Stabilization of It\^o Stochastic T-S Models via Line Integral and Novel Estimate for Hessian Matrices

TL;DR

This paper introduces a novel line integral Lyapunov function method for analyzing and stabilizing Itô stochastic T-S models, providing more general stability conditions and an effective control design approach.

Contribution

It presents a new stability analysis technique using line integral Lyapunov functions and a non-quadratic approach for stabilization of stochastic T-S models.

Findings

Derived more general stability conditions than quadratic Lyapunov methods

Developed a controller using cone complementarity linearization algorithm

Validated the approach with numerical examples demonstrating effectiveness

Abstract

This paper proposes a line integral Lyapunov function approach to stability analysis and stabilization for It\^o stochastic T-S models. Unlike the deterministic case, stability analysis of this model needs the information of Hessian matrix of the line integral Lyapunov function which is related to partial derivatives of the basis functions. By introducing a new method to handle these partial derivatives and using the property of state-dependent matrix with rank one, the stability conditions of the underlying system can be established via a line integral Lyapunov function. These conditions obtained are more general than the ones which are based on quadratic Lyapunov functions. Based on the stability conditions, a controller is developed by cone complementarity linerization algorithm. A non-quadratic Lyapunov function approach is thus proposed for the stabilization problem of the It\^o stochastic T-S models. It has been shown that the problem can be solved by optimizing sum of traces for a group of products of matrix variables with linear constraints. Numerical examples are given to illustrate the effectiveness of the proposed control scheme.