Jumps With Markov Switching System Stability Analysis

A large class of physical systems has variable structures subject to random changes, which many result from the abrupt phenomena such as component and interconnection failures or random communication delays in automobile vehicles. Systems with this character may be modeled as Markov jump systems. The state space of the systems contains both continuous-valued states that take values from a Euclidean vector space R~n and discrete- valued states that take values from a discrete finite set S. The transitions between the different regimes have to be considered as random; in other words, and the dynamics of the discrete states are given by a Markov jump process. Nowadays more and more people have paid attention to the stability analysis of the Markov jump systems and the study of the switching control. This dissertation devotes on the analysis of the exponential stability and the almost sure stability of the closed-loop systems. The main contributions of the research work presented in this dissertation are as follows:1. The almost sure stability of discrete-time jump linear systems with disturbance is studied by using the stochastic version of Lyapunov's second method. A sufficient condition for almost sure stability is derived. Some new simpler testable sufficient conditions for almost sure stability are obtained from the proposed sufficient condition.2. The exponential stability of a class of discrete time Markov switched linear systems with impulsive jump is studied. The given jump systems are composed of stable subsystems and unstable subsystems. Based on the average dwell time concept and by dividing the total activation time into the time with stable subsystems and the time with unstable subsystems, it is shown that if the average dwell time and the ratio of the expectation of activation time with stable subsystems to the expectation of activation time with unstable subsystems are properly large, the given jump system is exponentially stable with a desired stability margin even if there exists impulse at the switched moments. 3. The robust exponential stability of a class of discrete time-varying Markov jump systems with structure perturbations is studied. By the average dwell time method, it is shown that if the average dwell time and the ratio of the expectation of the activation time with stable subsystems to the expectation of the activation time with unstable subsystems are properly large, the given jump linear system is exponentially stable with a desired stability margin even if there exists impulse jump at the switching instances.4. The exponential stability is studied for a class of continuous time linear systems with a finite state Markov chain form process and the impulsive jump at switching moments. The given jump linear systems are composed of both Hurwitz stable and unstable subsystems. It is shown that if the average dwell time is chosen sufficiently large and the expectation of the activation time with unstable subsystems is relatively small compared with that of Hurwitz stable subsystems, exponential stability of a desired stability degree is guaranteed without the influence of the excitation signal. And the uniformly bounded result is realized for the case in which switched system is subjected to the impulsive effect of the excitation signal at some switching moments.5. A H_∞variable structure control is presented for singular Markov switched system with mismatched norm-bounded uncertainties and mismatched norm-bounded external disturbances. A sufficient condition guaranteeing the existence of linear switching surface is given based on the linear matrix inequality method (LMIs). It is shown that the sliding mode dynamic on the switching surface is regular, impulse-free, and stochastically stable and satisfies H_∞performance. A variable structure controller is designed to guarantee that the system trajectory is convergent to the linear switching surface.