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# Delay-Dependent Guaranteed Cost Control of an Interval System with Interval Time-Varying Delay

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 360928 (2009)

## Abstract

This paper concerns the problem of the delay-dependent robust stability and guaranteed cost control for an interval system with time-varying delay. The interval system with matrix factorization is provided and leads to less conservative conclusions than solving a square root. The time-varying delay is assumed to belong to an interval and the derivative of the interval time-varying delay is not a restriction, which allows a fast time-varying delay; also its applicability is broad. Based on the Lyapunov-Ktasovskii approach, a delay-dependent criterion for the existence of a state feedback controller, which guarantees the closed-loop system stability, the upper bound of cost function, and disturbance attenuation lever for all admissible uncertainties as well as out perturbation, is proposed in terms of linear matrix inequalities (LMIs). The criterion is derived by free weighting matrices that can reduce the conservatism. The effectiveness has been verified in a number example and the compute results are presented to validate the proposed design method.

## 1. Introduction

Recently, stability analysis, and control synthesis of the uncertainty interval systems and the time-delay system have been discussed extensively [1–7]. Literature [4, 5] has proposed a design method for a specific structure of single-input interval system, it has further been developed in [6], it has proposed a solution technique of an interval system stability and control synthesis by using a Riccati equation, but for the parameter matrix, it has constraint conditions of full column rank when the control input matrix is the interval matrix and matrix factorization requires the solving a square root. Time-delay is generally a source of instability in practical engineering systems; considerable attention has been paid to the problem of stability analysis and controller synthesis for time-delay systems. The guaranteed cost-control approach aims at stabilizing the systems while maintaining an adequate level of performance represented by the quadratic cost [8–10]. Fragility is a common dynamic problem and is caused by many factors; reduction in size and cost of digital control hardware results in limitations in available computer memory and word length capabilities of the digital processor [11–15]. Literature [7] studies the guaranteed cost control of the interval system but does not consider the time delay and the fragility or the results presented by the proposed matrix conditions. The existing approaches are all limited and conservative, and the proposed Riccati equation algorithm cannot be guaranteed to be convergent [8]. Very little open literature covering the research guaranteed cost control of an interval system with time delay has been published and the fragility problem has not been considered.

The control is an effective way for dealing with the disturbance uncertainty [16]. Since the delay-dependent results are less conservative than the delay-independent ones, especially when the delay time is small, it is necessary to discuss the delay-dependent guaranteed cost control for interval time-varying delay systems. Recently, a special type of time delay in practical engineering systems, that is, interval time-varying delay, was identified and investigated [17–20]. A typical example of systems with interval time-varying delay is networked control systems (NCSs). Employing the Lyapunov-Ktasovskii approach, literature [19–21] requires both the upper bound of the time-varying delay and additional information on the derivative of the time-varying delay, while literature [17, 18, 22, 23] has no restriction on the derivative of the time varying delay, which allows a fast time-varying delay. Moreover, the general model transformation and bounding technique may be the sources of conservatism results; to further improve the performance of delay-dependent stability criteria, much effort has been devoted recently to the development of the free weighting matrices method [24], in which neither the bounding technique nor model transformation is employed.

In this paper, we present a new method of dealing with the problem of the delay-dependent stability and guaranteed cost control of interval system with interval time-varying delay based on the LMIs [25]. This method employs Lyapunov-Krasovskii functional and free weighting matrices approaches, which are used to reduce the conservative result, guaranteeing that the closed-loop system gives a better dynamic performance.

## 2. Problem Formulation

Consider uncertainty interval system with time-varying delay

where , and denote the state vector, the control vector, and the disturbance input, respectively, and is the controlled output. The time-varying delay is a time-varying continuous function satisfying

where are known constants. Moreover, and the initial condition denotes a continuous vector-valued initial function of . is the state matrix. is the input matrix, is the disturbance matrix, and and are appropriate dimension constant matrices.

Interval matrix and vary with the parameters and can be shown as [3, 10]

where

Let

where , are known real matrices; denotes that the th component is with all other entries being zeros and can be factorized into . and can be described as an equivalent form:

where , and uncertainty matrices satisfy

The actual control input implemented is assumed to be , where are known constant dimension matrices and satisfies

where is controller gain perturbation uncertainty bound.

The cost function associated with this system is

where are given weighting matrices.

Substituting (2.1) into (2.10) the resulting closed-loop system is

Our controllers design objective is described as follows.

The closed loop system (2.10) is asymptotically stable with disturbance attenuation , nonfragility , if the following is fulfilled for all time-varying delay and admissible uncertainties satisfying (2.7) and (2.8).

(1) The closed close system (2.10) is asymptotically stable.

(2) The closed loop system (2.10) guarantees, under aeroinitial conditions,

for all nonzero .

(3) The closed-loop cost function satisfies .

Therefore, the objective of this paper is to design a nonfragile guaranteed cost controller in the presence of time-varying delay, time-varying parameter uncertainty of an interval system, and uncertainty of the controller. Also the controller guarantees disturbance attenuation of the closed loop system from to .

Lemma 2.1 (see [26]).

Let , and be matrices of appropriate dimensions and assume is symmetric, satisfying , then

if and only if there exists a scalar satisfying

## 3. Main Results

Defining ,, where and can be taken as the mean value and range of variation of the time-varying delay. First consider a delay-dependent stability for the following nominal system of (2.1) :

Using the Leibniz-Newton formula, we can write

and system (3.1) can be rewritten as

The following theorem presents a sufficient condition for the existence of the nonfragile guaranteed cost controller.

Theorem 3.1.

A control law is said to be a non-fragile guaranteed cost control associated with cost matrix and of appropriate dimensions for the system (3.1) and cost function (2.9) and given scalars and . Suppose that the disturbance input is zero for all times , if the following matrix inequality

where

holds for all admissible uncertainty (2.7), (2.8), and any satisfying (2.2). The closed-loop cost function satisfies

Proof.

Choose a Lyapunov function as

where and .

Case 1 ().

Taking the time derivative of along any trajectory of the closed-loop system (3.3) is given by

It is easy to see that

According to (3.2), for any matrices , with appropriate dimensions, the following equations hold:

Then we have

where

wheredenotes . With are some parameter matrices of appropriate dimensions. The asymptotic stability is achieved if and which is equivalent. From Schur complement, (3.4) holds if and only if and simultaneously.

From (3.11), we have

According to inequality (3.4), implies and , then

can be obtained so that

Therefore, the closed-loop system (3.1) is asymptotically stable. Furthermore, by integrating both sides of the above inequality from to and using the initial condition,

is obtained. Thus, Theorem 3.1 is true.

Case 2 ().

Similar to the above analysis, it is easy to see that are the leading minor of obtained (3.11) and (3.13), respectively. Therefore, system (3.1) is asymptotically stable if and onlyif (3.4) holds. The closed-loop cost function satisfies (3.6).

Case 3 ().

Similar to Case 1, one can prove that system (3.1) is asymptotically stable.

The objective of this paper is to develop a procedure for determining a state feedback gain matrix which contains controller gain perturbation such that the control law is a non-fragile guaranteed cost control of the system (2.1), cost function (2.9), and disturbance attenuation .

Theorem 3.2.

A control law is said to be a non-fragile guaranteed cost control associated with cost matrix , and of appropriate dimensions for the system (2.1) and cost function (2.9), disturbance attenuation , and given scalars and , if the following matrix inequality

where denotes , and

holds for all admissible uncertainty (2.7) and (2.8). The closed-loop cost function satisfies (3.6).

Proof.

It has been noticed that (3.17) implies (3.4). Therefore (3.17) ensures the asymptotic stability of the closed loop system (2.1). Under zero initial condition ,

can be introduced. Then for any nonzero ,

where and . The condition of implies . Therefore when , the closed loop system (2.1) is asymptotically stable with disturbance attenuation and nonfragility .

Theorem 3.3.

There exist non-fragile guaranteed cost controllers for the system (2.1) and the cost function (2.9), disturbance attenuation , , and , **,** if there exists scalars , , symmetric positive matrices and, of appropriate dimensions and a matrix such that

where denotes denotes , and

Furthermore, if is a feasible solution to the inequality (3.21), then is a non-fragile guaranteed cost controller of the system (2.1), where the feedback gain matrix is given by and the corresponding closed-loop cost function satisfies (3.6).

Proof.

Let ,.

By manipulating the left-hand side in inequality (3.17), it follows that the inequality (3.17) is equivalent to

where

where denotes .

By applying Lemma 2.1, the above inequality (3.23) holds for all , satisfying , if and only if there exists a constant , such that

It follows from the Schur complement that the above inequality is further equivalent to the following inequality:

where denotes , and

Let , . By manipulating the left-hand side in inequality (3.17) again, it follows that the above inequality is equivalent to

where

and

Applying Lemma 2.1 again, the inequality (3.28) holds for all satisfying , if and only if there exists a constant , such that

It follows from the Schur complement again that the above inequality is further equivalent to the following inequality:

and

By pre- and postmultiplying the left-hand side matrix in the above inequality by the matrix , respectively, and defining the matrix , , , , and, If we set then , and , and it can be concluded that the above matrix inequality is equivalent to (3.21). The proof is complete.

## 4. A Numerical Example

Consider system (2.1) with [17]

and By applying Theorem 3.3 with and , the controller gain is obtained:

where it is also a result :

For , the maximum allowable upper bound of the delay is which is larger than derived in Jiang and Han in [17]. This means that any satisfyies .

The associated upper bound over the closed-loop cost function is .

The obtained robust and non-fragile optimal guaranteed cost controller guarantees the asymptotic stability, disturbance attenuation, , and the upper bound of cost function .

## 5. Conclusion

This paper has considered the problem of delay-dependent stability and guaranteed cost control with interval time-varying delay for an interval system based on Lyapunov–Krasovskii functional approach. The delay-dependent stabilization criterion for guaranteed cost control has been formulated in terms of LMIs. The derivative of the interval time-varying delay is not a restriction, which allows a fast time-varying delay and alsois more close to practices control object. A numerical example has shown the effectiveness of the method.

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## Acknowledgment

This work is supported by the National Science Foundation of China (no. 60134010).

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Xiao, M., Shi, Z. Delay-Dependent Guaranteed Cost Control of an Interval System with Interval Time-Varying Delay.
*J Inequal Appl* **2009, **360928 (2009). https://doi.org/10.1155/2009/360928

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### Keywords

- Disturbance Attenuation
- Interval Matrix
- Guarantee Cost Control
- Interval System
- Admissible Uncertainty