Stability Analysis And Control Of Several Classes Of Singular Systems

Since the seventies of the20th century, singular system theory has gradually de-veloped into an most important branches of control theory. Since its strong application background, Singular system has attracted the attention of many researchers at home and abroad. There have been great development on fundamental theory and stability of singular systems, but there are still some limitations, such systems remains to be further developed.Based on singular systems and the control theory, this dissertation studies the prob-lems of stability and control of several classes of singular systems.The main contents of this dissertation include:Chapter1is the preface, the research background and research states of this disserta-tion are introduced. Firstly, the research background of singular systems theory, structure characteristics and research method are summarized, the development status and actual applications are introduced in detail. Then, the research states, methods and obtained results about switched, Markov jump, multi-agent and discontinuous nonlinear singular systems are presented. Finally, the main work of this dissertation is introduced.Chapter2studies a class of linear switched differential algebraic equations (DAEs). Case1:stable and unstable subsystems coexist. Asymptotic stability of the switched DAEs is obtained if the average dwell time is chosen sufficiently large and the total activation time of unstable subsystems is relatively small compared with that of stable ones. Sufficient conditions for stability of switched DAEs are presented based on the existence of suitable Lyapunov functions. The main result shows that asymptotic stability is preserved under switching with an average dwell time and an additional condition involving consistency projectors holds. Case2:switching signals with time delays. Based on average dwell time method and the merging switching signal technique, the switched DAEs achieve asymptotic stabllity when consistency projectors are involved.Chapter3discussed a class of singular systems with Makov jump. Firstly, a new nec-essary and sufficient condition is proposed in terms of strict linear matrix inequality, which guarantees the stochastic admissibility of the unforced Markovian jump singular systems. A bounded real lemma for discrete-time Markovian jump singular systems is derived and can be described by a strict matrix inequality. The results are more tractable and reliable in numerical computations than existing conditions.Then, the problem of the stability and stabilization of Markovian jump singular sys-tems with partly known transition probabilities is concerned, including two cases(continuous time and discrete time). By fully unitized the properties of the the transition rate ma-trix(TRM), and the convexity of the uncertain domains, a new sufficient and necessary condition in terms of strict linear matrix inequalities (LMIs) for the MJSS to be regu-lar, impulsive and stochastically stable is obtained. In continuous time case, based on the proposed stability criterion, a state feedback controller is designed, and a sufficient condition which guarantees the stochastic admissibility of the closed loop systems is given; In the discrete time case, not only a state feedback controller but also an output feedback controller is designed, and a sufficient condition such that the closed loop systems is stochastically admissible is also given.Finally, according to the different cases of the systems, several examples are given to demonstrate the effectiveness of the obtained results.Chapter4studies a class of multi-agent singular systems under switching topol-ogy. We extend the general linear systems to the singular systems, and discuss both the leaderless consensus problem and the leader-following consensus problem under switching network topology. Using the algebraic topology and singular systems theory, the multi-agent singular system can be decomposed into fast-slow subsystems, and we just need to consider the property of the slow subsystem, that is to design a state feedback control protocol, and get the consensusability of the slow subsystem. Then, the two consensus problems of the multi-agent singular systems are also solvable.Chapter5extends the LaSalle invariance principle and stability of conventional systems to discontinuous nonlinear singular systems. Firstly, the definitions of stabil-ity, asymptotical stability, and invariant set for such systems are proposed. Then, the LaSalle invariance principle is presented by citing the notion of E-invariant set and the generalized Lyapunov method, and sufficient conditions about the stability and asymp-totic stability of the discontinuous nonlinear singular systems are given. Furthermore, the application of the invariance principle is demonstrated by the given example.Chapter6summarizes the main results of this dissertation and point out the further research.