Robust delay-dependent H ∞ performance for uncertain neutral systems with mixed time delays

This paper is concerned with the problems of robust stability analysis and H ∞ performance for the uncertain neutral systems with mixed time delays. By constructing an augmented Lyapunov Krasovskii functional with quadruple integral terms, some sufficient criteria are provided in terms of linear matrix inequalities (LMIs). The goal of this paper is to establish some conditions to guarantee the system is robustly stable with a prescribed H ∞ performance level for all uncertainties.


Introduction
Nowadays many cases of the field for dynamic systems have been paid attention, such as biological models, epidemic models, transportation systems, aircraft, robotics, neural networks, and so on [2], furthermore the many cased was focused on the effect of delay-dependent stability analysis, since the systems with delays will be instability and poor performance. Therefore various stability and stabilization for dynamical systems with or without state delays have been intensively investigated by many researchers mathematics and control communities [5,11,20]. Stability criteria for dynamical systems with time delay are generally divided into two classes: delay-independent one and delay-dependent one. Delay-independent stability criteria tend to be more conservative, especially for small size delay, such criteria do not give any information on the size of a delay. On the other hand, delay-dependent stability criteria are concerned with the size of a delay and usually provide a maximal delay size.
Many problems of stability analysis for neutral time-delay systems have been interested, various improved delay-dependent criteria of the systems have been paid attention. The neutral system is the system that contains time delays on both state space and derivative at state space, the practical model of neutral systems occur as prey-predator model, electrical model. Moreover, the study of robustly stable of the neutral systems with parameter uncertainties aims to guarantee the range of time-delay that made the system still stable and the H ∞ control problem have been concerned . Hence, there are many techniques have been created to improved the method to get the effective results which is guarantee the stability properties for the systems [5,6,11,13,14,15,20 such that the closed-loop system is internally stable and its H ∞ − norm of the transfer function between the controlled output and the disturbances will not exceed a given level γ. Moreover, the studies H ∞ control systems with interval time-varying delays have been developed so the improvement of the theory of H ∞ control have extend the region to study. The problems about delay-dependent robust H ∞ for linear system with interval time-varying delay and restricted the derivative of the interval time-varying delay, that mean a fast interval time-varying delay is allowed [6,17]. The H ∞ performance for linear system with nonlinear perturbations have been paid attention moreover , in other hand, [10] showed the time derivative of the Lyapunov Krasovskii functional produced not only the strictly proper rational functions but also the nonstrictly proper rational functions of the time-varying delays with first-order denominators, which was fully handled using reciprocally convex approach [7,9].
From the many motivations above, this work investigates in robust stability analysis and applying the H ∞ performance method for our systems. The parameter uncertainties are bounded in magnitude as some inequality. Based on Lyapunov-Krasovskii theory which is containing in term quadruple integral functional, applying Leibniz-Newton formula, modified version of Jensen's inequality, Wirtinger-based integral inequality and linear matrix inequality techniques, then the stability criteria and the H ∞ performance for neutral system with interval time-varying delays are present in term linear matrix inequality.
Notations : The following notations will be used throughout for this paper. R + denotes the set of all real non-negative numbers; R n denotes the n-dimensional space with the vector norm · ; x denotes the Euclidean vector norm of x ∈ R n ; R n×r denotes the set of n × r real matrices; , 0], R n ) denotes the space of all continuous vector functions mapping [−b, 0] into R n , where b = max{h, r}, h 2 , r ∈ R + ; * represents the elements below the main diagonal of a symmetric matrix.

Preliminaries
Consider the system described by the following state equations of the form are known real constant matrices with appropriate dimensions.The delay h(t) and neutral delay τ (t) are time-varying continuous function that satisfies where h 1 , h 2 , τ, and µ are given real constants. Consider the initial functions The time-varying parameter uncertainties are ∆A(t), ∆B(t), ∆C(t), and ∆E ω (t) and satisfying where L ∈ R n×l , and G 1 , G 2 , G 3 , G 4 ∈ R l×n are given real constant matrices and F (t) ∈ R l×l is an unknown real time-varying function with appropriate dimension and bounded as F T (t)F (t) ≤ I. (1) is robustly exponentially stable, if there exist positive real constants k and N such that for each

Definition 1 The system
Definition 2 Given a scalar γ > 0, system (1) is said to be asymptotically stable with the H ∞ performance level γ, if it is asymptotically stable and satisfies the H ∞ − norm constraint Lemma 2 [8] For any positive definite matrix Z ∈ R n×n , scalars τ l , τ u > 0, vector function w : [τ l , τ u ] → R n such that the following integration are well defined, the inequality holds:

Proof 1 Construct a Lyapunov-Krasovskii functional as
and using Wirtinger-based integral inequality Lemma 1, theṅ The derivative of V 4 (t), then we geṫ Taking the time derivative of V 5 (t) yieldṡ using Lemma 2, we obtaiṅ Taking derivative of V 6 (t), it is obtained aṡ