**Delay Dependent Robust Exponential Stability Criterion for Perturbed and Uncertain Neutral Systems with Time Varying Delays**

**Issam AMRI**

Unit research LA.R.A., Ecole Nationale d’Ingnieurs de Tunis

ENIT, BP 37, 1002 Tunis, Tunisia

**Dhaou SOUDANI**

Unit research LA.R.A., Ecole Nationale d’Ingnieurs de Tunis

ENIT, BP 37, 1002 Tunis, Tunisia

**Mohamed BENREJEB **

Unit research LA.R.A., Ecole Nationale d’Ingnieurs de Tunis

ENIT, BP 37, 1002 Tunis, Tunisia

**Abstract**: This paper deals with the issue of robust exponential stability of uncertain neutral system with time varying delay and nonlinear perturbations. Using Lyapunov-Krasovskii functional, new sufficient delay dependent stability conditions have been derived in terms of Linear Matrix Inequalities LMIs solved using efficient convex optimization algorithms. Neither model transformation, nor estimating techniques for cross terms, nor free weighting matrices are involved in this work. Numerical examples are considered to show the efficiency of the proposed stability approach.

**Keywords**: Uncertain neutral systems; Time varying delays; Nonlinear perturbations; Exponential stability; Delay dependent stability; Linear matrix inequality

**Full text **

**CITE THIS PAPER AS**:

Isaam AMRI, Dhaou SOUDANI, Mohamed BENREJEB, **Delay Dependent Robust Exponential Stability Criterion for Perturbed and Uncertain Neutral Systems with Time Varying Delays, ***Studies in Informatics and Control*, ISSN 1220-1766, vol. 19 (2), pp. 135-144, 2010.

**1. Introduction**

The phenomena of time delays are frequently encountered in engineering dynamic systems, for instance in chemical process, in network control systems, in long transmission and so on. It is well known that the existence of time delays in a system may be a major source of instability, oscillations and poor performance. In view of this, considerable attention has been devoted to the problem of stability and robustness of time delays systems for several decades, see for example [1-27], and the references therein. The developed stability results can be classified into two types: delay dependent stability results, which are concerned with the size of the delay and usually give the maximum delay bounds for making the system stable, and the delay independent stability results, which can be applied with arbitrary delay’s size. Generally, delay dependent stability conditions are less conservative than delay independent ones, especially when the time delays are small.

In general, dynamical time delays systems models can be described as two types of functional differential equations. The first one concerns the retarded type which contains delays only in its states, whereas the second concerns the neutral system, being a special case of time delay system, which involved time delay in both state and state derivative simultaneously. Such system can be found in such places as population ecology [18], distributed networks containing lossless transmission lines [17], heat exchangers, robots in contact with rigid environments, etc. In recent years, stability issue in various neutral time delay systems have been widely investigated in many reports [8, 11, 13, 14, 16, 26]. Moreover, the stability for the systems with time varying delays will be an important focus that described systems more physical than the constant time delays cases [1, 3, 4, 6, 8, 14, 19, 20, 27]. It is well known that in practice, systems almost present some uncertainties and nonlinear perturbations. Thus, many methods have been proposed to deal with uncertainties and nonlinear perturbations in the literature, and much attention has been paid on robust stability analysis by using the Lyapunov-Krasovskii functional approach [1, 4, 6, 8, 11, 14, 15, 19-22, 26].

In [21], a parameterized neutral model transformation was utilized. Based on a model transformation technique, [6] presents delay dependent stability criteria by using a Lyapunov-Krasovskii functional approach. In [13], stability conditions for uncertain systems with time varying delay and nonlinear perturbations was developed by applying a descriptor model transformation and a decomposition technique of the delay term matrix. The estimation approaches for bounding the cross terms show in [14] may bring some conservatism. Furthermore, free weighting matrices have been employed in a lot of papers, such as [9], because they can increase the freedoms to search the Lyapunov matrices and reduce the conservatisms.

On another hand, the problem of exponential stability has also been considered by some researchers [1-4, 7-9, 11, 15, 16, 20, 23] because it is also important indices to get the convergence rates of prescribed time delay systems.

In these contexts, the issue of exponential stability for uncertain neutral systems with time varying delays and nonlinear perturbations remains open, which motivates [8] and this paper but by adding norm-bounded uncertainties to system under consideration.

The purpose of this paper is to investigate the problems of delay dependent robustly exponential stability for uncertain neutral system with time varying delays and nonlinear perturbations. We present a new sufficient condition based on a combination of Lyapunov-Krasovskii functional and Linear Matrix Inequalities LMIs which can be efficiently solved by standard convex optimization algorithms [5]. Neither model transformation, nor estimating techniques for cross terms, nor free weighting matrices are involved in this work. Some numerical examples are included to show the effectiveness of our approach and to illustrate the applicability of the developed results.

**Notations**

: Transpose of matrix | |

: symmetric positive definite matrix | |

: Maximum (minimum) eigenvalue of a symmetric matrix | |

: Euclidean vector norm | |

: Identity matrix | |

: Symmetric terms in a symmetric matrix |

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