Static Output Feedback Control for Positive Takagi–Sugeno Fuzzy Systems: A Successive Linear Programming Approach

Abstract

This article considers the control synthesis for positive nonlinear systems modeled by Takagi–Sugeno (T–S) fuzzy characterizations. A fuzzy static output feedback (SOF) controller is sought to assure the closed-loop stability and positivity with a 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 level. The favorable positivity enables us to adopt a linear co-positive function as the Lyapunov candidate, and we construct the essential conditions for the existence of fuzzy SOF controllers in the form of bilinear programming (BP) problem. To provide a direct and generic algorithm for solving the BP problem, we propose a vital approximation strategy by which the general bilinear constraints are replaced with a sequence of convex surrogate ones. Recognizing the fact that a fundamental promise of the proposed successive linear programming algorithm is a feasible initial solution, we also develop an iterative procedure to calculate an initial controller gain. Finally, we clarify the applicability of the developed scheme with two examples.