Stability Analysis of Discrete-Time Polynomial Fuzzy-Model-Based Control Systems With Time Delay and Positivity Constraints Through Piecewise Taylor Series Membership Functions

Abstract

This article proposes a novel membership-function dependent approach for the stability/positivity investigation and controller design of a nonlinear, discrete-time system with time delay which is represented by a polynomial fuzzy model to enhance the accuracy of the approximation. The polynomial fuzzy controller is designed based on imperfect premise matching design concept and a set of sum of square (SOS)-based conditions is formulated to check the positivity and stability of the control system. To relax the conservative SOS-based conditions, we employ piecewise Taylor series membership functions (PTSMFs) and introduce the membership function knowledge into the stability conditions. Slack matrices are used to represent the membership function and premise variable knowledge, which contains: 1) regional approximation error between PTSMFs and original membership functions; 2) regional bounds of PTSMFs; and 3) the property knowledge of interpolation membership function of PTSMFs and regional bounds of premise variables. A simulation example is finally given to verify our novel stability/positivity conditions and discrete-time polynomial fuzzy (DPF) controller design.