Research On Several Control Problems For Some Classes Of Markovian Jump Systems

In practical systems,since the system may be influenced by some factors,such as external disturbance,sensor or actuator failure and so on,it will result in the abrupt change for the parameters and the structure of the system.Markovian jump system is a good mathematical model to describe such system.In the decades of research on the Markovian jump system,a relatively complete theoretical system has gradually formed.In nature,the state variables of a class of systems need to take non-negative values,and scholars propose the positive system to describe it.Due to the specificity of positive systems,some theories of general systems are not suitable for positive systems.Therefore,the study of positive systems has formed a branch.On the other hand,the singular system has attracted the attention of scholars because it can better describe some practical systems.Since these two classes of systems may be affected by some factors,which will lead to the abrupt,change of the parameters and structure,in recent years,the research on positive Markovian jump systems and singular Markovian jump systems has attracted more and more attention of scholars.However,there are still some problems to be solved.Therefore,further improving the theoretical results of the Markovian jump system is an important task of this dissertation.Based on the existing results,this dissertation mainly focuses on several classes of Markovian jump systems with special form,namely positive Markovian jump systems,positive piecewise homogeneous Markovian jump systems,singular Markovian jump systems and singular semi-Markovian jump system,and studies the related control problems.The main content of this dissertation is as follows.For the nonlinear positive Markovian jump system with interval uncertainty,a pair of positive L1 state bounded observers is designed to estimate the unmeasurable states.The nonlinear system is dealt with the T-S fuzzy model technique.Firstly,the stochastic stability and L1 performance are analyzed by constructing a linear co-positive Lyapunov function.Since the system has interval uncertainty and disturbance,a pair of positive L1 fuzzy observers are designed to achieve the estimation of the system states all the time.Compared with the existing results,the proposed observer can handle the existence of uncertainty and disturbance,and the application scope is more extensive.In the implementation of the results,the conditions can be solved by using linear programming(LP)techniques rather than linear matrix inequality(LMI)techniques.Since the transition rate is finite piecewise,the jump process is finite piecewise homogeneous.This is different from the existing results about positive Markovian jump systems.Based on this situation,the problem of positive L1 filter design for T-S fuzzy positive piecewise homogeneous Markovian jump systems is studied.By using the linear co-positive Lyapunov function method,the stochastic stability and L1 performance of the system are analyzed when the transition rate of finite piecewise variation is described by stochastic variation and arbitrary variation,respectively.Finally,the filter designs in two cases are given.Compared with the existing results,the proposed method is simpler and has the advantage on computational complexity.By using the method of the reduced-order observer and sliding mode control,the control problem for singular Markovian jump systems is studied.The considered transition rate is time-varying.Firstly,the equivalent canonical form of the singular Markovian jump system is given.The integral sliding mode surface is designed by the estimated state of the reduced-order observer and the output of the equivalent canonical form.Secondly,the stochastic admissibility of sliding dynamics is analyzed,where the time-varying transition rate is handled by the quantization method.Finally,the sliding mode controller considering the nonlinear part of the system is designed.Compared with the existing results,the application scope of the proposed method is more extensive.The sliding mode control problem for T-S fuzzy singular semi-Markovian jump systems is studied.Firstly,the sliding surface design is performed.In order to improve the control performance of the system,an time-delay term is introduced.Then the admissibility analysis of sliding dynamics and the parameters of sliding mode surface are given,since the transition rate associated with the sojourn-time will cause the theorem conditions to be difficult to solve,the conditions that are easier to solve and less conservative are given.Finally,the sliding mode controller considering the transition rate is given.In the case that the derivative-term coefficient of the system depends on mode,the sliding mode control problem for the uncertain T-S fuzzy singular Markovian jump system is discussed.Firstly,a new continuous switching function is designed.Then the stochastic admissibility of the uncertain T-S fuzzy singular Markovian jump system is analyzed,and the strict linear matrix inequality conditions are given.Based on these conditions,the stochastic admissibility of sliding dynamics is discussed under the completely known transition rate and the uncertain transition rate,respectively.Finally,the sliding mode controller which can handle the uncertainty part is designed.