Controller Design And Analysis Of Stochastic Nonlinear Systems

Inverse optimal stabilization for a class of high-order stochastic nonlinear systems, and output-feedback stabilization for high-order stochastic nonlinear systems are considered in the paper, which is composed of the following two parts.1. The problem of Inverse optimal stabilization for a class of high-order stochastic nonlinear systems.Consider the following high-order stochastic nonlinear systems described by:dx_i=(x_i+1^pi+f_i(x_i))dt+g_i^T(x_i)dω,i=1,…,n-1,dx_n=(u^pn+f_n(x_n))dt+g_n^T(x_n)dω,where x = [x₁,…,x_n]^T∈Rⁿ and u∈R are the measurable state and control input, respectively;ω∈R^r is independent standard Wiener process vector; P_i≥1, i = 1,…,n, are odd integers. The functions f_i : Rⁱâ†'R and g_i : Rⁱâ†'R^r, i = 0,1,…, n, are assumed to be smooth which satisfies the following Assumption 1, vanished at the origin.Assumption 1: There are nonnegative smooth functions f_i1(x_i) and g_i1(x_i),i=1,…,n, such that |f_i(x_i)|≤(|x₁|^pi+…+|x_i|^pi)f_i1(x_i),|g_i(x_i)|≤(|x₁|^pi+…+|x_i|^pi)g_i1(x_i), i=1,…,n．p_i≥1,i=1,…,n,are odd integers satisfying Assumption 2.Assumption 2:p₁≥…≥p_n≥1．The objective of this part is to design a smooth state-feedback controller under Assumption 1 and 2, such that the closed-loop system is globally asymptotically stable in probability and inverse optimal stabilization in probability.2. The problem of output-feedback stabilization for high-order stochastic nonlinear systems.Consider the following stochastic nonlinear systems:dζ_i=ζ_i+1^p dt+φ_i^T(ζ_i)dω,i=1,…,n-1,dζ_n=v^pdt+φ_n^T(ζ_n)dω,y=ζ₁,where x = [x₁,…,x_n]^T∈Rⁿ, u∈R and y∈E are unmeasurable state vector, control input and measurable output, respectively;ω∈R^m is independent standard Wiener process vector; p > 0 is an odd integer,φ_i : Rⁱâ†'R^r, i = 1,…, n, are C¹ functions which satisfy the following Assumption 3 andφ_i(0) = 0.Assumption 2:φ_i(x_i),i= 1,…,n, there exists a real number k > 0 such that for all i= 1,…n,[φ_i((?)_i)|≤k|y|^p+1/2The objective is to design an observer and an output-feedback controller:(?)=(?)(x), u=μ(x),under Assumption 3, such that the closed-loop system is globally asymptotically stable in probability.