24 Scientific Technical Review, 2011,Vol.61,No.2,pp.24-34 UDK: 681.511.2.037:517.938 COSATI: 12-01 The Stability of Linear Continuous Singular and Discrete Descriptor Time Delayed Systems in the Sense of Lyapunov: An Overview Dragutin Debeljković 1) Sreten Stojanović 2) Tamara Nestorović 3) This paper gives a detailed overview of the work and the results of many authors in the area of Lyapunov stability of a particular class of linear systems. This survey covers the period since 1980 up to nowadays and has strong intention to present the main concepts and contributions during the mentioned period throughout the world, published in the respectable international journals or presented at workshops or prestigious conferences. Key words: linear system, continuous system, singular system, discrete system, descriptive system, system stability, time delayed system, Lyapunov stability. 1) University of Belgrade, Faculty of Mechanical Engineering, Dept. of Control Eng., Kraljice Marije 16, 11120 Belgrade, SERBIA 2) University of Niš, Faculty of Technological Engineering, Bulevar oslobođenja 124, 16000 Leskovac, SERBIA 3) Ruhr-University of Bochum, Bochum, GERMANY Introduction T should be noticed that in some systems we must consider their character of dynamic and static state at the same time. Singular systems (also referred to as degenerate, descriptor, generalized, differential - algebraic systems or semi – state) are those the dynamics of which is governed by a mixture of algebraic and differential equations. Recently, many scholars have paid much attention to singular systems and have obtained many good consequences. The complex nature of singular systems causes many difficulties in the analytical and numerical treatment of such systems, particularly when there is a need for their control. It is well-known that singular systems have been one of the major research fields of the control theory. During the past three decades, singular systems have attracted much attention due to the comprehensive applications in economics as the Leontief dynamic model Silva, Lima (2003), in electrical Campbell (1980) and mechanical models Muller (1997), etc. They also arise naturally as a linear approximation of systems models, or linear system models in many applications such as electrical networks, aircraft dynamics, neutral delay systems, chemical, thermal and diffusion processes, large-scale systems, interconnected systems, economics, optimization problems, feedback systems, robotics, biology, etc. Discussions of singular systems originated from 1974 with the fundamental paper of Campbell et al. (1974) and later with the anthological paper of Luenberger (1977). Since that time, considerable progress has been made in investigating such systems - see surveys, Lewis (1986) and Dai (1989) for linear singular systems, the first results for nonlinear singular systems in Bajic (1992). In the investigation of stability of singular systems, many results in sense of Lyapunov stability have been derived. For example, Bajic (1992) and Zhang et al. (1999) considered the stability of linear time-varying descriptor systems. Discrete descriptor systems are the systems the dynamics of which is covered by a mixture of algebraic and difference equations. In that sense, the question of their stability deserves great attention and is tightly connected with the questions of system solution uniqueness and existence. Moreover, the question of consistent initial conditions, time series and solution in the state space and phase space based on a discrete fundamental matrix also deserve a specific attention. In this case, the concept of smoothness has little meaning but the idea of consistent initial conditions being these initial conditions 0 x that generate solution sequence ( ) ( ) : 0 k k ≥ x has a physical meaning. Some of these question do not exist when the normal systems are treated. The problem of investigation of time delay systems has been exploited over many years. Time delay is very often encountered in various technical systems, such as electric, pneumatic and hydraulic networks, chemical processes, long transmission lines, etc. The existence of pure time lag, regardless of the fact that it is present in the control or/and the state, may cause an undesirable system transient response, or even instability. Consequently, the problem of the stability analysis for this class of systems has been one of the main interests for many researchers. In general, the introduction of time delay factors makes the analysis much more complicated. I