Asymptotically necessary and sufficient stability conditions for discrete-time Takagi–Sugeno model: Extended applications of Polyaʹs theorem and homogeneous polynomials

The stability and stabilization conditions of the nonlinear system in Takagi–Sugenoʹs form are considered. The homogeneously polynomially nonquadratic (HPNQ) Lyapunov functions and homogeneously polynomially parameterized (HPP) state feedback laws are adopted. By generalizing the procedure based on the Polyaʹs theorem, the asymptotically necessary and sufficient (ANS) stability and stabilization conditions in the case of HPNQ Lyapunov functions and HPP control laws are reformulated. The major contribution of this paper is to give the parallel results using the multiple indices, so that the slack matrices can be extensively utilized to improve the numerical efficiency. The effectiveness of the results is illustrated by the numerical examples.