Adaptive State-Feedback Stabilization of Stochastic High-Order Nonlinear Systems With Time-Varying Powers and Stochastic Inverse Dynamics

This article investigates adaptive state-feedback stabilization problem for a class of stochastic high-order nonlinear systems with unknown time-varying powers and stochastic inverse dynamics for the first time. The existence of stochastic inverse dynamics, unknown parameters, and time-varying powers makes stochastic high-order nonlinear systems essentially different from the related papers, which brings a series of obstacles to achieve the control objective. By virtue of the parameter separation principle, adaptive technique and some flexible algebraic methods, a novel adaptive state-feedback controller is designed to guarantee that the equilibrium of the closed-loop system is globally stable in probability. Finally, a simulation is provided to demonstrate the effectiveness of the control scheme.

I. Introduction

Consider the stochastic high-order nonlinear systems \begin{align*} dz &= f_{0}(z,x_1)dt+g_{0}^\top (z,x_1)d\omega \\ dx_i &= d_i(t,x)\lceil x_{i+1} \rceil ^{p_i(t)} dt+f_{i}(z,x,\theta)dt \\ &\quad\; +g_{i}^\top (z,x,\theta)d\omega, \quad i=1,\ldots,n-1, \\ dx_n &= d_n(t,x)\lceil u\rceil ^{p_n(t)}dt+f_{n}(z,x,\theta)dt \\ &\quad\; +g_{n}^\top (z,x,\theta) d\omega \tag{1} \end{align*} where is the system state with the initial value , is stochastic inverse dynamics and the initial value , is the control input, system powers , , are unknown time-varying functions, is an unknown constant vector, is an -dimensional standard Wiener process defined on the complete probability space with being a sample space, being a filtration, and being a probability measure. Functions , , , and , , are locally Lipschitz with , , , , are nonlinear control coefficients. For any , denotes its sign function, for .