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

H ∞ 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.H ∞ controller is designed via memoryless state feedback control and new sufficient conditions for the existence of theH ∞ 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.


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 0 ≤ 1 ≤ ( ) ≤ 2 and 1 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 .
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 2 Mathematical Problems in Engineering 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 [20][21][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 timevarying 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 delaydependent 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 ∫ −ℎ (ℎ − − )̇( )( ) ( = 1, 2) which allows us to use the matrix-based quadratic convex approach introduced in [20][21][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 Wirtingerbased 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.

Mathematical Model and Preliminaries
The following notations will be used in this paper: R + denotes the set of all nonnegative real numbers; R denotes the -dimensional space with the Euclidean norm ‖ ⋅ ‖; M × denotes the space of all matrices of ( × )-dimensions.
We introduce the following technical lemmas, which will be used in the proof of our results.
Before we introduce some useful integral inequalities, we denote Lemma 4 (see [20]). For a given scalar 1 ≥ 0 and any × real matrices 1 > 0 and 2 > 0 and a vectoṙ: [− 1 , 0] → R such that the integration concerned below is well defined, the following inequality holds for any vector-valued function where ] 3 ( ) is defined in (9).
We consider the following Lyapunov-Krasovskii functional candidate: ( , ,̇)   Proof. The proof is the same as in Theorem 9 by using Lyapunov-Krasovskii functional (62). The proof is omitted.

Numerical Examples
In this section, we provide numerical examples to show the effectiveness of theoretical results.    Figure 1 shows the trajectories of solutions 1 ( ) and 2 ( ) of system (1) without feedback control ( ( ) = 0) and Figure 2 shows the trajectories of solutions 1 ( ) and 2 ( ) of the system with feedback control ( ). Moreover, in Tables 1 and 2, by using Theorem 9, we give the minimum allowable value with 1 = −0.1, 2 = 0.1 and with 1 = 0.05, 2 = 0.1, for some given 1 and 2 , respectively.
Example 2. Consider the following nonlinear system with interval time-varying delays which was considered in [18]: = = 0.1, and 0 ≤ ( ) ≤ 2 , 1 ≤( ) ≤ 2 . By using LMI Toolbox in MATLAB, the LMIs in Corollary 10 are feasible. Table 3 shows the maximum allowable upper bound 2 with different values of 1 and 2 .

Conclusions
In this paper, we have investigated the ∞ control problem for a class of nonlinear systems with interval time-varying delay. A new Lyapunov-Krasovskii functional is constructed to obtain new delay-dependent sufficient condition for the ∞ control and asymptotic stability condition in terms of LMIs. Numerical examples are given to illustrate the effectiveness of the theoretical results.