Research On Input-to-state Stability (ISS) And Control Of Hybrid System

With the progress of science&technology and the development ofproductivity, the control systems, which are also becoming more andmore complex, are always of high nonlinearity and uncertainty. Oneimportant kind of the complex control systems is hybrid systems, whichwidely exist in the objective world. In recent years, the stability andcontrollability of the hybrid systems have been widely researched andapplied, and it has been widely recognized that the study of hybridsystems and their control are of great theoretical significance andpractical application value. The understanding of hybrid systems is one ofthe most important achievements in the nonlinear science. But at present,systematic and effective analysis and synthesis methods are still scarce inChina; therefore, the research on hybrid systems’ stability and control isof broad and far-reaching significance. This paper mainly discusses theinput-to-state stability and optimal control problems of the two typicalhybrid systems: impulsive systems and switched systems.Firstly, the research background and status of hybrid systems aresummarized; then the research status and development trends of the twotypical hybrid systems: impulsive hybrid systems and switched systems,are introduced; after that, the paper introduces some basic mathematicalknowledge, Lyapunov stability theory, input-to-state stability (ISS) andother relevant theoretical knowledge.Secondly, the paper investigates into the input-to-state stability(stability theorem) for the nonlinear impulsive hybrid systems withoutinput，and offers mathematical proof. Based on this, the paper proves theinput-to-state stability of the nonlinear impulsive hybrid system withinputs, and gives a numerical example to illustrate its validity. Fornonlinear impulsive hybrid systems with time-delay, the paper mainlyintroduces a sufficient condition of its input-to-state stability, and thenanalyzes the stability of different types of nonlinear impulsive hybridsystems with time-delay through simulations. The initial states and theinfluence of external inputs of systems on the system’s input-to-statestability are also discussed. Through simulations, it is verified that the input-to-state stability of the hybrid systems may be arrived, under theaction of impulses, although the coefficient matrices of the nonlinearimpulsive hybrid systems with time-delay aren’t Hurwitz matrices.Finally, the paper studies and offers the theorem for theinput-to-state stability of nonlinear switched systems, whose effectivenessis illustrated by a numerical example. Then the paper probes into theoptimal control of switched systems with impulsive effect, and offers amethod for the optimal control, verifying its effectiveness through anumerical example.