Over the past decades, the problem of control for 2-D discrete systems has drawn considerable attention. The main advantage of control is that its performance specification takes into account the worst-case performance of the system in terms of the system energy gain [1] . Based on this idea, many important results have been obtained in the literature [2] - [5] . Among these results, the problem of control and robust stabilization of 2-D discrete systems described by the Roesser model has been addressed in [2] . A solution to the problem of robust control for uncertain 2-D discrete systems represented by the general model (GM) via output feedback controllers has been presented in [3] . A 2-D filtering approach, based on the 2-D bounded real lemma, with an performance measure for 2-D discrete systems described by the Fornasini-Marchesini (FM) second model has been developed in [4] . The dynamic output feedback stabilization problem for a class of 2-D discrete switched systems represented by the FM second model has been addressed in [5] .

It is well known that delay is encountered in many dynamic systems and is often a source of instability, thus, much attention has been focused on the problem of stability analysis and controller design for 2-D discrete state-delayed systems in the last few years [6] - [25] . In [6] , the problem of stability analysis for 2-D discrete state-delayed systems in the GM has been considered and sufficient conditions for stability have been derived via Lyapunov approach. The problem of delay-dependent guaranteed cost control for uncertain 2-D discrete state-delayed system described by the FM second model has been presented in [7] . In [8] , the problem of robust guaranteed cost control for uncertain 2-D discrete state-delayed systems described by the FM second model has been considered. Several corrections in the main results of [8] have been made in [9] . In [10] , the guaranteed cost control problem via memory state feedback control laws for a class of uncertain 2-D discrete state-delayed systems described by the FM second model has been discussed. Robust reliable control of uncertain 2-D discrete switched state-delayed systems described by the Roesser model has been presented in [11] . The problem of positive real control for 2-D discrete state-delayed systems described by the FM second model via output feedback controllers has been addressed in [12] . In [13] , the problem of delay-dependent control for 2-D discrete state-delayed system described by the FM second model has been investigated. The problem of control for 2-D discrete state-delayed systems described by the FM second model has been studied in [14] and a method to design an optimal state feedback controller has been presented. Here, it may be mentioned that [14] considers the FM second model without uncertainties, but in the real world situation, the uncertainties in the system parameters cannot be avoided.

With this motivation, we consider the problem of robust optimal control for uncertain 2-D discrete state-delayed systems described by the GM. The approach adopted in this paper is as follows: We first establish a sufficient condition for the existence of g-suboptimal robust state feedback controllers in terms of a certain linear matrix inequality (LMI). Further, a convex optimization problem is introduced to select a robust optimal state feedback controller which minimizes the noise attenuation level g of the closed-loop system. Finally, two illustrative examples are given to demonstrate the effectiveness of the proposed technique.

The following notations are used throughout the paper:

real vector space of dimension n.

set of real matrices.

null matrix or null vector of appropriate dimension.

identity matrix of appropriate dimension.

transpose of matrix.

stands for a block diagonal matrix.

matrix positive definite symmetric.

matrix negative definite symmetric.

Consider the uncertain 2-D discrete state-delayed systems described by the GM [26] .

where are horizontal and vertical coordinates, , represent the state and control input, respectively, is the controlled output, is the noise input which belongs to and

The matrices, , , and are known constant matrices representing the nominal plant;, , and are constant positive integers representing delays. The matrices, , , , , , , and represent parameter uncertainties in the system matrices, which are assumed to be of the form

where , and are known structural matrices of uncertainty and is an unknown matrix representing parameter uncertainty which satisfies

(or equivalently,). (1e)

It is assumed that the system (1) has a finite set of initial conditions [6] , i.e., there exist two positive integers and, such that

Definition 1 [14] . The system described by (1) is asymptotically stable if with and the initial condition (2), where

Definition 2 [14] . Consider the system (1) with and the initial condition (2). Given a scalar and symmetric positive definite matrices the system (1) is said to have an noise attenuation if it is robustly stable and satisfies

where

and

The following well established lemmas are essential for the proof of our main results.

Lemma 1 [27] - [29] . Let and be given matrices. Then, there exist a positive definite matrix such that

for all satisfying if and only if there exists a scalar such that

Lemma 2 [30] . For real matrices of appropriate dimension, where and then if and only if

or equivalently

In this section, two examples illustrating the effectiveness of our proposed method are presented.

Example 4.1: Consider an uncertain 2-D discrete state-delayed system given by (1) and initial condition (2) with

We wish to design a robust optimal controller for the above system. Using the Matlab LMI toolbox [30] [32] , it is found that the optimization problem (29) is feasible for the present example and the optimal solution is given by

Thus, the robust optimal state feedback controller is obtained as

Figure 1 shows the frequency response from noise input to the controlled output for the closed-loop system (22) over all frequencies i.e., ,. The peak value of the frequency response is 0.5029, which is lower than the specified level of attenuation

Example 4.2: Consider the thermal processes in chemical reactors, heat exchangers and pipe furnaces [33]

[34] , which can be expressed by the following partial differential equation.

where is the temperature at space and time is the input function, and are the time delays, and are the space delays, and, , , , b are the real coefficients. Taking

, (34)

, (35)

(33) can be written in the following form:

where and, , is the integer function.

It is assumed that the surface of the heat exchanger is insulated and the heat flow through it is in steady state

condition, then we could take the boundary conditions as and, respectively.

Denoting it is easy to verify that (36) can be converted into the following 2-D state-delayed GM:

Let and the initial state satisfies the condition (2) with, To consider the problem of disturbance attenuation, the thermal process is modeled in the form (1) with

It is also assumed that the above system is subjected to the parameter uncertainties of the form (1c) and (1d) with

Now, using the Matlab LMI toolbox [30] [32] , it is found that the optimization problem (29) is feasible for the considered system and the optimal solution is obtained as

Thus, the robust optimal state feedback controller is given as

Figure 2 shows the frequency response from noise input to the controlled output for the closed-loop system (22) over all frequencies i.e., ,. The peak value of the frequency response is 0.5002, which is lower than the above obtained specified level of attenuation

In this paper, the problem of robust optimal control for a class of uncertain 2-D discrete state-delayed systems described by the GM has been studied. A sufficient condition for the existence of g-suboptimal robust state feedback controller has been derived in terms of the feasible solutions to a certain LMI. The desired robust optimal controller has been obtained by solving a convex optimization problem. Finally, two illustrative examples have been provided to demonstrate the applicability of the proposed approach.

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions.

Arun Kumar Singh,Amit Dhawan, (2016) Robust Optimal H_∞ Control for Uncertain 2-D Discrete State-Delayed Systems Described by the General Model. Journal of Signal and Information Processing,07,78-114. doi: 10.4236/jsip.2016.72011