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A key issue of practical importance in the two-dimensional (2-D) discrete system is stability analysis. Linear state-space models describing 2-D discrete systems have been proposed by several researchers. A popular model, called Forna- sini-Marchesini (FM) second model was proposed by Fornasini and Marchesini in 1978. The aim of this paper is to present a survey of the existing literature on the stability of FM second model.

There have been a continuously growing research interests in two-dimensional (2-D) systems due to their applications in various important areas such as multi-dimensional digital filtering, signal processing, seismographic data processing, thermal processes, gas absorption, water stream heating etc. [1-4]. In a 2-D discrete system, information propagates in two independent directions as a result of which the system dynamics may be represented as a function of two independent integer variables. Many researchers have made an attempt to describe the 2-D system dynamics in terms of linear state-space models for 2-D discrete systems [5-7]. The 2-D models that have received considerable attention are Roesser model [

Stability analysis and stabilization are the main issues in the design of any control system. Stability issues of 2-D systems have been considered by many researchers [8-18]. With the introduction of state-space models of 2-D discrete systems, various Lyapunov equations have emerged as powerful tools for the stability analysis and stabilization of 2-D discrete systems. Lyapunov based sufficient conditions for the stability of 2-D discrete systems have been studied in [19-26]. When the dynamics of practical systems are represented using state-space models, errors are inevitable as the actual system parameters would be different than the estimated system parameters, i.e., the model parameters. These errors arise due to the approximations made during the process modeling, differences in presumed and actual process operating points, change in operating conditions, system aging etc. Control designs based on these models, therefore, may not perform adequately when applied to the actual industrial process and may lead to instability and poor performances. This has motivated the study of robust control for the uncertain 2-D discrete systems. The aim of robust control is to stabilize the system under all admissible parameter uncertainties arising due to the errors around the nominal system. Many significant results on the solvability of robust control problem for the uncertain 2-D discrete systems have been proposed in [27-33].

The issue in robust control design is twofold: first is to design a robust controller to ensure the stability of uncertain systems and the other is to guarantee a certain performance level under the presence of uncertainties. The latter is called as guaranteed cost control problem which has the advantage of providing an upper bound on the closed-loop cost function (performance index). Consequently, a guaranteed cost controller not only stabilizes the uncertain system but also guarantees that the value of closed-loop cost function is not more than the specified upper bound for all admissible parameter uncertainties. Based on this idea, many significant results have been obtained for the uncertain 2-D discrete systems [34-39].

Study and analysis of 2-D discrete systems under the presence of noise is another research area of great interest where it is usually necessary to estimate the state variables from the system measurement data. One of the celebrated approaches is Kalman filtering [

The linear matrix inequalities (LMIs) have been evolved as a powerful technique to formulate various control designs [

In this paper, our main focus is on the FM second model which is one of the most investigated models for the study and analysis of 2-D discrete systems. A brief survey of the existing literature on the stability of the 2-D discrete systems described by FM second model has been presented in this paper. The paper is organized as follows: Section 2 presents the description of 2-D system described by FM second model. A brief survey and main results on the stability of FM second model has been discussed in Section 3. Finally, some concluding remarks are given in Section 4.

Throughout the paper the following notations are used: The Closed unit disc is represented by, represents the unit circle, denotes the closed unit bidisc. The superscript T stands for matrix transposition, denotes real vector space of dimension n, is the set of n ´ m real matrices, 0 denotes null matrix or null vector of appropriate dimension, is the identity matrix, diag (…) stands for a block diagonal matrix, G > 0 stands for the matrix G is positive definite, denotes determinant of a matrix, denotes induced matrix norm, stands for spectral radius of a matrix, stands for spectrum of a matrix, for a matrix pair, denote the set of all its generalized eigenvalues i.e., where is the set of complex numbers. Further, stands for the matrix, and represents Kronecker product of the matrices and.

Consider the following 2-D discrete system represented by FM second model [

where is an state vector, , , is input vector, , is a scalar output, and. It is understood that the above system has a finite set of initial conditions [_{ }such that

The equilibrium of system (1) is said to be globally asymptotically stable [

The transfer function of system (1) is given as

If we define,

then the state-space model (1) is asymptotically stable [

where.

The problem of asymptotic stability of system (1) has been studied by many researchers [51-59]. Lyapunov based sufficient condition for the stability of system (1) has been investigated in [

provided

Reference [

where is defined in (4a). Further, based on the 2-D Lyapunov equation approach, the problem of stability margin has also been studied in [

Studies in [

and

As noted in [

and

In [

and 2) there exist an symmetric matrix such that

provided

.

Here and are the horizontal and vertical radian frequencies, respectively.

The second criterion states that the system (1) is asymptotically stable if there exist symmetric matrices, and such that

and

At this point, readers are advice to observe the differences between (8) and (10). Furthermore, the relationship between 2-D Lyapunov approach and stability margin has also been investigated in [

Another 2-D Lyapunov equation, which is in a more general form, for asymptotic stability of system (1) has been presented in [

Further, it has been shown that the Lyapunov matrix Inequality (11) can be expressed in a succinct form using parallel addition of positive definite matrices and an equivalent condition of stability can be given as: The system (1) is asymptotically stable if there exist a pair of positive definite matrices, such that

where, ,

and is known as parallel addition of and. It is interesting to note that as a rough approximation, the terms and makes out the next time energy along the one-dimensional dynamics; it then follows that the sum of total next time energy should be less than the modified present energy represented by for the 2-D system (1) to be asymptotically stable. It has been further illustrated that in spite of its outward beauty, the Inequality (12) is rather complicated even for smaller values of.

In [

is stable (13a)

and

for all (13b)

Here the stability of is equivalent to. If we define the matrices

then, the necessary and sufficient condition for the stability is as follows: The 2-D system (1) is stable if and only if and additionally one of the following conditions holds:

1), for all. (15a)

2), for all. (15b)

3), for all

and

4)

and

where

and

.

Further, the authors in [

An LMI based necessary and sufficient condition for the positive FM second model has been presented in [

1) The LMI

is feasible with respect to the diagonal matrix.

2) The LMI

is feasible with respect to the diagonal matrix.

The Lyapunov based sufficient condition for the stability of system (1) under shift delays has been discussed in [

where and are delay matrices. Based on condition (17), the problem of robust stability and stabilization of 2-D discrete shift-delayed system described by the FM second model has also been addressed.

In [

In [

A review on the stability of 2-D discrete systems described by FM second model has been presented in this paper. The Lyapunov based approach has emerged as a popular approach to study the stability properties of such systems. The 2-D Lyapunov based stability conditions discussed so far in literature are only sufficient conditions. The Lyapunov based necessary and sufficient condition for the stability of 2-D discrete systems remains an open and challenging problem.

The authors wish to thank the associate editor and the anonymous reviewers for their constructive comments and suggestions.