With Siiss Inverse Dynamic Stochastic Nonlinear System Stability Studies And Controller Design

When global stabilization controllers are designed by Lyapunov stabilization theory for stochastic nonlinear system, the closed-loop systems usually can be transformed into exceedingnonlinear functions even for simple time-unvarying linear systems. And the input-to-outputsignals 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.The contributions of this work are characterized by the following novel features:1.Motivated by the concept of integral input-to-state stability (iISS) in deterministic systems and stochastic input-to-state stability (SISS) using Lyapunov function in stochastic systems, a concept of stochastic integral input-to-state stability (SilSS) using Lyapunov function is first introduced, two important properties of SilSS are obtained: (i) SilSS is strictly weaker than SISS using Lyapunov function; (ii) SilSS is stronger than the minimumphaseproperty. However, only under the minimum-phase assumption, there is no dynamic output feedback control law for global stabilization in probability.2.Almost sure boundedness, a reasonable and stronger concept than boundedness in probability, is introduced. The purpose of introducing the concept is to prove the boundednessand convergence of some signals in the closed-loop control system.3.Some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed-loop system are established.4.Considering the stochastic nonlinear systems with SilSS inverse dynamics described in the following form: a unifying framework is proposed to design a dynamic output feedback control law, which drives the states to the origin almost surely while maintaining all the closed-loop signals bounded almost surely.