Polynomial Fuzzy Observed-State Feedback Stabilization via Homogeneous Lyapunov Methods

Abstract

A homogeneously state-dependent polynomial Lyapunov function for an observed-state feedback polynomial fuzzy control system is proposed. From which an advanced sufficient sum of squares (SOS) condition formulated in terms of state-dependent matrix inequalities to achieve fuzzy stabilization is derived. Three main features, compared with existing nonhomogeneously polynomial methods, are as follows: 1) homogeneous Lyapunov function and its time derivative are nicely linked via the Hessian matrix; 2) removing the nonconvex obstacles on $\dot{P}(x)$ when deriving a nonhomogeneous Lyapunov stabilization condition $\dot{V}(x)$ ; 3) separation principle holds; since common $P$ falls out as a special case, the proposed homogeneous SOS approach is placed ahead of the existing SOS-based methods found in polynomial fuzzy stabilization controls. To verify the problem-solving theories of the homogeneous polynomial method announced, four examples are demonstrated to show the promising features of the approach adduced. Finally, an Appendix illustrating the solving procedure is added for clarification on how the examples are solved via SOS.