Finite-time Stability Analysis And Control For Several Classes Of Markovian Jump Systems

In view of different methods are applied to solving both discrete-time and continuous-time systems in the practical application and physical phenomenon, hybrid systems run stably and the better control results are satisfying. As a class of special hybrid systems, Markovian jump systems enriched and has developed the theory of control systems. Due to its widely applications in industrial process control, aerospace craft,medical, electric power and economy, the Markovian jump systems has been taken into account by scholars and abroad. Up to now, a large number of results on robust control and estimation have been given in the literature. It should be mentioned that, in the existing literature, the transition probability matrix is always given as completely known in most obtained results. However, due to the fact that the exists of external factors, it is very difficult and expensive to obtain all the information of conditional probabilities in the Markov chain. Therefore, it is more general to investigate the Markovian jump systems with partly known transition probability matrix. More recently, in some applications, the assumption of transition probability matrix to be constant is not consistent with reality, it is very important to investigate the Markovian jump systems with time-varying Markov chain. The transition probability matrices mentioned above are important in both theory and real applications.On the other hand, Lyapunov asymptotically stability is always employed over an infinite-time interval to deal with the state behaviors of systems. Compare with asymptotically stable controller, finite-time controller can ensure that the systems achieve the control performance of fast convergence in an finite-time interval. Finite-time stability has been scholars’ focus for its better robustness and disturbance attention properties.Therefore, this thesis mainly concern about finite-time stability, finite-time stabilization and finite-time H∞control for Markov jump systems with transition probability matrix:1. Finite-time H∞control for a class of Markovian jump systems with time-varying delays. By employing the property of convex function and Lyapunov-like function with four integral, some less conservative sufficient conditions are derived on finite-time stability. Based on the finite-time stability condition, the finite-time H∞controller of the considered Markovian jump systems is studied.2. Finite-time H∞control for a class of Markovian jump systems with modedependent time-varying delay. By using the new augmented Lyapunov-like function with more general decomposition approach, a novel sufficient condition for finite-time boundedness with an H∞performance index is derived. Based on the derived condition, the reliable H∞control problem is solved. It is easy to see that neither bounding technique for cross terms nor model transformation is involved, the obtained result is expected to be less conservative.3. Finite-time H∞filtering for a class of discrete-time Markovian jump systems with partly unknown transition probabilities. By introducing some slack matrix variables in terms of probability identity, a less conservative bounded real lemma is derived to ensure that filtering Markovian jump systems is finite-time stable. Finally, the existence criterion of the desired filter is obtained such that the corresponding filtering error system is finite-time bounded with a guaranteed H∞performance index. It should be pointed out that, the LMI solutions can be obtained by solving the eigenvalue problem with respect to solution variables, which is a convex optimization problem and can be checked readily by standard algorithms such as the interior-point method.4. Finite-time boundedness of a class of discrete-time Markovian jump systems with piecewise-constant transition probabilities subject to average dwell time switching. Another set of useful regime-switching models has been given for both fixed transition probability Markov switching models and time-varying transition probabilities. Based on the knowledge of average dwell time, Leibniz-Newton formula and multiple Lyapunov function, a novel sufficient condition for finite-time boundedness of H∞filtering is derived and the system trajectory stays within a prescribed bound.