A Sum of Squares Approach to Stability Analysis of Polynomial Fuzzy Systems

This paper presents a sum of squares (SOS) approach to stability analysis of polynomial fuzzy systems. Our SOS approach provides two innovative and extensive results for the existing LMI approaches to Takagi-Sugeno fuzzy systems. First, we propose a polynomial fuzzy model that is a more general representation of the well-known Takagi-Sugeno fuzzy model. Second, we derive stability conditions based on polynomial Lyapunov functions that contain quadratic Lyapunov functions as a special case. Hence, stability analysis discussed in this paper is more general than that based on the existing LMI approaches to Takagi-Sugeno fuzzy systems. The stability conditions in the proposed approach can be represented in terms of SOS and are numerically (partially symbolically) solved via the recent developed SOSTOOLS. To illustrate the validity and applicability of the proposed approach, two analytical examples are provided. The first example shows that our approach provides more relaxed stability results than both the existing LMI approaches and a polynomial system approach. The second example illustrates the utility of our approach in comparison with a piecewise Lyapunov function approach.