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

- Medina Rigoberto
^{1}Email author

**2009**:240707

**DOI: **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.

## 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

- (a)
- (b)(3.2)
- (c)

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

## References

- Lakshmikantham V, Leela S, Martynyuk AA:
*Stability Analysis of Nonlinear Systems, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 125*. Marcel Dekker, New York, NY, USA; 1989:xii+315.Google Scholar - Niamsup P, Phat VN:
**Asymptotic stability of nonlinear control systems described by difference equations with multiple delays.***Electronic Journal of Differential Equations*2000,**2000**(11):1–17.MathSciNetGoogle Scholar - Sun YJ, Hsieh JG:
**Robust stabilization for a class of uncertain nonlinear systems with time-varying delay: Razumikhin-type approach.***Journal of Optimization Theory and Applications*1998,**98**(1):161–173. 10.1023/A:1022645116123MATHMathSciNetView ArticleGoogle Scholar - Phat VN, Kiet TT:
**On the Lyapunov equation in Banach spaces and applications to control problems.***International Journal of Mathematics and Mathematical Sciences*2002,**29**(3):155–166. 10.1155/S0161171202010840MATHMathSciNetView ArticleGoogle Scholar - Recht B, D'Andrea R:
**Distributed control of systems over discrete groups.***IEEE Transactions on Automatic Control*2004,**49**(9):1446–1452. 10.1109/TAC.2004.834122MathSciNetView ArticleGoogle Scholar - Sasu B, Sasu AL:
**Stability and stabilizability for linear systems of difference equations.***Journal of Difference Equations and Applications*2004,**10**(12):1085–1105. 10.1080/10236190412331314178MATHMathSciNetView ArticleGoogle Scholar - Feliachi A, Thowsen A:
**Memoryless stabilization of linear delay-differential systems.***IEEE Transactions on Automatic Control*1981,**26**(2):586–587. 10.1109/TAC.1981.1102653MATHMathSciNetView ArticleGoogle Scholar - Benabdallah A, Hammami MA:
**On the output feedback stability for non-linear uncertain control systems.***International Journal of Control*2001,**74**(6):547–551. 10.1080/00207170010017383MATHMathSciNetView ArticleGoogle Scholar - Alabau F, Komornik V:
**Boundary observability, controllability, and stabilization of linear elastodynamic systems.***SIAM Journal on Control and Optimization*1999,**37**(2):521–542. 10.1137/S0363012996313835MATHMathSciNetView ArticleGoogle Scholar - Boukas EK:
**State feedback stabilization of nonlinear discrete-time systems with time-varying time delay.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(6):1341–1350. 10.1016/j.na.2006.01.020MATHMathSciNetView ArticleGoogle Scholar - Bourlès H:
**Local -stability and local small gain theorem for discrete-time systems.***IEEE Transactions on Automatic Control*1996,**41**(6):903–907. 10.1109/9.506248MATHView ArticleGoogle Scholar - Chukwu EN:
*Stability and Time-Optimal Control of Hereditary Systems, Mathematics in Science and Engineering*.*Volume 188*. Academic Press, New York, NY, USA; 1992:xii+508.Google Scholar - Sontag ED:
**Asymptotic amplitudes and Cauchy gains: a small-gain principle and an application to inhibitory biological feedback.***Systems & Control Letters*2002,**47**(2):167–179. 10.1016/S0167-6911(02)00191-3MATHMathSciNetView ArticleGoogle Scholar - Trinh H, Aldeen M:
**On robustness and stabilization of linear systems with delayed nonlinear perturbations.***IEEE Transactions on Automatic Control*1997,**42**(7):1005–1007. 10.1109/9.599983MATHMathSciNetView ArticleGoogle Scholar - Gaĭshun IV:
**Controllability and stabilizability of discrete systems in a function space on a commutative semigroup.***Difference Equations*2004,**40**(6):873–882.MATHView ArticleGoogle Scholar - Megan M, Sasu AL, Sasu B:
**Stabilizability and controllability of systems associated to linear skew-product semiflows.***Revista Matemática Complutense*2002,**15**(2):599–618.MATHMathSciNetView ArticleGoogle Scholar - Gil MI, Medina R:
**The freezing method for linear difference equations.***Journal of Difference Equations and Applications*2002,**8**(5):485–494. 10.1080/10236190290017478MATHMathSciNetView ArticleGoogle Scholar - Jiang Z-P, Lin Y, Wang Y:
**Nonlinear small-gain theorems for discrete-time feedback systems and applications.***Automatica*2004,**40**(12):2129–2136.MATHMathSciNetGoogle Scholar - Karafyllis I:
**Non-uniform in time robust global asymptotic output stability for discrete-time systems.***International Journal of Robust and Nonlinear Control*2006,**16**(4):191–214. 10.1002/rnc.1037MATHMathSciNetView ArticleGoogle Scholar - Karafyllis I:
**Non-uniform robust global asymptotic stability for discrete-time systems and applications to numerical analysis.***IMA Journal of Mathematical Control and Information*2006,**23**(1):11–41.MATHMathSciNetView ArticleGoogle Scholar - Bylov BF, Grobman BM, Nemickii VV, Vinograd RE:
*The Theory of Lyapunov Exponents*. Nauka, Moscow, Russia; 1966.Google Scholar - Shahruz SM, Schwartz AL:
**An approximate solution for linear boundary-value problems with slowly varying coefficients.***Applied Mathematics and Computation*1994,**60**(2–3):285–298. 10.1016/0096-3003(94)90110-4MATHMathSciNetView ArticleGoogle Scholar - Vinograd RE:
**An improved estimate in the method of freezing.***Proceedings of the American Mathematical Society*1983,**89**(1):125–129. 10.1090/S0002-9939-1983-0706524-1MATHMathSciNetView ArticleGoogle Scholar - Gel'fand IM, Shilov GE:
*Some Questions of Differential Equations*. Nauka, Moscow, Russia; 1958.MATHGoogle Scholar - Daleckii Yul, Krein MG:
*Stability of Solutions of Differential Equations in Banach Spaces*. American Mathematical Society, Providence, RI, USA; 1971.Google Scholar - Berezansky L, Braverman E:
**Exponential stability of difference equations with several delays: recursive approach.***Advances in Difference Equations*2009,**2009:**-13.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.