In the past decades, research on two-dimensional (2-D) discrete systems has rapidly increased due to their extensive practical applications in circuits analysis [1] , digital image processing [2] , signal filtering [3] and thermal power engineering [4] , etc. Thus, the study of 2-D systems is an attractive problem and a number of results have been presented in the literature. Among these results the stability analysis of 2-D discrete systems has been given in [5] - [15] . In [5] , the problem of robust stability analysis and stabilization for 2-D discrete uncertain systems described by the Fornasini-Marchesini (FM) second model has been studied and the sufficient conditions for 2-D discrete uncertain systems to be robustly stable have been given in terms of linear matrix inequalities (LMIs). A necessary and sufficient condition for the stability of 2-D discrete systems described by the FM first model has been derived in [13] . The problem of stability for linear discrete 2-D singular general model (GM) has been discussed in [14] and a sufficient condition for the internal stability of the 2-D singular GM has been derived in terms of LMIs. In [15] , the robust stability analysis problem for a class of uncertain 2-D discrete systems described by the FM second model has been studied and sufficient conditions in terms of LMIs have been derived to ensure the robust stability of the uncertain 2-D discrete systems. Lately, the solution to the guaranteed cost control problem for 2-D discrete uncertain systems has been presented by many authors [16] - [22] .

In recent years, the H_¥ control problem for 2-D discrete systems has gained a great deal of interest and many important results have been obtained [23] - [26] . A major advantage of H_¥ control is that its performance specification takes account of the worst- case performance for system in terms of the system energy gain. This is appropriate for system robustness analysis and robust control with modeling uncertainties and disturbances than other performance specifications, such as the LQ-optimal control specification [23] . In [24] , the solutions for the H_¥ control and robust stabilization problems for 2-D systems described by the Roesser model using the 2-D system bounded realness property have been presented. The problem of H_¥ static output feedback control for 2-D discrete systems described by the Roesser model and the FM second model has been addressed in [25] . In [26] , the problem of robust H_¥ control for uncertain 2-D discrete systems described by the GM via output feedback controllers has been investigated.

Since, delay is encountered in many dynamic systems and is often a source of instability, much attention has been focused on the problem of stability analysis and controller design for 2-D discrete state-delayed systems in the last decades. Presently, the stability results for 2-D discrete state-delayed systems fall in two groups: delay-inde- pendent stability conditions [27] - [35] and delay-dependent ones [36] - [45] . The former refers to the stability conditions which do not depend on delay; the latter contains information on the size of delay. Generally speaking, the delay-dependent stability condition is less conservative especially when the sizes of the delays are small [45] . The problem of stability analysis for 2-D discrete state-delayed systems in the GM has been considered in [33] and sufficient condition for the stability has been derived via Lyapunov approach. In [27] , a solution to the problem of delay-independent H_¥ control for 2-D state-delayed systems described by the FM second model has been presented. The problem of robust reliable control for a class of uncertain 2-D discrete switched systems with state delays and actuator faults represented by a model of Roesser type has been studied by [28] . In [35] , the problem of robust guaranteed cost control via memoryless state feedback for uncertain 2-D discrete state-delayed systems described by the FM second model has been considered. Several technical errors that have occurred in the main results of [35] were corrected in [30] . A solution to 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 presented in [31] . The problem of delay?dependent guaranteed cost control via memoryless state feedback for uncertain 2-D discrete state-delayed systems described by the FM second model has been studied in [43] . In [41] , a delay-range dependent H_¥ control and stabilization problem for a class of uncertain 2-D state-delayed systems described by the Roesser model has been proposed. The delay-dependent H_¥ control problem via a delay-dependent bounded real lemma for a class of 2-D state-delayed systems described by the FM second model has been addressed in [40] . Recently, the problems of delay- dependent H_¥ control for 2-D discrete state-delayed systems described by the FM second model and Roesser model have been considered in [42] [44] , respectively and a delay-dependent optimal state feedback H_¥ controller has been obtained for both the models. It may be mentioned here that the criteria presented in [42] [44] do not consider the uncertainty in system parameters. In real-time applications, the parametric uncertainties cannot be ignored as they are the main source of instability and poor system performance. Hence, the delay-dependent robust optimal H_¥ control for uncertain 2-D discrete state delay systems is an important and challenging problem. However to the best of authors’ knowledge, the delay-dependent robust optimal H_¥ control problem for uncertain 2-D discrete state delay systems represented by the GM which is structurally distinct from FM second model and Roesser model has not been addressed so far in the literature.

This paper, therefore, investigates the problem of delay-dependent robust optimal H_¥ control for a class of uncertain 2-D discrete state delay systems described by the GM. The approach adopted in this paper is as follows: We first derive an LMI-based sufficient condition for the existence of delay-dependent g-suboptimal state feedback robust H_¥ controllers in terms of feasible solution to a certain LMI. Further, a convex optimization problem with LMI constraints is formulated to design a delay-dependent robust optimal H_¥ controller which minimizes the H_¥ noise attenuation g of the closed-loop system. The paper is organized as follows. Section 2 formulates the problem of delay- dependent robust H_¥ control for a class of uncertain 2-D discrete state delay systems described by the GM and recalls some useful results. In Section 3, a solution to the problem of delay-dependent robust optimal H_¥ control is presented. An example illustrating the potential of the proposed technique is given in Section 4.

Notations:

Throughout the paper, the following notations are used: Rⁿ denotes real vector space of dimension n; is the set of n ´ m real matrices; the superscript T stands for matrix transposition; 0 denotes null matrix or null vector of appropriate dimension; I is the identity matrix of appropriate dimension; G < 0 stands for the matrix G which is symmetric and negative definite; diag{….} stands for a block diagonal matrix; denotes the Euclidean vector norm. The norm of a 2-D signal is given by

, where is said to be in if.

Consider an uncertain 2-D discrete state delay system described by the GM [46] .

where

and are horizontal and vertical coordinates, respectively, , are state and control input, respectively, is the controlled output, is the noise input with bounded energy i.e., it belongs to. The matrices, , , and are known constant matrices represent- ing the nominal plant;, , and are unknown 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)

For system (1), suppose a finite set of initial conditions [33] , i.e., there exist positive integers and, such that

Denote we first present the definition of asymptotic stability for the system (1).

Definition 1. [42] The system (1) is asymptotically stable if with

, and the initial condition (2).

Introduce the following state feedback controller

Applying the controller (3) to system (1) will result in the closed-loop system:

where

, ,.

T o investigate the delay-dependent control problem, we first define the vectors such that

The following well known lemmas are needed in the proof of our main result.

Lemma 1. [42] [44] For any matrices and, and any integer, the following summation inequality holds

where

Lemma 2. [5] Let, , and be real matrices of appropriate dimension with satisfying then

holds for all satisfying, if and only if there exist such that

The performance measure for system (1) with the initial condition (2) is defined as follows:

Definition 2. [42] Given a scalar, integers, , , and symmetric positive definite weighting matrices. The closed-loop system (4), formed by system (1) with the initial condition (2) and state- feedback controller (3), is said to have an noise attenuation g for any delays, , , satisfying, , and if it is robustly stable and satisfies

where

and, , , are upper bounds of delays in states.

In the case when the initial condition is known to be zero, then the performance measure (8) reduces to

Using the 2-D Parseval’s theorem [3] , (9) is equivalent to

where denotes the maximum singular value of the corresponding matrix and

is the transfer function from the disturbance input to the controlled output for the system (4).

The objective of this paper is to design a controller of the form (3) such that the closed-loop system (4) is asymptotically stable and the norm of the transfer function (11) from the noise input to the controlled output for the closed-loop system (4) is smaller than g. Such controller is said to be a g-suboptimal state feedback robust controller for the uncertain 2-D discrete state delay system (1) with any state delays, , , satisfying, , and.

In this section, we first present a delay-dependent approach to solve the control problem for the uncertain 2-D discrete state delay system (1) with any state delays, , , satisfying, , and. Based on Lemma 1, a sufficient condition for the existence of g-suboptimal state feedback robust controllers is obtained in terms of an LMI. Further, a delay-dependent state feedback robust optimal controller is obtained by solving a convex optimization problem which minimizes the noise attenuation g of the closed-loop system (4).

Theorem 1. The closed-loop system (4), formed by system (1) with the initial condition (2) and state-feedback controller (3), is robustly stable and has a specified noise attenuation g, if there exist matrices and symmetric positive definite matrices satisfying, , , , , such that

where

,

,

,

,

,

,

,

,

,

,

,

.

Proof. To prove that the closed-loop system (4) with is asymptotically stable, we define a Lyapunov-Krasovskii function

where

and, , , , and. Thus, it is explicit that.

Along any trajectory of the system (1) with, the increment is given by

Applying (5), we get

Now applying Lemma 1, we get the following summation inequalities

and

where

Now, substituting (15)-(18) in (14) yields

Applying Schur complement, it follows from matrix inequality (12) that

Thus, from (19b), it implies that

holds for any delays, , and satisfying, , and.

Hence, the closed-loop system (4) with is asymptotically stable.

Next, we establish the performance measure of the closed-loop system (4) for

We consider

where

It follows from inequality (12) that

Summing the inequality (22) over gives

Now,

Thus, by using (25) in (23), we get

Since, , , , , and, therefore, it follows from Definition 2 that the closed-loop system (4) guarantees the noise attenuation. This completes the proof of Theorem 1.

In the following, we will show that the above derived sufficient condition for existence of delay-dependent robust controllers can be transformed to the solvability of an LMI.

Theorem 2. Consider the closed-loop system (4) with the initial condition (2). Given scalars, and integers, , and, if there exist matrices, and symmetric positive definite matrices such that

where

,

,

,

,

,

,

,

then the closed-loop system (4) with, , and has a specified noise attenuation g and the controller (3) with is a g-suboptimal state feedback robust controller for the uncertain 2-D discrete state delay system (1).

Proof. It follows from matrix inequality (12) that, and are reversible. So, let us assume that

Pre-multiplying and post-multiplying both sides of matrix inequality (12) by

and its transpose, respectively, we get

Denoting, , , , , , , , , , , and, we can re-write (29) as

where

Using (1d) and (1e), (30) can be expressed as

where

,

,

Therefore, using Lemma 2, (31) can be rearranged as

The equivalence of (32) and (27) follows trivially from Schur complements. This completes the proof of Theorem 2.

Theorem 2 provides a parameterized representation of a set of g-suboptimal robust state feedback controllers (if they exist) in terms of feasible solutions to the LMI (27). Based on Theorem 2, the design problem of a delay-dependent robust optimal controller can be formulated as

which ensures the minimization of noise attenuation of the closed-loop system (4).

In this section, we present an application example to demonstrate the effectiveness of our proposed result.

Example 1. In this example, we shall illustrate the applicability of Theorem 2 to the control of thermal process in heat exchanger [4] [47] which can be expressed by the partial differential equation with time and space delays:

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

(34) can be expressed 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.

Denote, it is easy to verify that (36) can be converted into the following discrete form:

Next, consider the problem of delay-dependent control for uncertain two- dimensional (2-D) discrete state-delayed systems characterized by (37). 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

Using the MATLAB LMI toolbox [48] [49] , it is found that the optimization problem (33) is feasible for the present example and the optimal solution is given by

and a delay-dependent optimal state feedback controller

When, , , , the frequency response from the disturbance input to the controlled output for the system (4) is shown in Figure 1. It is apparent from Figure 1 that the maximum value of is 0.5081, which is below the specified level of attenuation.

A solution to delay-dependent robust optimal control problem for a class of uncertain 2-D discrete state delay systems described by the GM with norm-bounded uncertainties has been presented. A sufficient condition for the existence of delay-de- pendent g-suboptimal state feedback robust controllers has been established under the LMI framework. A delay-dependent state feedback robust optimal controller is obtained through a convex optimization problem which can be solved by using MATLAB LMI toolbox [48] [49] . Finally, an application example has been provided to illustrate the effectiveness of the proposed technique.

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

Singh, A.K., Tandon, A. and Dhawan, A. (2016) Delay-De- pendent Robust H_¥ Control for Uncertain 2-D Discrete State Delay Systems Described by the General Model. Circuits and Systems, 7, 3645-3669. http://dx.doi.org/10.4236/cs.2016.711308