Input-to-state Stability And Dissipative Control Of A Class Of Descriptor Systems

As we all know, descriptor systems are a more widespread form compared with normal ones. Descriptor systems not only reflect the intrinsic structure of the actual system more conveniently, clearly and accurately, and depict the nature of it, but they have a wider range of practical applications and more important theoretical significance. Descriptor systems have caught many scholars’ attention, both at home and abroad, since they were proposed. And the research results and methods of normal systems have been extended to descriptor systems to make the theory of descriptor systems gradually growing mature.Among many indicators of a system, stability is undoubtedly the most funda-mental and important one. Therefore research on stability and related issues has been one of the hot issues of control theories. However, for systems with disturbance input, it is not enough to study stability only. In 1989, Sontag ED first proposed the concept of input-to-state stability. Then his studies become an important branch of nonlinear system stability. It is an effective way to describe the robust stability of nonlinear systems with input disturbances. Broadly speaking, input-to-state sta-bility means that when the system is bounded input, the system state is bounded; when the input is very small, the system state is very small too.In 1970s Willems J C proposed a new theory-dissipative systems theory whose essence is that there is a non-negative energy function (storage function), so that the system energy loss is always less than the energy supply. Dissipative systems theory is strongly associated with the analysis of stability of a system. In the process of studying stability of a system, the theoretical basis most commonly used is Lyapunov stability theory. During the study, to construct a suitable Lyapunov function is a difficult problem. However, under certain conditions energy function of dissipation systems can serve as a Lyapunov function. It makes the dissipation theory play an important role in the study of stability of a system.Based on the above, a number of issues are analyzed and studied for a class of nonlinear descriptor system. The main contributions of this dissertation are summarized as follows:(A) Input-to-state stability of continuous descriptor systems is studied. The input-to-state stability theory of normal continuous systems is developed to descrip-tor systems. Based on Lyapunov stability theory and linear matrix inequalities, existence and uniqueness of solutions of descriptor systems and the input-to-state stability criterion are given. And sufficient conditions of input-to-state stability of a class of nonlinear systems with a special structure are presented. Then the prob-lem of index for 1 is study. Finally, the method of state feedback controller design to make the closed-loop system input-to-state stability is given. And simulation examples are given to illustrate the effectiveness of the conclusion.(B) Input-to-state stability of discrete time descriptor systems is investigated. The concept of input-to-state stability for standard state-space systems is extended to descriptor systems. Based on the input-to-state stability theory of normal dis-crete systems and Lyapunov stability theory, a sufficient condition for a class of nonlinear descriptor systems to be input-to-state stability is derived by using linear matrix inequality approach. Then the relationship between input-to-state stability and admissibility of linear discrete descriptor systems is discussed. Furthermore, a numerical example demonstrates the proposed results.(C) Partial state observer design for a class of continuous descriptor systems is studied. Necessary and sufficient conditions are derived for the existence of partial state observers. Then based on Lyapunov stability theorem, design methods of partial state observers are given by Cayler-Hamilton theorem and generalized inverse theory for normal and special descriptor systems respectively. Finally, numerical examples are given to illustrate the effectiveness of the observer design.(D) The dissipative and passivity control problems of continuous and time-delay descriptor systems are studied. Using LMI method combined with multiple Lyapunov stability method sufficient conditions of strictly dissipative problems are presented in the two cases of Q> 0 and Q≤0 for quadratic form provision (Q, S, R). Then by using Schur complement lemma, the state feed-back controller and deriva-tive proportional feedback controller have been designed to make the closed-loop systems strictly dissipative. Furthermore the sufficient condition of passivity and the controller design of a system are considered. Finally, a numerical example is presented to demonstrate the validity of the proposed methods.