Stochastic Nonlinear Systems Control Study

When global stabilization controllers are designed by Lyapunov stabilization theory for stochastic nonlinear system,the closed-loop systems usually can be transformed into exceeding nonlinear functions even for simple time-unvarying linear systems.And the input-to-output signals given by the closed-loop systems usually are more complicated stochastic process, so the difficulties of the investigation for control theory are "stochastic" and "nonlinear".It is because of this basic reason that many typical and basal problems of control theory are unfathomed for a long time.Thus,the investigation for the control theories of stochastic nonlinear systems is a challenge and worthy to investigate,which is very important in actual application. In this paper,the problems of state-feedback stabilization for several classes of stochastic nonlinear systems are investigated.The main contributions include:1.For a class of high-order stochastic nonlinear systems with stochastic inverse dynamics which are neither necessarily feedback linearizable nor affine in the control input,this paper investigates the problem of state-feedback stabilization for the first time.Under some weaker assumptions,a smooth statefeedback controller is designed,which ensures that the closed-loop system has an almost surely unique solution on[0,∞),the equilibrium at the origin of the closed-loop system is globally asymptotically stable in probability,and the states can be regulated to the origin almost surely.2.This chapter investigates the adaptive state-feedback stabilization problem for a class of high-order stochastic non-linear systems with unknown lower and supper bounds for uncertain 10 control coefficients.Under some weaker and reasonable assumptions,a smooth adaptive state-feedback con- troller is designed,which guarantees that the closed-loop system has an almost surely unique solution on[0,∞),the equilibrium of interest is globally stable in probability and the states can be regulated to the origin almost surely.3.This chapter investigates the adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization.By using the parameter separation lemma and some flexible algebraic techniques, and choosing an appropriate Lyapunov function,a smooth adaptive statefeedback controller is designed,which guarantees that the closed-loop system has an almost surely unique solution for any initial state,the equilibrium of interest is globally stable in probability,and the state can be regulated to the origin almost surely.4.An adaptive state-feedback stabilization is investigated for a class of high-order stochastic nonlinear systems(6.1.1)in which the upper bound of the function f_i(·)depends on the state x_i+1besides the states x₁,…,x_i and z.By choosing an appropriate Lyapunov function,a smooth adaptive statefeedback controller is designed,which guarantees that the closed-loop system has an almost surely unique solution on[0,∞),and the equilibrium at the origin is globally stable in probability.5.Together with the Input-to-State Stability of certain case,the concept of Input-to-State practical Stability(ISpS)in probability and the Lyapunov stability criterion of ISpS in probability are given in this paper,which will play an essential role in the controller design and stability,analysis of stochastic nonlinear systems.