Robust ℓ₁-Controller Design for Discrete-Time Positive T–S Fuzzy Systems Using Dual Approach

Abstract

In this article, a new approach is proposed for stability analysis and controller design of nonlinear discrete-time positive systems by means of the Takagi&#x2013;Sugeno fuzzy model. The closed-loop stability and the positivity constraint are guaranteed by synthesizing a linear co-positive Lyapunov function and by applying the parallel distributed compensation controller. In contrast to the state-of-the-art approaches for ensuring the <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-stability of the positive system which are based on bilinear matrix inequalities, the proposed optimal robust control design under <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-induced performance is derived based on linear programming framework. It has been shown that the computational complexity of the proposed optimization problem can be effectively reduced. Finally, a numerical example and the Leslie population model are adopted to show the capabilities of the proposed method.