Theory And Applications Of Several Discontinuous Dynamical Systems

System Theory arises in various systems ranging from huge systems such as social economical systems to micro systems such as controlled movement of an atom. System Theory has played such an important role that it greatly promoted the development of both social and industrial productions. With the development of System Theory, continuous systems have been well studied so far. However, the area of discontinuous dynamical systems (DDS) is still to be fully explored. A large number of problems are to be solved. As new problems require new approaches, the analysing strategy of discontinuous systems is different from continuous ones. On the other hand, as engineering applications enrich System Theory, problems arising from the real world motivate the research of DDS theory.The two major problems in System Theory are system analysis and system synthesis. The former aims at understanding the system properties in terms of both quality and quantity. The latter, which is based on the former, tries to control system behavior so that it behaves as demanded. The work on DDS in this thesis will follow the general logic which begins with analysis and ends up with synthesis. New theories for analysis and synthesis are generalized from concrete studies of specific systems. In view of the variety of discontinuous systems, it is hard to get every detail involved in this thesis. Only some typical systems will be discussed and a generalized theoretical framework is developed from these systems. Nonlinear digital filter (NDF) and variable structure systems (VSS) are studied as two typical DDS in this thesis. The algebraic approach and geometrical approach are developed during the study of NDF and discrete time VSS. The concept of differential inclusion for discontinuous retarded functional differential equations is introduced. LMI method and eigenvalue analysis are further developed to investigate the stability of time-delayed DDS. Furthermore, continuous approximation of DDS is studied at a later stage. Theories discussed in this thesis are applied to engineering systems. The work in this thesis is depicted in a more detailed way below:Firstly, the algebraic approach is briefly introduced and further developed. The dynamics of a second-order NDF is explored. New algebraic conditions are proposed with the aid of dynamic symbolic sequence, and they are applied to a second-order NDF with periodic input and discrete time VSS, to understand their switching dynamics. Periodic switching dynamics is systematically studied. The new results derived are more general than the existing results. Following these, a generic algebraic strategy is presented to deal with general discrete time piecewise linear systems.Secondly, geometrical approach to study the dynamic behavior of discrete-time DDS is presented in detail. For a second-order NDF, this method can find almost all the admissible periodic dynamic behaviors. Furthermore, depending on the countability of the partition of state space, a result on the existence of aperiodic switching is given. Similar conclusions are drawn for discrete time VSS using this approach. In addition, the author gives the geometrical logic for the analysis of general discrete time piecewise linear systems.After that, the interaction between time delay and sliding mode in discontinuous time delay systems is studied in terms of both quality and quantity. A strategy for continous approximation is proposed. The area of discontinuous time delay systems is reletively new. Sliding mode, chattering and stability of time-delayed DDS are investigated. Specifically, chattering and stability of time-delayed VSS are studied by eigenvalue analysis and convergence of system state to sliding surface. LMI toolbox is employed. Conditions for stability are developed and the estimation of the width of the switching band is given. Following these, theory of continuous approximaiton of DDS is expanded to time-delayed DDS. Time-delayed VSS is thoroughly explored as a typical model and conditions for setting system parameters with given time delay are derived. The band where the approximation happens is described. Solution decomposition is utilized to have a thorough understanding of system dynamics within the band and condition on chattering avoidance is proposed.Finally, theories in this thesis are applied to several information systems and industrial systems to demonstrate the significance.In conclusion, new strateties and new theories for the analysis and synthesis of DDS are proposed in this thesis. Although the systems studied are specific, a deep analysis has been conducted. The theories developed in this thesis can easily be generalized, which will pave the path for further research. The theories have been applied to practical systems and more applications are in prospect.