Polynomial Fuzzy-Model-Based Control Systems: Stability Analysis Via Piecewise-Linear Membership Functions

• Published in: IEEE transactions on fuzzy systems Vol. 19; no. 3; pp. 588 - 593
• Main Author: Lam, H K
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.06.2011; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic stability; Control systems; Fuzzy; Fuzzy logic; Fuzzy set theory; Fuzzy-model-based (FMB) control; Mathematical analysis; Mathematical models; Numerical stability; piecewise-linear membership functions (PLMFs); polynomial fuzzy systems; Polynomials; Stability; Stability analysis; Stability criteria; Studies; sum-of-squares (SOS); Systems stability; Thermal stability
• ISSN: 1063-6706; 1941-0034
• DOI: 10.1109/TFUZZ.2011.2118215

This paper presents the stability analysis of polynomial fuzzy-model-based (PFMB) control systems using the sum-of-squares (SOS) approach. The PFMB control system under consideration requires that the polynomial fuzzy model and polynomial fuzzy controller share neither the same premise membership functions nor the same number of fuzzy rules. This class of PFMB control systems offers a greater design flexibility to the polynomial fuzzy controller. However, due to the imperfectly matched membership functions, it usually produces more conservative stability conditions by following the traditional stability-analysis approach for the FMB control systems. To facilitate the stability analysis, piecewise-linear membership functions (PLMFs) are proposed, which offer a nice property that the grades of membership are governed by a finite number of sample points. Thus, it allows the PLMFs to be brought to the SOS-based stability conditions, which are applied to the PFMB control systems with the specified PLMFs rather than any shapes. The system stability can be examined by checking only the PFMB control system at the sample points. It is worth mentioning that the PLMFs, which are not necessarily implemented physically, are a mathematical tool to carry out the stability analysis. To verify the stability-analysis result, a simulation example is given to demonstrate the effectiveness of the proposed approach.