Local stability analysis and estimation of domains of attraction for nonlinear systems via Takagi-Sugeno fuzzy modeling

This paper addresses the problem of nonquadratic stability and estimation of domains of attraction for continuoustime nonlinear systems represented by Takagi-Sugeno (T-S) fuzzy models. The T-S fuzzy model is obtained by the sector nonlinearity approach and represents exactly the nonlinear system in a compact set of the state-space. Sufficient linear matrix inequality (LMI) conditions to access local stability are obtained using homogeneous polynomially parameter-dependent Lyapunov functions and slack variables introduced through the Finsler’s Lemma. The time-derivatives of the membership functions are handled by means of a polytopic representation of the gradient of the membership functions. As a technical novelty, whenever possible, the nonlinear terms of the gradient are handled through a Taylor series expansion, avoiding the definition of new (generally overbounded) uncertain parameters that increase the numerical complexity. Finally, a numerical example is presented to show the superiority of the approach in terms of estimating larger domains of attraction when compared with previous methods from the literature.