Polynomial static output feedback H _∞ control via homogeneous Lyapunov functions

In this paper, we study a polynomial static output feedback (SOF) stabilization problem with H _∞ performance via a homogeneous polynomial Lyapunov function (HPLF). It is shown that the quadratic stability ascertaining the existence of a single constant Lyapunov function becomes a special case. With the HPLF, the proposal is based on a relaxed two-step sum of square (SOS) construction where a stabilizing polynomial state feedback gain K(x) is returned at the first stage and then the obtained K(x) gain is fed back to the second stage, achieving the SOF closed-loop stabilization of the underlying polynomial fuzzy control systems. The SOS equations obtained thus effectively serve as a sufficient condition for synthesizing the SOF controllers that guarantee polynomial fuzzy systems stabilization. To demonstrate the effectiveness of the proposed polynomial fuzzy SOF H _∞ control, benchmark examples are provided for the new approach.