- Research Article
- Open Access

# Stabilization of Discrete-Time Control Systems with Multiple State Delays

- Medina Rigoberto
^{1}Email author

**2009**:240707

https://doi.org/10.1155/2009/240707

© Medina Rigoberto. 2009

**Received:**16 March 2009**Accepted:**21 June 2009**Published:**21 July 2009

## Abstract

We give sufficient conditions for the exponential stabilizability of a class of perturbed time-varying difference equations with multiple delays and slowly varying coefficients. Under appropiate growth conditions on the perturbations, combined with the "freezing" technique, we establish explicit conditions for global feedback exponential stabilizability.

## Keywords

- Lyapunov Function
- Exponential Stabilizability
- Nonlinear Control System
- Global Exponential Stability
- Multiple Delay

## 1. Introduction

and is a given vector-valued function, that is,

asymptotically stable in the Lyapunov sense.

if the evolution operator generated by is stable, then the delay control system (1.1)-(1.2) is asymptotically stabilizable under appropiate conditions on (see [4, 8, 14]). For infinite-dimensional control systems, the study of stabilizabilization is more complicated and requires sophisticated techniques from semigroup theory.

The concept of stabilizability has been developed and successfully applied in different settings (see, e.g., [9, 15, 16]). For example, finite- and infinite-dimensional discrete-time control systems have been studied extensively (see, e.g., [2, 5, 6, 10, 17–20]).

as a particular case of the system (1.1), with its time dependence "frozen" at time Thus, in this paper it is shown that if each frozen system is exponentially stabilizable and the rate of change of the coefficients of system (1.1) is small enough, then the nonautonomous system (1.1)-(1.2) is indeed exponentially stabilizable.

The purpose of this paper is to establish sufficient conditions for the global exponential feedback stabilizability of perturbed control systems with both time-varying and time-delayed states.

Our main contributions are as follows. By applying the "freezing" technique to the control system (1.1)-(1.2), we derive explicit stabilizability conditions, provided that the coefficients are slowly varying. Applications of the main results to control systems with many delays and nonlinear perturbations will also be established in this paper. This technique will allow us to avoid constructing the Lyapunov functions in some situations. For instance, it is worth noting that Niamsup and Phat [2] established sufficient stabilizability conditions for the zero solution of a discrete-time control system with many delays, under exponential growth assumptions on the corresponding transition matrix. By contrast, our approach does not involve any stability assumption on the transition matrix.

The paper is organized as follows. In Section 2 we introduce notations, definition, and some preliminary results. In Section 3, we give new sufficient conditions for the global exponential stabilizability of discrete-time systems with time-delayed states. Finally, as an application, we consider the global stabilization of the nonlinear control systems.

## 2. Preliminaries

where is a variable matrix.

where is the Hilbert-Schmidt (Frobenius) norm of ; that is,

is true, and will be useful to obtain some estimates in this work.

Theorem 2 A (11,Theorem 3.7).

holds for every nonnegative integer , where is the spectral radius of , and .

Remark 2.1.

In general, the problem of obtaining a precise estimate for the norm of matrix-valued and operator-valued functions has been regularly discussed in the literature, for example, see Gel'fond and Shilov [24] and Daleckii and Krein [25].

The following concepts of stability will be used in formulating the main results of the paper (see, e.g., [26]).

Definition 2.2.

Definition 2.3.

for any solution of (1.4) with the initial conditions (1.2).

Definition 2.4.

Remark 2.5.

The control is a feedback control of the system.

Definition 2.6.

System (1.1) is said to be globally exponentially stabilizable (at ) by means of the feedback law (2.1) if there is a variable matrix such that the zero solution of (1.4) is globally exponentially stable.

## 3. Main Results

Now, we are ready to establish the main results of the paper, which will be valid for the system (1.1)-(1.2) with slowly varying coefficients.

subject to the initial conditions (1.2), where is a given integer and is a variable matrix.

Proposition 3.1.

Suppose that

Proof.

This proves the global stability of the zero solution of (3.1)–(1.2).

with small enough, where is a solution of (3.1).

Applying the above reasoning to (3.14), according to inequality (3.3), it follows that is a bounded function. Consequently, relation (3.13) implies the global exponential stability of the zero solution of system (3.1)–(1.2).

is not an easy task. However, in this section we will improve the estimates to these formulae.

Proposition 3.2.

then the zero solution of system (3.1)–(1.2) is globally exponentially stable.

Proof.

Relation (3.26) proves the global stability of the zero solution of system (3.1)–(1.2). Establishing the exponential stability of this equation is enough to apply the same arguments of the Proposition 3.1.

Theorem 3.3.

- (i)
- (ii)
and

- (iii)

then system (1.1)-(1.2) is globally exponentially stabilizable by means of the feedback law (2.1).

Proof.

where

According to (i), (ii), and (iii), the conditions (b) and (3.17) hold. Furthermore, condition (3.28) assures the existence of a matrix function such that condition (3.18) is fulfilled. Thus, from Proposition 3.2, the result follows.

where the minimum is taken over all matrices satisfying (i), (ii), and (iii).

Corollary 3.4.

then the system (1.1)-(1.2) is globally exponentially stabilizable by means of the feedback law (2.1).

Hence, Theorem 3.3 implies the following corollary.

Corollary 3.5.

Then system (3.32)-(1.2), under condition (a), is globally exponentially stabilizable by means of the feedback law (2.1).

Example 3.6.

where and

If and are small enough such that for some and we have then by Theorem 3.3, system (3.35)-(3.36), under conditions (3.37) and (3.38), is globally exponentially stabilizable.

where ( ) are variable matrices,

Theorem 3.7.

for some positive numbers and

is asymptotically stable.

Theorem 3.8.

then system (3.51)-(3.52) is globally exponentially stabilizable by means of the feedback law (2.1).

Proof.

where

Thus, by reasoning as in Theorem 3.3, and using the estimates established in Proposition 3.2, the result follows.

## Declarations

### Acknowledgment

This research was supported by Fondecyt Chile under Grant no. 1.070.980.

## Authors’ Affiliations

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