Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies

Abstract

In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.