Session Keynote: Lyapunov Function Candidates for Descriptor Systems: Problems and Solutions

This paper presents in a systematic way problems encountered in the construction of Lyapunov function candidates for descriptor systems. The solutions to some of the major difficulties in the application of Lyapunov’s direct method to descriptor systems are presented. Some new results regarding the extension of Lyapunov’s direct method tied to the construction of the Lyapunov functions are given. Also, the algebraic necessary and sufficient conditions for some specific properties of motions of descriptor systems are developed. INTRODUCTION AND BACKGROUND It is a tradition to consider the equations describing internal system dynamics of continuous time systems in the normal form of the so-called state equations ? (1) where is the ordinary time derivative operator, and where , o and ? denote the time, the input vector and the state vector, respectively. Systems governed by the model (1) we usually call the state variable systems. However, there are physical systems for which the state models do not exist. It was recognized that the more natural models for these cases go beyond the classical state-variable description (1). These models are given by R (2) where may be allowed. Systems with such models we denote as implicit. In the special case the models (2) contain the canonical form of models linear in , and they are of the form (3) It should be pointed out that the matrix can even be a rectangular one. Systems governed by (3) are known as descriptor, as well as singular, semi-state, generalized state-space or differential-algebraic systems. STABILITY ANALYSIS OF DESCRIPTOR SYSTEMS The surveys of some of the results concerning both continuous and discrete descriptor and general implicit systems can be found, for example, in the books [1]-[3], [6]-[7] and [10] and in the special issues of the journal Circuits, Systems and Signal Processing [8]-[9]. The systematic introduction and presentation of part of the results relating to the general application of Lyapunov’s direct method (LDM) for the analysis of DS is given in [2], [3]. For some other results on stability of DS, particularly with regard to mechanical DS, see [12]-[16]. It has been shown in [2], [4], that specific structure of DS may lead to several problems regarding the construction of Lyapunov function candidates (LFCs) for the intended qualitative analysis. These problems do not appear in the application of the LDM to systems in the normal forms (1). This paper aims to contribute to the general methodology relating to the construction of the LFCs for descriptor systems. We present in a systematic way problems encountered in the construction of LFCs for continuous-time DS, as well as the solutions to some of the major difficulties in the application of the LDM. Some new results regarding the extension of the LDM tied to the construction of LFCs are given. PROBLEM STATEMENT AND SIGNIFICANCE: LDM AND THE ROLE OF LFC There are essentially two problems that relate to the preliminary construction of the LFC: the problems that appear in the process of evaluation of the total time derivative (TTD) of an LFC along the motions of (2) and (3). the structure and properties of the matrix in (3) The application of the LDM requires the selection of a concrete LFC. When the exact analytical form of an LFC is known and when the necessary and sufficient conditions that ensure the appropriate qualitative concept of motion are expressed directly in terms of system and LFC parameters, then we say that the construction of the LFC is given. Unfortunately, in the general case the main drawback of the LDM is the conceptual nonexistence of a systematic procedure for the practical construction of an LFC for the qualitative concept of interest (one of possible general solutions for this is proposed in this paper). There are only a few results of the general nature on the construction of the LFC for DS (see references in [2]). So far there are no global answers on how to select an LFC for DS. Thus any general hint that concerns the construction of an LFC is of particular importance. The aim of this presentation is to provide some suitable solutions of this problem in a very specific way by the evaluation of the derivative of an LFC along the solutions of DS. This poses practical problems in finding a TTD of an LFC for DS, and, also, directly restricts the classes of useful LFCs. To make this more evident let the matrix and the functions and in (3) be sufficiently smooth and such that the systems considered possess the continuous and differentiable in solutions in some domain. For simplicity, let be an arbitrary function of and , which is continuously differentiable, and let it be an LFC for systems (2) or (3). In any qualitative analysis of the solutions of systems considered by means of the LDM, specific properties of the LFC are required. In what follows the TTD of along the solutions of any of the systems (1) to (3) we will denote by P In all cases, the calculation of the TTD P of the LFC , along the system motion , has to be found. In general, this is not possible in a direct way for systems (2) or (3), unless depends on in a specific manner. As is well known, P can be found without the knowledge of solutions if the system analyzed is in the normal form (1) [17, p. 12]. Using the same argument, the P of the LFC along the motion is given by