$\mathcal{l}_{1}$ Filtering for Positive Takagi–Sugeno Fuzzy Systems via Successive Linear Programming

Abstract

This paper studies the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {l}_{1}$</tex-math></inline-formula> filtering problem for positive Takagi-Sugeno (T-S) fuzzy systems. A pair of filters are sought to assure stability and positivity of filtering error systems with guaranteed <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {l}_{1}$</tex-math></inline-formula> performance indices. For this purpose, a pair of linear co-positive Lyapunov functions that capture the positivity characteristics of filtering error systems are adopted to derive sufficient conditions for the existence of filters, which boils down to the bilinear programming problem. To solve this dilemma, we propose a convex approximation strategy, and thus a sequence of convex surrogates are developed to reasonably replace the non-convex constraints, based on which a successive linear programming algorithm is constructed. Therefore, the filters can be achieved by solving a battery of linear programmings. Finally, the efficiency of the designed filtering scheme was justified by simulation validations.