Study On Stability For Switched Systems And Related Problems With Time-Delay

Switched systems constitute a special class of hybrid systems which contain both con-tinuous dynamics and discrete dynamics, and a rule that orchestrates the switching amongthem. Due to the existence of the switching rule, their dynamics may become very com-plicated. The involvement of the switching mechanism brings new insights as well as newchallenges in systems & control and related fields. Meanwhile, research results of switchedsystems also make significant reference to general hybrid systems in terms of ideas, methodsand theories. On the other hand, uncertainties and time-delays are frequently encountered inmany practical engineering systems, and are often the sources of instability and performancedeterioration. Due to the interaction among them and switching rules, the behaviors of suchsystems are very complex. Mechanism of system evolution is far from clear and many funda-mental problems are still either unexplored or less well understood. Therefore, the study ofswitched systems deserves investigation for theoretical development as well as for practicalapplications.This dissertation devotes on the study of the switched systems and related problemswith time-delay, including the stability analysis and robust control/filtering synthesis. Themain contributions of the research work presented in this dissertation can be summarized asfollows.As a primary issue, the stability of switched systems has received much attention withinthe last decades. It is well known that the stability of switched systems depends not onlyon the dynamics of each subsystem but also the properties of switching rules. The study ofthe stability properties of switched systems gives rise to many interesting and challengingmathematical problems. In Chapter 2, we outline some of these problems, review progressmade in solving these problems in a number of diverse communities, and comment on theinherent difficulties of determining stability of switched systems. We also review some prob-lems that remains open and present fundamental observations, which provide a sound basisof the dissertation. In Chapter 3, the switching rule design methodology is investigated for the robust stabi-lization of uncertain discrete-time switched systems. A switching rule is proposed combinedwith the switched static output feedback controllers to ensure 1) quadratic stabilization ofthe resulting closed-loop system; 2) pole location inside a circle for each linear mode of op-eration. The switching rule is state-dependent and do not rely on any uncertainties. All theresults are expressed in terms of linear matrix inequalities (LMIs), which can be easily testedwith efficient algorithms.Chapter 4 is concerned with the robust H∞filtering problem for polytopic switchedlinear discrete-time systems. Combined the mode-switching idea with Finsler's Lemma,and further utilized the parameter-dependent result, a new characterization of the asymptoticstability with a specifical performance level bound for the filtering error system has been ad-dressed. The existence condition of such filter is formulated in terms of dilated linear matrixinequalities. This directly leads to performance improvement and reduction of conservative-ness in the filtering solution. General comparisons with other methods in terms of the degreeof conservatism and computational complexity are made via numerical examples.In the remainder, we focus on the related problems with time-delay. Chapter 5 dealswith the problem of H∞output feedback control for switched linear discrete-time systemswith time-varying delay. A delay-dependent stability analysis is implemented by employingfinite sum equality based on quadratic terms, a new method of estimating the upper boundon the finite sum. In combination with a switched quadratic Lyapunov functional for theunderlying system, both static and dynamic H∞output-feedback controllers are designedrespectively, under which the resulting closed-loop system under arbitrary switching signalsis asymptotically stable with a prescribed H∞noise attenuation level. The results are furtherextended to uncertain switched systems. Finally, numerical examples are given to illustratethe effectiveness of the approach and its advantage over existing methods.In Chapter 6, the problem of exponential stability for linear continuous-time switchedsystems with mixed delays is considered. Under the circumstance that both stable and unsta-ble subsystems coexist, sufficient condition for exponential stability is developed for a classof switched delay systems. By using the Halanay inequality, the estimation on the decay(growth) rate of the candidate Lyapunov function is first presented. Then, a piecewise Lya-punov function is constructed. Based on a new average dwell time scheme and under thecondition that the activation time ratio between stable subsystems and unstable ones is notless than a specified constant, the delay-dependent stability condition is derived in terms ofLMIs. Numerical examples show that the new criterion is effective and furthermore, enrich the scope of research.Finally, in Chapter 7, the results of the dissertation are summarized. Some interestingand challenging problems are also pointed out, which deserve further study.