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

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.


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.
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][25][26][27][28][29][30][31][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][14][15][16]21].In [24], the problem of robust stability and stabilization of 2D statedelayed 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 H 2 /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 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 where 0 ≤ i, j ∈ Z are horizontal and vertical coordinates, respectively; x(i, j) ∈ R n is the local state vector; A 1 , A 2 , A d1 , A d2 are the known real constant n × n matrices; ΔA 1 , ΔA 2 , ΔA d1 , ΔA d2 are the unknown real n × n matrices representing parametric uncertainties in the state matrices; d 1 , d 2 are constant positive integers representing delays along vertical direction and horizontal direction, respectively.The generalized overflow characteristic is given by where With the appropriate selection of L, L 1 , and L 2 , (2a) characterizes the common types of overflow arithmetics employed in applications such as saturation ( The uncertain matrix ΔA is defined in the normbounded form [13-16, 21-24, 27-29] as where H, E 1 , E 2 , E d1 , E d2 are known real constant matrices with appropriate dimensions and F is an unknown real matrix satisfying In the uncertainty structure given by (3a)-(3b), the matrices H, E 1 , E 2 , E d1 , E d2 characterize how the uncertain parameters in F enter the state matrices.The matrix F can always be restricted as (3b) by appropriately selecting H, E 1 , E 2 , 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 that 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][42][43].

Main Result and Its Corollaries
Consider two ranges for L 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]).Let Σ, Γ, F, and M be real matrices of appropriate dimensions with M satisfying M = M T ; then for all F T F ≤ I if and only if there exists a positive scalar ε such that Next, suppose C = [c kl ] ∈ R n×n is a matrix characterized by where it is understood that, for n = 1, C corresponds to a scalar γ > 0. Thus, corresponding to n = 3, the matrix C takes the form where α kl > 0, β kl > 0, k, l = 1, 2, 3 (k / = l).Now, we have the following result.
Next, we prove the following result.
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 becomes Now, as a consequence of the presented approach, we have the following corollaries.
The following result can easily be obtained as a direct consequence of Theorem 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, P 1 , 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.

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 = L 1 = L 2 = 1) is globally asymptotically stable if there exist n × n positive definite symmetric matrices P, P 1 and a positive definite diagonal matrix D, and a positive scalar ε such that Pertaining to the saturation nonlinearity [21, Theorem 2] leads to the following result.
Theorem 7 (Paszke et al. [24]).Consider the system (1a)-( 4) with no nonlinearity and no uncertainty; then system becomes The null solution of the system described by (40) is globally asymptotically stable if there exist n×n positive definite symmet-ric matrices P, P 1 , Q 1 , and Q 2 such that Now, we have the following propositions.Proof.It is not difficult to show that, with ( Thus, condition (42) leads to (38a).
Proof.Using Schur complement, (39a) can equivalently be expressed as 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.
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 = L 1 = L 2 = 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).

Illustrative Examples
To illustrate the effectiveness of the presented results, we now consider the following examples.
Figure 1 shows the trajectories of the two state variables for the present example with x i, j = 0, ∀i ≥ 30, j = −1, 0, x i, j = 0, ∀ j ≥ 30, i = −1, 0, (51b) 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.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.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.

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].