Abstract
In this paper, the stability of polynomial-fuzzy-model-based (PFMB) systems equipped with mismatched interval type-2 (IT2) membership functions is investigated. Unlike the membership-function-independent methods, the information and properties of IT2 membership functions are considered in the stability analysis and contained in the stability conditions in terms of sum-of-squares (SOS) based on the Lyapunov stability theory. Three methods, demonstrating their own advantages, are proposed to conduct the stability analysis for the IT2 PFMB control systems. In the first one, we divide the operating domain into subdomains and then conduct the stability analysis incorporating the information and properties of the IT2 membership functions in subdomains. Through this approach, the stability conditions can be further relaxed compared with the membership-function-independent analysis. Polynomial functions are adopted in the second method to approximate the IT2 membership functions. The advantage of this method compared with the first one is that richer information of IT2 membership functions is considered without increasing the number of SOS conditions. In the third one, we combine the advantages of both the first and the second method offering a new approach which utilizes the information and properties of the lower and upper IT2 membership functions in subdomains through simpler polynomial approximation functions. It can be shown that more relaxed stability conditions can be obtained compared with the first two methods. Numerical examples and simulations are presented to verify the effectiveness of the proposed methods.
Highlights
Background KnowledgeT YPE-1 fuzzy set theory was first proposed by Zadeh in 1965 [1], which has been widely applied in domestic and industrial fuzzy control approaches
Given that the interval type-2 (IT2) membership functions are continuous in general cases, when taken into the stability analysis, it will lead to infinite stability conditions that are not practical to be solved numerically
We propose various techniques to bring the information of membership functions into the stability analysis, which avoids turning the number of stability conditions into infinite, but still can achieve more relaxed stability conditions
Summary
T YPE-1 fuzzy set theory was first proposed by Zadeh in 1965 [1], which has been widely applied in domestic and industrial fuzzy control approaches. It should be noted that when the requirement of the same rule set is removed, the results of stability analysis can be very conservative as the permutations of membership functions used in the PDC design approach cannot be applied due to the imperfectly matched membership functions. In [14], staircase-shaped functions were adopted to approximate the original membership functions in the stability analysis of FMB control systems, which allows adding the approximated membership functions into the stability conditions to make them membership function dependent, which leads to more relaxed stability analysis results The sum-ofsquares (SOS) approach is widely used in the stability analysis of polynomial-fuzzy-model-based (PFMB) control systems. To the best knowledge of the authors’ knowledge, there has been some research on the IT2 fuzzy control systems, the issues of stability analysis and control synthesis on IT2 PFMB control systems are rarely investigated
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