Research On Stability Analysis And Pareto Optimal Control Of Stochastic Systems

This thesis mainly investigates the stability analysis of nonlinear stochastic discrete-time systems and the Pareto optimal control of stochastic It?o systems.Stability is one of the most important dynamic characteristics of a system,and we first need to research the stability analysis when design a controller in infinite horizon for the considered sys-tem.However,in most of exiting literatures,the stability criteria of nonlinear stochastic discrete-time systems were given via stochastic inequalities rather than deterministic inequalities,which are difficult to be verified in practice.Besides,Pareto game is an important cooperative game,which has been applied in a wide field,such as chemical technology,economic management,wireless communication,signal processing and so on.While few scholars researched the Pareto optimize about the stochastic systems disturbed by external perturbation.In this thesis,by introducing an efficient discrete operator,we first obtain a kind of stability criteria for the nonlinear stochastic discrete-time systems based on deterministic inequalities,which provides a new method for studying the sta-bility of discrete nonlinear stochastic systems.On the other hand,by we first solve the Pareto optimal problems for the stochastic systems with external disturbances in com-bination with the Nash game and Pareto game.The main contributions of the research work are as follows:1. The stability and stabilization of the nonlinear discrete-time stochastic systems have been researched.On one hand,for the general nonlinear discrete-time stochastic systems,the criteria about the p-th moment exponential stability and the locally/globally asymptotic stability in probability have been presented based on a new difference operator which do not depend on the mathematical expectation of the state trajectory but only the mathematical expectation of the noise.It is worth to notice that the obtained criteria are all given via deterministic function inequalities that can be easily tested.On the other hand,for a class of quasi-linear discrete-time stochastic control systems,both state-and output-feedback asymptotic stabilization are studied,for which,sufficient conditions are presented in terms of linear matrix inequalities.2. The stability analysis for discrete-time stochastic time-delayed systems with multi-plicative noise is studied.The H-representation technique is used to make the connection between the stochastic time-delay systems with common time-varying coefficients and the standard deterministic time-varying systems.A necessary and sufficient condition for the considered system satisfying the exponentially stable in mean square sense is given based on the spectral operator method.Then,by applying the so-called“frozen”technique,it is shown that the stability of a“frozen”system implies that of the corresponding slowly time-varying system.Two different augmented system methods are used to deal with the time delay terms in our considered system.3. The Pareto optimal control problem with H_?constraint is studied for the linear stochastic systems in finite horizon.Firstly,a stochastic bounded real lemma with any deterministic initial condition is obtained,which is an improvement for most of existing results with zero initial condition in stochastic systems.Secondly,the two-person non-zero sum Nash game and the Pareto game has been applied to research the Pareto optimal strategy of the multi-controller system disturbed by external disturbances.The necessary and sufficient condition for the Pareto optimal control with H_?constraint existing is presented.And we give a sufficient condition for obtaining Pareto solution of every controller under the Pareto efficient strategy and worst-case external disturbance working on a stochastic system with only state-dependent noise.At last,a practical example illustrates the effectiveness of our results.4. The Pareto optimal control problem of the mean-field stochastic systems is in-vestigated in finite horizon.Firstly,combing the stochastic mean-field theory,we give a stochastic bounded real lemma with certain stochastic initial condition which satisfies some statistical characteristics.Secondly,the mean-field forward-backward stochastic differential equation is applied to solve the mean-field linear quadratic Pareto optimal control problem with undefined cost function.Thirdly,we present a necessary and suffi-cient condition for obtaining the Pareto optimal strategy under H_?constraint based on coupled Riccati equations,and a sufficient condition of the existence of Pareto solution is presented for the linear mean-field system with Pareto optimal strategy and worst-case disturbance.Finally,the conclusions and some topics for future work are given.