝐀2,𝐀𝐱(𝑖,𝑗+1)(34a)𝐀=1𝐀2,(34b)Δ𝐀1Δ𝐀2𝐄=𝐇𝐅𝐄,𝐄=1𝐄2𝐅,(34c)𝑇𝐅𝐈.(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 𝛼𝑘𝑙,𝛽𝑘𝑙(𝑘,𝑙=1,2,,𝑛(𝑘𝑙)),𝜀, and 𝑛×𝑛 positive definite symmetric matrices 𝐏and𝐏1such that 𝐏𝐏1𝟎𝟎𝐏1𝐄+𝜀𝑇𝐄𝐀𝑇𝐂𝐂𝟎𝑇𝐀𝐏𝐂𝐂𝑇𝐂𝑇𝐇𝟎𝐇𝑇𝐂𝜀𝐈<𝟎,(35) 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 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 𝐏𝐏1𝟎𝟎𝐏1𝐄2𝜀𝐿𝑇𝐄𝟎𝐀2𝐿𝑇𝟎𝐃𝟎(1𝐿)22𝐿𝐃+𝐏(1+𝐿)2𝐿𝐃𝟎𝐀(2𝐿𝐃1+𝐿)2𝐿𝐃𝐃𝐃𝐇𝟎𝟎𝐇𝑇𝐃𝜀𝐈<𝟎.(36)

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(𝑘𝑙)), 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 𝐏𝐏1𝟎𝟎𝐏1𝐄+𝜀𝑇𝐄𝐀𝑇𝐃𝐃𝟎𝐀𝐏2𝐃𝐃𝐇𝟎𝐇𝑇𝐃𝜀𝐈<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 matrices𝐏𝟏, 𝐏=[𝑝𝑘𝑙] such that𝑝𝑘𝑘𝑛𝑙=1,𝑙𝑘||𝑝𝑘𝑙||,𝑘=1,2,,𝑛,(38a)𝐏𝐏𝐀1𝐏𝐀2𝐀𝐏𝐇𝑇1𝐏𝐏+𝐏1+𝜀𝐄𝑇1𝐄1𝜀𝐄𝑇1𝐄2𝟎𝐀𝑇2𝐏𝜀𝐄𝑇2𝐄1𝐏1+𝜀𝐄𝑇2𝐄2𝟎𝐇𝑇𝐏𝟎𝟎𝜀𝐈<𝟎.(38b)

Theorem 6 (Singh [14]). Suppose there exists a matrix 𝐖=[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 satisfying𝐏𝐏1𝟎𝟎𝐏1𝐄+𝜀𝑇𝐄𝐀𝑇(𝐔+𝐕𝐖)𝟎𝐔+𝐖𝑇𝐕𝐀𝐏2𝐔𝐕𝐖𝐖𝑇𝐕𝐔+𝐖𝑇𝐕𝐇𝟎𝐇𝑇(𝐔+𝐕𝐖)𝜀𝐈<𝟎,(39a)where 𝑤𝑘𝑘𝑤=𝑛1,𝑘=1,2,,𝑛,(39b)𝑘𝑙=𝜆𝑘𝜆,𝑘,𝑙=1,2,,𝑛(𝑘𝑙),(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)=𝐀1𝐱(𝑖+1,𝑗)+𝐀2𝐱(𝑖,𝑗+1)+𝐀d1𝐱𝑖+1,𝑗𝑑1+𝐀d2𝐱𝑖𝑑2.,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=𝐏𝐏1𝐐1𝟎𝟎𝟎𝟎𝐏1𝐐2𝟎𝟎𝟎𝟎𝐐1𝟎𝟎𝟎𝟎𝐐2𝐀𝑇1𝐀𝑇2𝐀𝑇𝑑1𝐀𝑇𝑑2𝐏𝐀1𝐀2𝐀𝑑1𝐀𝑑2>𝟎.(41)

Now, we have the following propositions.

Proposition 1. Corollary 1 implies Theorem 4.

Proof. With 𝛼𝑘𝑙=𝛽𝑘𝑙(𝑘,𝑙=1,2,,𝑛(𝑘𝑙)) 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=𝐂𝑇=𝐏,(42)(35) reduces to (38b). Condition (42) along with (9a)–(9c) and 𝐿=𝐿1=𝐿2=1 yields𝑝𝑘𝑘=𝑐𝑘𝑘=𝑛𝑙=1,𝑙𝑘𝛼𝑘𝑙+𝛽𝑘𝑙𝑝,𝑘=1,2,,𝑛,(43a)𝑘𝑙=𝑐𝑘𝑙=𝑐𝑙𝑘=𝛼𝑘𝑙𝛽𝑘𝑙,𝑘,𝑙=1,2,,𝑛(𝑘𝑙),(43b)where 𝛼𝑘𝑙>0,𝛽𝑘𝑙>0,𝑘,𝑙=1,2,,𝑛(𝑘𝑙). Using (43a)-(43b), we obtain 𝑛𝑙=1,𝑙𝑘||𝑝𝑘𝑙||=𝑛𝑙=1,𝑙𝑘||𝛼𝑘𝑙𝛽𝑘𝑙||<𝑛𝑙=1,𝑙𝑘||𝛼𝑘𝑙+𝛽𝑘𝑙||=𝑝𝑘𝑘,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 as𝐏𝐏1𝟎𝟎𝐏1𝐄+𝜀𝑇𝐄𝐀𝑇(𝐔+𝐕𝐖)𝟎𝐔+𝐖𝑇𝐕𝐀𝐏2𝐔𝐕𝐖𝐖𝑇𝐕𝐔+𝐖𝑇𝐕𝐇𝟎𝐇𝑇(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 𝑛2. In this case, let us choose the positive scalars 𝛼𝑘𝑙 and 𝛽𝑘𝑙 given by (9c) as𝛼𝑘𝑙=𝑢𝑘+(𝑛1)𝑣𝑘+𝑣2(𝑛1)𝑘𝜆𝑘2,𝑘,𝑙=1,2,,𝑛(𝑘𝑙),(46a)𝛽𝑘𝑙=𝑢𝑘+(𝑛1)𝑣𝑘𝑣2(𝑛1)𝑘𝜆𝑘2,𝑘,𝑙=1,2,,𝑛(𝑘𝑙),(46b)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) 𝑐𝑘𝑘=𝑢𝑘+(𝑛1)𝑣𝑘𝑐,𝑘=1,2,,𝑛,(47a)𝑘𝑙=𝑣𝑘𝜆𝑘,𝑘,𝑙=1,2,,𝑛(𝑘𝑙).(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 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. 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=Δ𝐀𝑑1=Δ𝐀𝑑2=𝟎. In this case, (14) can be rearranged as 𝐱Δ𝑉(𝐱(𝑖,𝑗))=𝐱(𝑖+1,𝑗)𝐱(𝑖,𝑗+1)𝑖+1,𝑗𝑑1𝐱𝑖𝑑2,𝑗+1𝑇𝐘𝐱𝐱(𝑖+1,𝑗)𝐱(𝑖,𝑗+1)𝑖+1,𝑗𝑑1𝐱𝑖𝑑2,,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,𝑗𝑑1)=𝐱(𝑖𝑑2,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) with𝐀1=1.0520.10,𝐀2=𝐀0.001000.001,(49a)𝑑1=𝐀𝑑2=0𝐄0.001000.001,𝐇=0.01,(49b)1=𝐄2=𝐄𝑑1=𝐄𝑑2=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. 𝛼12=62.4706,𝛼21𝛽=298.8216,12=146.1559,𝛽21,𝐏=693.5676,𝐏=237.7874448.6233448.62331610.91=,𝐐30.531917.749717.7497308.64351=,𝐐15.713510.640610.6406182.59972=,19.887710.145710.1457178.9414𝜀=289.0811.(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),𝐱1111180(51a)(𝑖,𝑗)=𝟎,𝑖30,𝑗=1,0,𝐱(𝑖,𝑗)=𝟎,𝑗30,𝑖=1,0,(51b)𝐱(𝑖,0)=,0𝑖<30,𝐱(0,𝑗)=,0𝑗<30.(51c)

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=00.010,𝐇=0.01,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. 𝛼12=61.6039,𝛼21𝛽=237.6665,12=104.1096,𝛽21,𝐏=536.3285,𝐏=181.5949331.7387331.73871179.51=,25.84948.04288.0428244.9546𝜀=333.0229.(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.

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Copyright © 2011 Anurita Dey and Haranath Kar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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