Saturation Control Of Markov Jump Positive Systems

Positive systems are a special kind of systems.This kind of systems have received exten-sive attention in the fields of social economy and biomedicine due to the positive characteriza-tion of the systems model.Markov jump positive systems are a kind of multi-modal stochastic systems,which consists of some positive subsystems and a Markov chain.Such systems are suitable for modeling population dynamics,urban water management networks,communica-tion networks and so on.In practical systems,actuator saturation is unavoidable due to some physical limitations.Therefore,it has significant theoretical and practical significance for the control and synthesis of Markov jump positive systems with actuator saturation.This thesis mainly studies the robust control and non-fragile control of Markov jump positive systems with actuator saturation.The thesis will be divided into the following aspects:The first part is a introduction.The research background and significance of the subject are discussed.The recent developments of positive systems,Markov jump positive systems and actuator saturations are summarized.The research methods used in this thesis are introduced around the systems model.Finally,the main research contents of this thesis are summarized.The second part studies the robust saturation control of Markov jump positive system-s.Firstly,a cone invariant set is proposed.A stochastic co-positive Lyapunov functional is constructed for the systems.Sufficient conditions for stochastic stabilization based on linear programming approach are obtained.The conservativeness of controller design is reduced by using a gain matrix decomposition method.Then,the method is extended to the system with interval and polytopic uncertainties,respectively.Furthermore,an effective method for esti-mating the attraction domain is established by solving an optimization problem.Finally,two numerical examples are given to show the validity of the proposed design approach.The third part investigates the non-fragile saturation control of Markov jump positive sys-tems.By constructing a co-positive Lyapunov function,sufficient conditions for stochastic stabilization of continuous-time systems with additive gain uncertainty and partial known tran-sition probability are obtained.A new non-fragile controller and attraction domain gain design method is proposed.Then,this method is used to study the non-fragile saturation control prob-lem of discrete-time Markov jump positive systems.This method not only solves the non-fragile stabilization problem of the saturated positive system,but also reduces the conservative design of the controller.Finally,two numerical simulations are given to verify the effectiveness of the results.The fourth part considers the non-fragile saturation control of nonlinear Markov jump pos-itive systems.Firstly,sufficient conditions for positivity of nonlinear systems are established.Then,by using a nonlinear co-positive Lyapunov-Krasovskii functional,sufficient conditions for saturation stabilization of nonlinear Markov jump positive systems with multiplicative and additive gain uncertainties are proposed,respectively.The design method of state-feedback controller and auxiliary controller is solved based on linear programming and matrix decompo-sition method.Then,an optimization condition is established to estimate the attraction domain.Finally,two numerical examples are given to verify the effectiveness of the method.The fifth part:Conclusions and prospects.First of all,the conclusions of this thesis are summarized.Then,some thoughts and ideas for the future are proposed.