A Novel Sliding Mode Control for a Class of Stochastic Polynomial Fuzzy Systems Based on SOS Method

In this paper, a novel robust controller for continuous stochastic polynomial fuzzy systems is investigated. The aim of the proposed method is to eliminate the restrictive assumptions that the local input matrix <inline-formula> <tex-math notation="LaTeX">${B}_{{i}}$ </tex-math></inline-formula> must be uniform and the sliding mode surface proposed did not consider the stochastic perturbations, which are required in most existing results. At the same time, the proposed method could handle the system with matched external disturbances. First, a novel vector integral sliding mode surface (VISMS) is constructed according to the basis matrix <inline-formula> <tex-math notation="LaTeX">$\mathbb {B}({x})$ </tex-math></inline-formula>. The sliding mode surface parameter matrix <inline-formula> <tex-math notation="LaTeX">$\mathbb {K}({x})$ </tex-math></inline-formula> can be obtained through the provided sum of squares conditions. Second, by using an improved Lyapunov method and a new proposed lemma, a novel sliding mode control law is designed to keep the state of the closed-loop systems on the VISMS approximately since the initial time. Third, a practical example and a numerical one are provided to illustrate the validity of the proposed approach.