Stability analysis of dynamic nonlinear interval type-2 TSK fuzzy control systems based on describing function
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METHODOLOGIES AND APPLICATION
Stability analysis of dynamic nonlinear interval type-2 TSK fuzzy control systems based on describing function Zahra Namadchian1 · Assef Zare1
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper focuses on the limit cycles prediction problem to discuss the stability analysis of dynamic nonlinear interval type-2 Takagi–Sugeno–Kang fuzzy control systems (NIT2 TSK FCSs) with adjustable parameters. First, in order to alleviate computational burden, a simple architecture of NIT2 TSK FCS using two embedded nonlinear type-1 TSK fuzzy control systems (NT1 TSK FCSs) is proposed. Then, describing function (DF) of NIT2 TSK FCS is obtained based on the DFs of embedded NT1 TSK FCSs. Subsequently, integrating the stability equation and parameter plane approaches provides a solution to identify the limit cycle and the asymptotically stable regions. Moreover, particle swarm optimization technique is applied to minimize the limit cycle region. Furthermore, for robust design, a gain-phase margin tester is utilized to specify the minimum gain margin (GMmin ) and phase margin (PMmin ) when limit cycles can arise. Finally, two simulation examples are considered to validate the advantages of the presented method. Keywords Interval type-2 Takagi–Sugeno–Kang fuzzy control systems · Stability · Limit cycle · Gain margin · Phase margin · Describing function
1 Introduction Takagi–Sugeno–Kang (TSK) fuzzy logic controller (FLC) is among the most well-known structure of FLCs which has been utilized in many applications such as biomedical engineering, robotics, and marketing; since TSK FLC has efficient performance in learning accuracy based on its structure, the fuzzy rules are equipped with functional-type consequences instead of fuzzy terms and also are associated with mathematical knowledge in control processes. Thus, its behavior can be analyzed applying conventional control theory (Jafari et al. 2018). On the other hand, as commonly discussed, type-2 fuzzy system (T2FS) as an extension of T1FS is applicable to handle uncertainties. Moreover, to deal with complex computations of T2FSs, interval type-2 fuzzy systems (IT2FSs) are introduced, while all secondary degree Communicated by V. Loia.
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Zahra Namadchian [email protected] Assef Zare [email protected]
1
Department of Electrical Engineering, Gonabad Branch, Islamic Azad University, Gonabad, Iran
of membership functions is equal to one. Hence, with taking advantages of the above-mentioned properties of three kinds of FLCs, IT2 TSK FS has received much attention in the fields of modeling, control system, and stability analysis (see Zhao et al. 2019; Zhao and Dian 2017, 2018; Hailemichael et al. 2018; Sun et al. 2017; Gao et al. 2017; Xun et al. 2015; Enyinna et al. 2015 and references therein). For example, an adaptive IT2 TSK fuzzy system with a supervisory mode has been presented to control and stabilize a certain class of nonlinear fractional order systems (Jafari et al. 2018). An IT2 TSK FLC to enhance the c
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