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This paper considers the problem of delay-dependent robust optimal
H
_{∞}
control for a class of uncertain two-dimensional (2-D) discrete state delay systems described by the general model (GM). The parameter uncertainties are assumed to be norm-
bounded. A linear matrix inequality (LMI)-based sufficient condition for the existence of delay-dependent
g
-suboptimal state feedback robust H
_{∞}
controllers which guarantees not only the asymptotic stability of the closed-loop system, but also the H
_{∞}
noise attenuation
g
over all admissible parameter uncertainties is established. Furthermore, a convex optimization problem is formulated to design a delay-de
pendent state feedback robust optimal H
_{∞}
controller which minimizes the H
_{∞}
noise attenuation
g
of the closed-loop system. Finally, an illustrative example is provided to demonstrate the effectiveness of the proposed method.

In the past decades, research on two-dimensional (2-D) discrete systems has rapidly increased due to their extensive practical applications in circuits analysis [

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 [_{¥} 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 [_{¥} 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 [_{¥} 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 [_{¥} 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 [_{¥} 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 [_{¥} control for 2-D discrete state-delayed systems described by the FM second model and Roesser model have been considered in [_{¥} controller has been obtained for both the models. It may be mentioned here that the criteria presented in [_{¥} 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: R^{n} denotes real vector space of dimension n;

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

where

and

where

(or equivalently,). (1e)

For system (1), suppose a finite set of initial conditions [

Denote

Definition 1. [

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

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

Lemma 1. [

where

Lemma 2. [

holds for all

The

Definition 2. [

where

and

In the case when the initial condition is known to be zero, then the

Using the 2-D Parseval’s theorem [

where

is the transfer function from the disturbance input

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

In this section, we first present a delay-dependent approach to solve the

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

where

Proof. To prove that the closed-loop system (4) with

where

and

Along any trajectory of the system (1) with

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

Hence, the closed-loop system (4) with

Next, we establish the

We consider

where

It follows from inequality (12) that

Summing the inequality (22) over

Now,

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

Since,

In the following, we will show that the above derived sufficient condition for existence of delay-dependent robust

Theorem 2. Consider the closed-loop system (4) with the initial condition (2). Given scalars

where

then the closed-loop system (4) with

Proof. It follows from matrix inequality (12) that

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

and its transpose, respectively, we get

Denoting

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

which ensures the minimization of

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 [

where

(34) can be expressed in the following form:

where

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

Denote

Next, consider the problem of delay-dependent

To consider the problem of

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 [

and a delay-dependent optimal

When

A solution to delay-dependent robust optimal

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