An SOS-Based Sliding Mode Controller Design for a Class of Polynomial Fuzzy Systems

This paper investigates the problem of designing a robust controller for continuous polynomial fuzzy systems with external disturbances. The input matrix <inline-formula><tex-math notation="LaTeX">$\mathbf {B}_i(x)$</tex-math></inline-formula> of the considered system is not equal to one another. To solve this problem, first, we rewrite every column vector of input matrix <inline-formula><tex-math notation="LaTeX">$\mathbf {B}_i(x)$</tex-math></inline-formula> using the vectors of the basis matrix <inline-formula><tex-math notation="LaTeX">$\mathbb {B}(x)$</tex-math></inline-formula>. Second, by use of the column vector of <inline-formula><tex-math notation="LaTeX">$\mathbb {B}(x)$</tex-math></inline-formula>, the subsliding mode surface <inline-formula><tex-math notation="LaTeX">$s_p(t)$</tex-math></inline-formula> is designed. Based on the rewritten system, a novel sliding mode surface <inline-formula><tex-math notation="LaTeX">$s(t)$</tex-math></inline-formula> is devised. The sliding mode surface parameter matrix can be characterized in terms of the solution of the provided sum of squares conditions. Then, a sliding mode controller is proposed to stabilize the considered system. It could make the state of the considered system drive onto <inline-formula><tex-math notation="LaTeX">$s(t)$</tex-math></inline-formula> and avoid the state escaping from the surface in the subsequent time. A lemma is provided to obtain the final result. Finally, practical and numerical examples are provided to demonstrate the validity of the proposed approach.