Study On Practical And Finite-time Stability Of Several Classes Of Descriptor Systems

Stability is one of the essential problems in control theory. Based on Lyapunov stabil-ity and other theories, researchers studied descriptor systems and gave significant results on strongly absolute stability, input-state stability, etc. However, exploration on practical stability, finite-time stability of descriptor systems and design of corresponding controllers is premature. By using Lyapunov stability theory, Razumikhin technique, fuzzy control theory, linear ma-trix inequalities (LMIs), adaptive control theory and neural network, this dissertation explores practical stability and finite-time stability in descriptor systems. Fuzzy controllers, adaptive controllers, dissipative controllers and observer-based controllers are devised respectively for such descriptor systems.The main results are as follows.(1) Since research of practical stability for nonlinear time-delay systems is rarely published, this dissertation studies practical stability and its controllability for a class of nonlin-ear discrete time-delay systems via Razumikhin-type theorems and Lyapunov stability theory. Sufficient conditions of practical stability for the class of nonlinear time-delay systems are given and proved.(2) Research on finite-time stability in descriptor systems is premature. Following recent studies of finite-time stability theory and fuzzy control theory, this dissertation studies finite-time stability for a class of nonlinear descriptor systems with non-zero initial state. First, using fuzzy control theory and Lyapunov function approach, necessary and suffi-cient conditions for finite-time boundedness are devised. Comparison between the given sufficient condition and existing results in recent papers is discussed. Second, an auxil-iary fuzzy system is constructed for nonlinear descriptor systems with a fitting-precision variable devised. On this basis, fuzzy state-feedback and static output-feedback con-trollers are designed. The proposed algorithm reduces the demand of computation for similar problems.(3) Single and dual fuzzy models are adopted for a class of fuzzy descriptor systems with known norm-bounded uncertainty and external disturbance. For such descriptor systems, finite-time robust dissipative control is studied. Using linear matrix inequalities, a suffi-cient condition is given to confirm that the system satisfies finite-time robust dissipativity. Fuzzy controllers based on parallel distributed compensation (PDC) and non-parallel dis-tributed compensation (non-PDC) are designed. In the solving process, feasible regions of corresponding LMIs are compared. Performance comparison between PDC and non- PDC controllers is also discussed. With the proposed method compared with some ex-isting results in this area, the simulation shows that our approach has better performance and robustness with respect to uncertain dynamics and disturbances, which provides a reference basis for future use of the given method.(4) Being different from most of current works on boundedness discussion, under the defi-nitions of practical stability and finite-time stability, this dissertation use the conditions of time intervals and information of desired state boundary to devise sufficient condition for practical stability and finite-time robust stability. Following the sufficient conditions, system can be practical stable or finite-time stable, alternatively. Using adaptive control theory and LMIs, controllers based on PDC and non-PDC are presented, assuring that system trajectory reach the desired bounded set.