Research Article | Open Access

# Control for Nonlinear Systems with Time-Varying Delay Using Matrix-Based Quadratic Convex Approach

**Academic Editor:**Qingling Zhang

#### Abstract

control problem for nonlinear system with time-varying delay is considered by using a set of improved Lyapunov-Krasovskii functionals including some integral terms, and a matrix-based on quadratic convex, combined with Wirtinger's inequalities and some useful integral inequality. controller is designed via memoryless state feedback control and new sufficient conditions for the existence of the state feedback for the system are given in terms of linear matrix inequalities (LMIs). Numerical examples are given to illustrate the effectiveness of the obtained result.

#### 1. Introduction

The phenomena of time delays are often encountered in many practical systems such as process control systems, manufacturing systems, networked control systems, and economic systems. The existence of these delays may be the source of instability and serious deterioration in the performance of the closed-loop systems. In real world systems especially, the delay should be assumed to be time-varying satisfying and is not necessarily restricted to be 0, namely, interval time-varying delay. Stability analysis of time-delay system has been investigated extensively in the past decades [1â€“25].

As of time delays, it is well known that the nonlinear perturbations can also cause instability and poor performance of practical systems. Therefore, the stability problem of time-delay systems with nonlinear perturbations has received increasing attention; see [9, 13, 16] and the references cited therein.

control problem has been widely used to minimize the effects of the external disturbances. The purpose of the problem is to design an controller to robustly stabilize the systems while guaranteeing a prescribed level of disturbance attenuation in the sense for the systems with external disturbances. A delay-dependent controller ensures asymptotic stability and a prescribed performance level of the closed-loop systems. The performance indexes and the upper bound of the delay are usually two performance indexes to be used to evaluate the conservatism of the derived condition. The conservatism of the delay-dependent control is measured by the allowable delay size or performance level bound obtained.

Recently, an improved robust stability and performance analysis criterion has been reported [3â€“5, 7, 10â€“12, 15, 17, 18]. In [17], control problem for uncertain linear system with state delay and parameter uncertainties has been studied, but the time-varying delay is only bounded above by a constant. control problem for system with interval time-varying delay has been considered in [7], by employing free weighting matrices approach. However, some useful terms in estimating the derivative of Lyapunov-Krasovskii functional are ignored which might lead to some conservatism. In [15], control problem for nonlinear systems with interval time-varying delay has been studied by using Jensenâ€™s inequality to estimate some integral terms of Lyapunov-Krasovskii functional and deriving delay-dependent sufficient condition for the existence of control by using reciprocally convex combination technique.

In the study of time-delay system, several approaches have been proposed in order to reduce conservatism. For example, it is well known that choosing appropriate Lyapunov-Krasovskii functional and using improved bounding techniques to estimate time-derivative of Lyapunov-Krasovskii functional lead to improvement of stability region. Furthermore, free weighting matrices approach; delay decomposition approach; and convex optimization and reciprocally convex optimization techniques have been widely used to reduce conservatism of stability criterion; see [3, 7, 8, 10, 11, 13, 14, 16]. Recently, the so-called matrix-based quadratic convex approach has been introduced to derive stability criterion for time-delay system which was shown to reduce conservatism; for example, it gives better maximum allowable upper bound for time-varying delay than some other existing approaches; see [20â€“22].

Motivated by the above discussions, in this paper, matrix-based quadratic convex approach will be used to study control problem for system with interval time-varying delay and nonlinear perturbations. To the best of our knowledge, this is one of the first reports of such investigation. By introducing new augmented Lyapunov-Krasovskii functional which has not been considered yet in stability analysis of control problem, a delay-dependent stability criterion and performance analysis are derived in terms of linear matrix inequalities (LMIs). This new Lyapunov-Krasovskii functional consists of integral terms of the form â€‰â€‰ which allows us to use the matrix-based quadratic convex approach introduced in [20â€“22]. With the use of this new Lyapunov-Krasovskii functional, matrix-based quadratic convex approach combined with some improved bounding techniques for integral terms such as Wirtinger-based integral inequality [14, 20], some new cross terms will be introduced which enhance the feasible stability criterion. Through two numerical examples, it is shown that the obtained stability criterion may give a larger maximum allowable upper bound of time-varying delay than some existing results.

#### 2. Mathematical Model and Preliminaries

The following notations will be used in this paper: denotes the set of all nonnegative real numbers; denotes the -dimensional space with the Euclidean norm ; denotes the space of all matrices of -dimensions.

denotes the transpose of matrix ; is symmetric if ; denotes the identity matrix; denotes the set of all eigenvalues of ; .

, ; denotes the set of all -valued continuous functions on ; denotes the set of all the -valued square integrable functions on .

Matrix is called semipositive definite if , for all ; is positive definite if for all ; means . The symmetric term in a matrix is denoted by .

Consider the following system with time-varying delays and control input:where is the state; is the control input; is a disturbance input; and is the observation output. The delay is time-varying continuous function which satisfies Let . The nonlinear functions , satisfy the following growth condition:

*Definition 1. *Given , the control problem for system (1) is to seek if there exists a memoryless state feedback controller such that we have the following.(i)The zero solution of the closed-loop system, where ,â€‰is asymptotically stable.(ii)The performance â€‰of the closed-loop system (5) is guaranteed for all nonzero and a prescribed under the condition . In this case, we say that the feedback control asymptotically stabilizes the system.

We introduce the following technical lemmas, which will be used in the proof of our results.

Lemma 2 (see [14]). *For a given matrix , the following inequality holds for any continuously differentiable function :**where *

*Remark 3. *It is clear to see that the inequality in Lemma 2 provides a tighter lower bound for than Jensenâ€™s inequality since for . Thus, inequality (7) is an improvement over Jensenâ€™s inequality.

Before we introduce some useful integral inequalities, we denote

Lemma 4 (see [20]). *For a given scalar and any real matrices and and a vector such that the integration concerned below is well defined, the following inequality holds for any vector-valued function and matrices and satisfying , **where is defined in (9).*

Lemma 5 (see [21]). *Let be a continuous function satisfying . For any real matrix and a vector such that the integration concerned below is well defined, the following inequality holds for any and real matrices , satisfying , ,**where**with () being defined in (9).*

Lemma 6 (see [20]). *Let be a continuous function satisfying . For any real matrix and a vector such that the integration concerned below is well defined, the following inequality holds for any real matrix satisfying ,**where and**with () being defined in (9).*

*Remark 7. *In Lemma 6, when , the inequality reduces to similar one in [14]. It contains slack matrix variable with dimension comparing to slack matric variable with dimension introduced in [8].

Lemma 8 (see [20]). *Let , , and be real symmetric matrices and a continuous function satisfy , where and are constants satisfying . If , then**or*

#### 3. Main Results

In this section, we give a design of memoryless feedback control for system (1). First, we present delay-dependent asymptotical stabilizability analysis conditions for the nonlinear system with time-varying delay (1). Now, we operate the matrix-based quadratic convex approach with the integral inequalities in [20] to formulate a new stability criterion for system (1). For our goal, we choose the following Lyapunov-Krasovskii functional: where denotes the function defined on the interval . Set , , , andwhere , , , , , , and are real matrices to be determined. Before introducing the main theorem, for simplicity, we set

Theorem 9. *Given , then system (1) is asymptotically stabilizable and satisfies for all nonzero if there exist positive definite matrices , , , , , , , , , , , , , , and such that the following hold:**where and**with () denoting the th row-block vector of the identity matrix and**Moreover, the feedback control is given by*

*Proof. *Taking the derivative of along the solution of system (1), we obtainFrom (4) and Cauchy inequality, we get the following inequalities:Similarly, By using the following identity relation: we obtain the following:From (27) and (29), (32), is estimated aswhereWith the consideration of the three terms of , we obtained the following inequality: Therefore, the estimation of is as follows:where is defined in (22). Similarly, is estimated as whereBy Lemmas 5 and 6, we obtain the following: where and Thus,where is given in (23). From Lemmas 2 and 4, we obtain from which it follows thatwhere Hence, from (41) and (43), we obtainwhere . From (33), (36), and (45), we obtain along the solution of system (1) as where Therefore, we havewhere and is defined in (24). Observe that may be rewritten aswhere and , are -independent real matrices. By Lemma 8, if and the inequalities in (21) hold, then , , . Moreover, may be rewritten as a convex combination of as follows:where and , are -independent real matrices. By utilizing the Schur complement lemma, it follows from (21), (50), and (51) that holds, from which it follows from inequality (48) thatLetting and fromthere exists a scalar such that Therefore, system (1) with is asymptotically stable. To complete the proof of theorem, next we consider the performance . By assuming that , , it follows from definition of thatFrom (52), we obtainFrom estimations of and in (48) and (55), we obtain Integrating both sides of the above equation from to , we get It follows thatTherefore, under zero initial condition , , by letting in (59), we getwhich gives . This completes the proof.

When , , and , the nonlinear function is , and (1) reduces to