Asymptotically necessary and sufficient stability conditions in the sense of homogeneously polynomially nonquadratic Lyapunov functions for discrete-time Takagi-Sugeno systems

This paper considers stability of discrete-time nonlinear systems in Takagi-Sugeno´s form. This problem has been studied over twenty years with many sufficient conditions, and asymptotically necessary and sufficient (ANS) conditions in the sense of common quadratic Lyapunov function, having being obtained. This paper considers general forms of homogeneously polynomially nonquadratic Lyapunov (HPNQL) function and homogeneously polynomially parameterized (HPP) state feedback laws. By generalizing the procedure based on Polya´s theorem, which has been studied previously in different context, ANS stability conditions in the sense of HPNQL function are obtained.