XI International SAUM Conference on Systems, Automatic Control and Measurements Niš, Serbia, November 14 th -16 th , 2012 Finite-time stability and stabilization of singular time delay systems S. B. Stojanovic + D. LJ. Debeljkovic ++ and D. S. Antic +++ + Department of Mathematical and Engineering Sciences, University of Nis, Faculty of Technology, Bulevar oslobodjenja 124, 16000 Leskovac, Serbia Phone: (381) 16-247-203, Fax: (381) 16-242-859, E-Mail: ssreten@ptt.rs ++ Department of Control Engineering, University of Belgrade, Faculty of Mechanical Engineering, Kraljice Marije 16, 11120 Beograd, Serbia Phone: (381) 11-33-70-266, Fax: (381) 11-337-0364, E-Mail: ddebeljkovic@mas.bg.ac.rs +++ Department of Control Engineering, University of Nis, Faculty of Electronic Engineering, Aleksandra Medvedeva 14, 18000 Nis, Serbia Phone: (381) 18-529-100, Fax: (381) 18-588-399, E-Mail: Dragan.Antic@elfak.ni.ac.rs Abstract - In this paper, finite-time stability analysis and stabilization for linear singular time-delay systems are considered. Sufficient condi- tion for the systems to be regular, impulsive-free and finite-time stable is derived via linear matrix inequalities. In the derivation of delay- independent finite-time stability conditions, we define an appropriate Lyapunov like functional. Then the synthesis problem is addressed and a state feedback controller is designed in terms of nonlinear matrix inequality, which can be numerical solved using the cone complemen- tarity linearization algorithm. Key words: singular systems, time-delay, finite-time stability, stabiliza- tion, linear matrix inequality I. INTRODUCTION A singular system describes a natural representation for physical systems. In general, the singular representation consists of differential and algebraic equations, and hence it is a generalized representation of the state-space system. The class of singular systems is more appropriate to de- scribe the behaviour of some practical systems like power systems [1], electrical systems [2], social economic systems [3], and chemical systems [4]. It is well known that study of singular systems is much more complicated than that of regular ones. When we consider the control design of practical systems (chemical engineering systems, lossless transmission lines, large-scale electric network control, aircraft attitude con- trol, flexible arm control of robots, etc.), time-delay often appears in many situations. When a time-delay is small, it can be ignored. If it is large, however, it may cause instabil- ity in the system. In general, the dynamic behavior of con- tinuous-time singular systems with delays is more compli- cated than that of system without any time-delay because the continuous time-delay system is infinite dimensional. For this reason, over the past decades, there has been in- creasing interest in the stability analysis for singular time- delay systems and many results have been reported in the literature [5–10]. Often Lyapunov asymptotic stability is not enough for practical applications, because there are some cases where large values of the state are not acceptable, for instance in the presence of saturations. For this purposes, the concept of the finite-time stability (FTS) and practical stability are used. A system is said to be FTS if, once a time interval is fixed, its state does not exceed some bounds during this time interval. A little work has been done for the finite-time stability and stabilization of singular time-delay systems. Some results on FTS and practical stability can be found in [11-13] (singular systems) and [14] (singular time-delay systems). However, according to the author's knowledge, there is no result available yet on finite-time stability and stabilization for a class of linear time-delay systems using linear matrix inequality. In this paper, a new sufficient condition of finite-time stability for continuous singular time-delay systems is de- rived. Then, by utilizing the skill of matrix theory, a simple and efficient approach is proposed to design state feedback controller which ensures the closed-loop singular time- delay systems to be regular, impulse-free and finite-time stable. II. PRELIMINARIES AND PROBLEM FORMULATION The following notations will be used throughout the pa- per. Superscript “T” stands for matrix transposition. n  de- notes the n-dimensional Euclidean space and nm   is the set of all real matrices of dimension n m  . 0 X  means that X is real symmetric and positive definite and X Y  means that the matrix X Y  is positive definite. In symmetric block matrices or long matrix expressions, we use an asterisk (*) to represent a term that is induced by symmetry. { } diag stands for a block-diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. Consider a continuous singular system with state delay described by:      d Ex t Ax t Axt Bu t      (1) with a known compatible vector valued function of the initial conditions   , 0 xt φt t      (2)