Research On Stability Analysis And Related Control Problems For Periodically Time-Varying Descriptor Systems

In recent years, the research on periodic descriptor systems is one of the most concerned questions. There are two important central problems. First, the system can be divided into two subsystems by using a restricted equivalent transformation, then study and analyze the performance of the system respectively;Second, the admissibility of the system and other related problems can be studied by solving the generalized Lyapunov inequality. With the enlargement of the singular systems application, the study on periodic descriptor systems theory is being probed further progressively. We discussed the stability and related control problems for periodic descriptor systems. Combing the standard periodical system theory with robustness control theory and H_∞ control theory, using Lyapunov method, monodromy matrices and LMI, this thesis has given a further investigation about the stability analysis and control problems of the above systems. The main contributions are as follows:(1) For the periodic continuous singular systems, the necessary and sufficient condition was presented for impulse-free and asymptotic stability of periodic singular system by establishment of generalized Lyapunov inequality. Based on this work, the necessary and sufficient condition for the systems to be quadratic admissibility was also studied and the controller design method was given. These results are useful for further investigation of singular periodical systems.(2) For linear uncertain periodic descriptor systems, the necessary and sufficient condition of robust stability was presented, and a kind of state feedback robust controllers was given. Then, the relationship between quadratic stability and robust stability of the systems was discussed.(3) For linear periodic continuous descriptor systems, he state feedback H_∞ control problem in two special cases were discussed. Under two common assumptions, the equivalenceof this problem and the feedback Hx control problem for standard state-space linear periodicalsystems was derived. By means of differential Riccati inequality, the necessary and sufficient condition for this problem and the existence of a kind of controller were given.(4) For continuous singular periodic systems, the problem of impulsive controllability was discussed. The direct characterizations of the above problem were given directly in terms of the original system coefficients and their derivatives, and the explicit criterions were presented by using operator expression and the rank of Gramian matrices.