Research Article | Open Access

Wei Zhu, "A Sufficient Condition for Asymptotic Stability of Discrete-Time Interval System with Delay", *Discrete Dynamics in Nature and Society*, vol. 2008, Article ID 591261, 7 pages, 2008. https://doi.org/10.1155/2008/591261

# A Sufficient Condition for Asymptotic Stability of Discrete-Time Interval System with Delay

**Academic Editor:**Manuel De La Sen

#### Abstract

The asymptotic stability of discrete-time interval system with delay is discussed. A new sufficient condition for preserving the asymptotic stability of the system is presented by means of the inequality techniques. By mathematical analysis, the stability criterion is less conservative than that in previous result. Finally, one example is given to demonstrate the applicability of the present scheme.

#### 1. Introduction

The stability analysis of interval system is very useful for the robustness analysis of nominally stable system subject to model perturbations. Therefore, there has been considerable interest in the stability analysis of interval system in literature ([1–15], and references therein). In general, those approaches can be classified into two categories: the first is the polynomial and the second is the matrix approach. However, due to information transmission between elements or systems, data computation, natural property of system elements, and so forth, time delays also inherently exist in controlled systems and therefore must be integrated into system models. The stability analysis for interval systems with delays becomes more complicated. In [6], a sufficient condition for the stability of discrete-time systems is given in terms of pulse-response sequence matrix. In [11], based on the Gersgorin theorem, the stability testing problem for continuous and discrete systems including a time delay is discussed.

The objective of this paper is to deal with the asymptotic stability of a discrete-time interval system with delay. Based on the inequality techniques [16], a new sufficient condition for preserving the asymptotic stability of the system is presented. By mathematical analysis, the stability criterion is less conservative than that in previous result. An example is given to compare the proposed method with one reported.

#### 2. System Description and Notations

Consider the discrete-time interval system with delay described by where delay is a positive integer, , , are the interval matrices described as , , , are bounded.

In the sequel, the following notations will be used: : the space of -dimensional (nonnegative) real column vectors; : the set of (nonnegative) real matrices; : the spectral radius of matrix ; : each pair of corresponding elements of and satisfies the inequality “," where or ; : for , ; : the vector (or matrix) obtained by replacing each entry of by its absolute value; : , where are nonnegative integers; if , we define , where is the empty set; if , we write as ; : the set of matrices obtained by exchanging corresponding column () of () and (), so there are matrices in each , where , and denotes the factorial of .

#### 3. Main Result

In order to prove our main result, we first need the following technical lemmas.

Lemma 3.1 (see [17, Theorem 8.3.1]). * If , then there is a nonnegative vector , such that . *

So it is clear that is not empty by Lemma 3.1.

Lemma 3.2. *Let , satisfy that ** If** then there
exists a constant such that** for some .*

*Proof. *Since , using continuity, there must be a sufficiently small
constant such that
Let so we have By (3.1), we have Since , we derive that

We next show that for any , If this is not
true, then there must be a positive constant and some
integer such that By using (3.4)
and (3.8), we obtain that which
contradicts the first inequality of (3.10). Thus (3.9) holds for all . Therefore, we have and the proof
is completed.

Theorem 3.3. *
For any , , if the inequality** holds, then the
discrete-time interval system (2.1) is asymptotically stable.*

*Proof. *Let , , where satisfying are integers
and or is equivalent
to that or is empty, respectively.
Obviously, and are finite
sets.

By the definitions of and , we can obtain matrices by exchanging
the corresponding columns of and (if , then , and matrices by exchanging
the corresponding columns of and (if , then such that the
following inequalities hold: So together with (2.1), we have From the above,
we see that and depend only on
the position of the negative components of and , respectively.

Then, from (3.15), we derive So we have or

Since , , by the definitions of and again, (3.17) and
(3.18), for any , we can find corresponding matrices , , such that In view of
condition (3.13), we obtain that Thus, by
Lemma
3.2 and (3.19),
(3.20), for any , there exist constants and some , , such that Set , , , , , obviously, and are independent
of any choice of , so by (3.21), we derive that which implies
that the conclusion of the theorem holds.

*Remark 3.4. *By the meanings of and , we know that , , . So there are matrices in the
sets and , respectively. Furthermore, if number is even, that
is, , then the equality () for is transformed
to be () and then contains only different
matrices. Therefore, condition (3.13) can be verified easily and quickly by computer
software (such as MATLAB).

Corollary 3.5 (see [11, Theorem IV]). *
The discrete-time
interval system (2.1) is asymptotically stable if the following condition is
satisfied: ** where matrices and are defined as , , .*

*Proof. *Clearly, for any , , the inequality holds, then from [18] (i.e., for , if , then ) and associated
with (3.23), we have Therefore, system (2.1) is asymptotically stable in terms
of Theorem 3.3.

#### 4. Illustrative Examples

*Example 4.1. *Consider the discrete-time
interval system (2.1) with delay and For this case, By simple calculation, we have for and So we have

Therefore, the system (2.1) is asymptotically stable by
means of Theorem 3.3.

In what follows, the simulation result is illustrated
in Figure 1.

*Remark 4.2. * If [11, Theorem IV] is applied to
Example 4.1, we obtain where , are defined by
Corollary 3.5, that is,
[11, Theorem IV]. Then that is,
[11, Theorem IV] cannot be applied. So the sufficient condition (3.13) proposed
in this paper is less conservative than condition (3.23) proposed by [11].

#### 5. Conclusion

In this paper, we have investigated the asymptotic stability of discrete-time interval system with delay. A new sufficient condition for preserving the asymptotic stability of the system is developed. By mathematical analysis, the presented criterion is to be less conservative than that proposed by [11]. So, the result of this paper indeed allows us to have more freedom for checking the stability of the discrete-time interval systems with delay. From the proposed example, it is easily seen that the criterion presented in this paper for the stability of the discrete-time interval system with delay is very helpful. We believe that the present scheme is applicable to robust control design.

#### Acknowledgments

The author wishes to thank the editor and the referees for their helpful and interesting comments. The work is supported by National Natural Science Foundation of China under Grant 10671133 and the Doctor’s Foundation of Chongqing University of Posts and Telecommunications A2007-41.

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#### Copyright

Copyright © 2008 Wei Zhu. 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.