Variable Structure Control Of A Class Of Markov Jump Systems

Markov jump linear systems(MJLSs)are defined as a family of linear systems subject to randomly jumping parameters,where the jumps are modeled by the transitions of a Markov chain,and are usually used to model plants with component failures,sudden environment dis-turbances,changes in subsystems interconnections,and so on.For MJLSs,there are two com-ponents in the state vector,a part of the state that takes values continuously(x?Rn)and another part of the state that takes discrete values rt?l.Because Markov jump linear systems have an extensive application in the fields of economy systems,flight control systems,robot systems and so on,many fruitful results about stability,control and sliding mode control are obtained in recent years.A most common assumption in existing works is made that the modes are available and the information for transition probability matrix are completely known.Although some ex-isting results concern uncertain transition probability matrix,the techniques used to deal with the uncertainties are based on robust control methods for linear system.As we know,in practise,the transition probabilities may be obtained from tests or measurements.Because of restrict-ing of measuring conditions,some elements of transition probabilities matrix may change in in an interval or cannot be measured,which leads that we cannot directly utilize the methods of completely known transition probabilities to study stochastic stable and sliding mode control problem for Markov jump systems and stochastic admissible and sliding mode control problem descriptor Markov jump systems.Enlightened by the above observations,and based on some existing works of others,this dissertation respectively investigates the stochastic stable and sliding mode control problem for Markov jump systems subject to partially unknown information and the stochastic admissible and sliding mode control problem descriptor Markov jump systems subject to partially unknown information.Firstly,the system under consideration is more general,which covers the systems with completely known and completely unknown transition probabilities as two special cases-the latter is hereby the switched linear systems under arbitrary switching.Moreover,in contrast with the uncertain transition probabilities studied recently,the concept of partly unknown tran-sition probabilities proposed in this paper does not require any knowledge of the unknown elements.The sufficient conditions for stochastic stability or admissible and sliding mode con-troller of the underlying systems are derived via LMIs formulation.The main contributions are given as follows:Chapters 1-2 firstly summarize and analyze the development and main research methods for Markov jump linear systems.Preliminaries about the considered problem are also given.Chapter 3 investigates the problems of stochastic stability and the variable structure con-trol of a class of continuous-time Markov jump linear system with partially unknown transition probabilities.The method proposed is more general and includes two special cases where tran-sition probabilities are completely known and completely unknown.Firstly,by using linear matrix inequalities(LMIs)approach,we come up with a laconic sufficient condition to insure the stochastic stability of reduced order systems.Then,a succinct method is given to obtain stochastic stable sliding mode surface.Furthermore,we design a sliding mode controller to guarantee the convergence of the closed-loop system's state trajectories to the desired sliding switching in finite time and eliminate the chattering for all subsequent time,At last,three ex-amples are provided to demonstrate the advancement of our method.Chapter 4 investigates the variable structure control problems for a class of descriptor Markov jump systems subject to partially unknown transition probabilities.Firstly,a suffi-cient condition,under which such type descriptor Markov jump systems subject to partially unknown transition are stochastically admissible,is presented by virtue of the strictly linear matrix inequality technique.Then a sliding surface function,in the light of both system states and inputs,is presented for descriptor Markov jump systems subject to partially unknown tran-sition probabilities and a dynamic sliding mode controller is synthesized,which ensures the reachability of predefined sliding surface in finite time.It is also shown that the stochastic ad-missibility of the overall closed loop systems can be determined by checking the feasibility of a series of LMIs.Finally,an illustrative example on DC motor is provided to demonstrate the effectiveness of the theoretical results.At last,the results of the dissertation are summarized and further research topics are pointed out.