Asymptotically necessary and sufficient stability with respect to nonquadratic Lyapunov function for Takagi-Sugeno model

This paper proposes novel stability conditions of nonlinear systems in Takagi-Sugeno´s form. This problem has been studied over twenty years with many sufficient conditions. Recently, asymptotically necessary and sufficient conditions are obtained, which are preferred with respect to common quadratic Lyapunov function. This paper considers general forms of homogeneously polynomially nonquadratic Lyapunov function and homogeneously polynomially parameterized state feedback laws. By generalizing the procedure based on the Polya´s theorem, which has been studied previously in different context, asymptotically necessary and sufficient stability conditions with respect to nonquadratic Lyapunov function are obtained. The results are novel also in that the number of conditions is minimized with respect to homogenously polynomially parameter-dependent solutions.