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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 401572, 8 pages

http://dx.doi.org/10.1155/2013/401572

## Input-to-State Stability of Singularly Perturbed Control Systems with Delays

^{1}School of Mathematics and Statistics, Chongqing Three Gorges University, Wanzhou 404000, China^{2}School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China

Received 16 December 2012; Revised 9 April 2013; Accepted 14 May 2013

Academic Editor: Kang Liu

Copyright © 2013 Yongxiang Zhao et al. 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.

#### Abstract

We study the input-to-state stability of singularly perturbed control systems with delays. By using the generalized Halanay inequality and Lyapunov functions, we derive the input-to-state stability of some classes of linear and nonlinear singularly perturbed control systems with delays.

#### 1. Introduction

The stability properties of control systems are an important research field. The concept of input-to-state stability (ISS) of the control systems was proposed by Sontag [1]. Since then, the ISS of the control systems has been widely studied (cf. [2–12]), and most of the obtained results are often based on the Lyapunov functions.

Singularly perturbed control systems are a special class of control systems which is characterized by small parameters multiplying the highest derivates. Recently, many attentions have been devoted to the study of singularly perturbed systems, in particular, to their stability properties. Saberi and Khalil [13] investigated the asymptotic and exponential stability of nonlinear singularly perturbed systems. They obtained a quadratic-type Lyapunov function as a weighted sum of quadratic-type Lyapunov functions of the reduced and the boundary-layer systems. They used the composite Lyapunov function to estimate the degree of exponential stability and the domain of attraction of stable equilibrium point. Corless and Glielmo [14] obtained some results and properties related to exponential stability of singularly perturbed systems. They pointed out that, if both the reduced and the boundary-layer systems are exponentially stable, then, provided that some further regularity conditions are satisfied, the full-order system is exponentially stable for sufficiently small value . Liu et al. [15] derived the exponential stability criteria of singularly perturbed systems with time delay. Christofides and Teel [11] obtained a type of total stability for the input-to-state stability property with respect to singular perturbations under the assumptions that the reduced system is ISS and the boundary-layer system is uniformly globally asymptotically stable. Tian [16, 17] discussed the analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations. There are some results about the stability of numerical methods for control systems (cf. [18, 19]).

The previous studies have mainly focused on the exponential stability of singularly perturbed systems with or without delays and the ISS of singularly perturbed control systems without delay. There are no results about the ISS of delay singulary perturbed control systems. In this paper, we study the ISS of some classes of delay singularly perturbed control systems. By using the generalized Halanay inequality and the Lyapunov functions, we obtain the sufficient conditions under which these delay singularly perturbed control systems are input-to-state stable.

#### 2. Preliminary

We introduce the following symbols (cf. [8, 11, 15]).(1) denotes the standard Euclidean norm of a vector, and denotes the norm of an matrix . A matrix is bounded means that .(2) denotes the transpose of the matrix , denotes the th eigenvalue of the matrix , and denotes the real part of .(3)The matrix means that is positive-definite. The vector means each component .(4)A real matrix with for all is an -matrix if is nonsingular and .(5)For any measurable locally essentially bounded function , .(6)A function : is a -function if it is continuous, strictly increasing, and .(7)A function : is a -function if, for each fixed , the function , and for each fixed , the function is decreasing and as .

Lemma 1 (see [20]). *Let be an matrix whose elements are continuous functions defined on the time interval and the following assumptions hold:
**Then there exists a positive-definitive matrix such that the following algebraic Lyapunov equation holds:
**
where are constants, is the identity matrix, and is bounded.*

The following generalized Halanay inequality will play a key role in studying the ISS for the system (9).

Lemma 2 (generalized Halanay inequality (see [16, 17])). *Suppose
**
Here is a non-negative real-value continuous function, , , , and are continuous with , , and for and . Then
** where and is defined as
*

Lemma 3. *Let and be matrix-value functions, ,, and let be vector functions of dimensions . Assume that *(i)* is bounded;*(ii)* with ;*(iii)*;
*(iv)*,
** for , where , ,. Then the following estimate holds
**
where is defined as
*

* Proof. *Let . Then

Moreover, by the conditions (i)–(iii), the estimate (6) can be derived as a consequence of (3)–(5) and (8).

Consider the delay singularly perturbed control systems where is the “time,” and are the state variables, is the control input which is locally essentially bounded, is the singular perturbation parameter, and is a constant time delay. The sufficiently smooth mapping has bounded derivatives and . and are given vector-functions and the derivative of exists.

*Definition 4. *The delay singularly perturbed control system (9) is ISS if there exist -functions : and -functions such that, for any initial functions and each essentially bounded input , the solution of (9) satisfy
where are the solutions of (9), .

#### 3. Linear Systems

In this section, we are concerned with ISS of the following linear delay singularly perturbed control systems as a special class of (9): Here we let , , , , and for simplicity; , , , , , and are smooth matrix functions of , and is nonsingular for every . Now, we introduce some assumptions.

*Assumption 5. * There exist positive constants and such that, for for all ,

From Assumption 5 and Lemma 1, we can easily show that there exist the differentiable positive-definite matrices and such that where are identity matrices, respectively, [21] shows that Assumption 5 guarantees that Reference, for every , (13a), (13b) have unique positive-definite solutions and given by respectively. It follows from the boundness and the positive-definiteness of and that there exist positive constants such that

*Assumption 6. * There exist bounded functions such that
where

*Assumption 7. *(1) There exists a positive number such that is an -matrix;

(2) ;

(3) with ;

(4) ,

where

Theorem 8. *If Assumptions 5–7 hold, then the delay singularly perturbed control system (11) is input-to-state stable for .*

* Proof. *Let , . For the derivative of along the trajectory of (11), we have
For the derivative of along the trajectory of (11), we have

From of Assumption 7, we can derive and the following inequalities for and :
It follows from Lemma 3 that there exist positive constants such that
where and is defined by
By the definition of and the positive-definiteness of , we have
where and . Moreover,
Thus, (25) and the inequality
imply that
where , . The proof is complete.

#### 4. Nonlinear Systems

In this section, we are concerned with ISS of the following nonlinear delay singularly perturbed control systems as a special class of (9): where , , , and . Assume that (28) has a unique equilibrium at the origin and the functions and are smooth enough and the derivative of exists.

*Assumption 9. * There exist positive constants for all such that

If Assumption 9 holds, then there exist the differentiable positive-definite matrices and such that where are identity matrices, respectively. It follows from the boundness and the positive-definiteness of and that there exist positive constants such that

*Assumption 10. * There exist bounded functions such that
where

*Assumption 11. *(1) There exist a positive number such that is an -matrix;

(2) ;

(3) with ;

(4) ,

where

Theorem 12. *If Assumptions 9–11 hold, then the delay singularly perturbed control system (28) is input-to-state stable for .*

* Proof. *Let , . For the derivative of along the trajectory of (28), we have
For the derivative of along the trajectory of (28), we have

From of Assumption 11, we can derive and the following inequalities for :
It follows from Lemma 3 that there exist positive constants such that
where and is defined as in (23).

By the definitions of , the positive-definiteness of , and the similar proof process to that of Theorem 8, we can obtain
where , , and . The proof is complete.

#### 5. Examples

*Example 1. *Consider the following linear delay system as an application of Theorem 8:
Let , , . Then
So we obtain the matrices
If we require that the constant satisfies ; that is, , then, we can take such that it is easy to show that the conditions in Assumptions 5–7 will be satisfied for any . Moreover, by Theorem 8, the system (40) is ISS for .

*Example 2. *Consider the following nonlinear delay system as an application of Theorem 12:

Let , , and . Then So we obtain the matrices If we require that the constant satisfies ; that is, , then, we can take such that it is easy to show that the conditions in Assumptions 9–11 will be satisfied for any . Moreover, by Theorem 12, the system (44) is ISS for .

#### 6. Conclusion

In this paper, we have studied the input-to-state stability of two classes of the linear and nonlinear delay singularly perturbed control systems. The generalized Halanay inequality and the Lyapunov function play important roles in obtaining the stability results. The sufficient conditions of input-to-state stability for delay singularly perturbed control systems are given.

#### Acknowledgment

This work is supported by projects NSF of China (11126329, 11271311, 11201510), NSF of Hunan Province (09JJ3002), Projects Board of Education of Chongqing City (KJ121110). The authors express their sincere thanks to the referees for their useful comments and suggestions, which led to improvements of the presentation.

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