$$L_{1}$$ -Induced Static Output Feedback Controller Design and Stability Analysis for Positive Polynomial Fuzzy Systems

Abstract

The aim of this paper is to study the control synthesis and stability and positivity analysis under \(L_1\)-induced performance for positive systems based on a polynomial fuzzy model. In this paper, not only the stability and positivity analysis are studied but also the \(L_{1}\)-induced performance is ensured by designing a static output feedback polynomial fuzzy controller for the positive polynomial fuzzy (PPF) system. In order to improve the flexibility of controller implementation, imperfectly matched premise concept under membership-function-dependent analysis technique is introduced. In addition, although the static output feedback control strategy is more popular when the system states are not completely measurable, a tricky problem that non-convex terms exist in stability and positivity conditions will follow. The nonsingular transformation technique which can transform the non-convex terms into convex ones successfully plays an important role to solve this puzzle. Based on Lyapunov stability theory, the convex positivity and stability conditions in terms of sum of squares (SOS) are obtained, which can guarantee the closed-loop systems to be positive and asymptotically stable under the \(L_{1}\)-induced performance. Finally, in order to test the effectiveness of the derived theory, we show an example in the simulation section.