Copyright © IFAC Time Delay Systems, New Mexico, USA, 2001 IFAC Pu blications www.elsevier.com/locate/ifac ON DELAY DEPENDENT ROBUST STABILITY OF NEUTRAL SYSTEMS S.A. Rodrlguez·, J.-M. Dion·, L. Dugard· and D. Ivanescu· • Laboratoire d'Automatique de Grenoble (INPG CNRS UJF) ENSIEG, BP 46, 38402, St. Martin d'Heres, FRANCE. Author to be contacted: Jean Michel Dion e-mail: Jean-Michel.Dion@inpg.fr • Laboratoire HEUDIASYC, Universite de Technologie de Compiegne, BP 529, 60205, Compiegne, FRANCE. Abstract: This paper investigates the robust stability of linear neutral delay systems. The uncertain time delay neutral systems under consideration are described by functional differential equations of neutral type with norm bounded nonlinear uncertainties and unknown constant delays. The novelty here is to consider a delay dependent robust stability approach leading to less conservative results than when dealing with robust stability independent of the delay time. The analysis is performed via Lyapunov-Krasovskii functional approach. Sufficient conditions for delay-dependent robust stability are given in terms of the existence of positive definite solutions of linear matrix inequalities. The proposed stability analysis extends previous results obtained either for delay-dependent stability without robustness issues or for robust delay-independent stability. Copyright © 2001IFAC Keywords: Time-Delay System, Neutral System, Robust Stability, Linear Matrix Inequalities. 1 INTRODUCTION A great variety of systems can be modeled by retarded functional differential equations [Kol- manovskii and Myshkis 1999], i.e. the future states depend not only on the present, but also on the past history. Aftereffect is a natural component of the dynamic systems in many fields: mechanics, physics, biology, chemistry, economics, information theory, etc. Even if the system itself does not have internal delays, closed loop systems may involve de- lay phenomena, because of actuators, sensors and computation time [Richard 1998]. The theory and mathematical tools for such systems have been sig- nificantly developed in, for instance, [Bellman and Cooke 1963], [Hale and Verduyn Lunel 1993], [Kol- manovskii and Myshkis 1999]. Among time-delay systems, an interesting class is the class of neutral systems characterized by the fact that the delay argument occurs also in the derivative of the" state variables". Some examples of such neu- tral systems are given in [Brayton 1976], [Niculescu and Brogliato 1995], [Logemann and Townley 1996], [Mounier et al. 1997], [Bellen et al. 1999], [Hu and Davison 2000]. Several works have been concerned with the sta- bility analysis of neutral systems either in the time domain approach, see for example: [Slem- rod and Infante 1972], [Hale and Verduyn Lunel 1993], [Niculescu and Brogliato 1995], [Richard et al. 1997], [Verriest and Niculescu 1997], or in the frequency domain approach, see for example: [Chen 1995], [Verriest and Niculescu 1997]. In the above mentioned studies, the attention was mainly focused in giving conditions for delay independent stability. These conditions are conser- vative in the case where the delays are unknown. It is then of interest to consider delay-dependent st.abilit.y analysis. Some result.s have been given in the frequency domain [Chen 1995] and in the time domain [Kolmanovskii 1996], [I viinescu et al. 2000]. In practice, the model parameters are not pre- cisely known and it is of interest to study the robust- ness of the stability with respect to parameter un- certainties, see [Li and de Souza 1997], [Kharitonov 1998]. Some neutral systems can be represented by uncertain models, see for instance [Kolmanovskii and Myshkis 1999], (lossless transmission line mod- els may have uncertain parameters). The objective of this paper is to study the sta- bility analysis of linear neutral systems in a delay- dependent framework incorporating robustness is- sues. The delays are assumed to be unknown and constant and the uncertainties may be time vary- ing and nonlinear. One obtains sufficient delay- dependent stability conditions via the Lyapunov- Krasovskii functional approach. The main result is expressed in terms of an easy to check Linear Matrix Inequality. This paper is organized as follows: Section 2 gives some preliminaries and states the problem. Model transformation is discussed in section 3. The main stability result is given in section 4. Some final re- marks end the paper. NOTATION. Here I m is the identity matrix of dimension m. x E Rn, 11 . 11 denotes the Eu- clidean norm of x. For a real number r > 0, C T = C([-r,O],Rn) will be the Banach space of continuous vector functions cp : [-r, 0] -7 Rn with the supremum norm Ilcplle = SUP_T<t<O Ilcp(t)ll· C T •V denotes the open set cp E C T with ITcpllc < v. ". " on the variable and ft(-) denote the right-hand derivative at t. = C 1 ([-r, 0], Rn) denote the Banach space of continuous differentiable functions cp : [-r, 0] -7 Rn with ep E C T and the norm IIcpllc ' = IIcpllc + lIeplle· The function Xt denote the restriction of x to interval [t - r, t] so that Xt is an element of C T defined by Xt((J) = x(t + 0) for -r 0 0. 83