Approximation performance of the nonlinear hybrid fuzzy system based on variable universe

Psychonomic Bulletin & Review, Sep 2016

Variable universe is through a group of nonlinear contraction expansion factors online timely to adjust the input domain, so that the input domain is subdivided as detailed as possible at surrounding the expected control points. It is characteristic of constant rules, quick response, and high stability precision. In this paper, a nonlinear hybrid fuzzy system was established by introducing adjusting parameters to combine Mamdani fuzzy system and Takagi–Sugeno (T–S) fuzzy system. The input–output expression of the nonlinear hybrid fuzzy system based on variable universe was deduced according to the hybrid inference rules and contraction expansion factors. Furthermore, the second-order approximation of the nonlinear hybrid fuzzy system based on variable universe on the second-order continuously differentiable function was proved by multiple Taylor formula and maximum norm, and a sufficient condition for the approximation of the nonlinear hybrid fuzzy system was disclosed when adjusting parameters and approximation accuracy are known. Finally, the simulation example confirmed that the established nonlinear hybrid fuzzy system based on variable universe has better approximation performance than other systems under different adjusting parameter meanings.

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Approximation performance of the nonlinear hybrid fuzzy system based on variable universe

Granul. Comput. Approximation performance of the nonlinear hybrid fuzzy system based on variable universe Guijun Wang 0 1 Yu Li 0 1 Xiaoping Li 0 1 0 The Second Middle School , Zhanjiang 524022 , China 1 School of Mathematics Science, Tianjin Normal University , Tianjin 300387 , China Variable universe is through a group of nonlinear contraction expansion factors online timely to adjust the input domain, so that the input domain is subdivided as detailed as possible at surrounding the expected control points. It is characteristic of constant rules, quick response, and high stability precision. In this paper, a nonlinear hybrid fuzzy system was established by introducing adjusting parameters to combine Mamdani fuzzy system and TakagiSugeno (T-S) fuzzy system. The input-output expression of the nonlinear hybrid fuzzy system based on variable universe was deduced according to the hybrid inference rules and contraction expansion factors. Furthermore, the secondorder approximation of the nonlinear hybrid fuzzy system based on variable universe on the second-order continuously differentiable function was proved by multiple Taylor formula and maximum norm, and a sufficient condition for the approximation of the nonlinear hybrid fuzzy system was disclosed when adjusting parameters and approximation accuracy are known. Finally, the simulation example confirmed that the established nonlinear hybrid fuzzy system based on variable universe has better approximation performance than other systems under different adjusting parameter meanings. Contraction expansion factors; Variable universe; Fuzzy rules; Nonlinear hybrid fuzzy systems; Approximation 1 Introduction Since linguistic variable and approximate reasoning were proposed for the first time by Zadeh in 1975, fuzzy system and its approximation research have become an important field of many scholars at home and abroad. It has been applied successfully in automatic control, communication engineering, and space technology fields (Buckley 1993; Castro 1995; Hassine et al. 2003; Kosko 1994; Wang and Mendel 1992; Zeng and Singh 1995). Generally, Mamdani fuzzy system and T–S fuzzy system are two most representative ones. Liu and Li (2000, 2001) studied the approximation performance of fuzzy system of Lebesgue integrable function for the first time by introducing piecewise linear function, proving T–S fuzzy system is a universal approximator with respect to p-integrable functions. Recently, Wang and Duan (2012) combines Mamdani and T–S fuzzy system by using adjusting parameters to establish a generalized hybrid fuzzy system and reduce its rules based on different hierarchical methods of input variables. In Wang et al. (2014), K-integral norm was redefined by quasi-subtraction operator, and the generalized Mamdani fuzzy system was proved still having approximation performance on one type of integrable function under the significance of K-integral norm. These research results play an important role in further exploration of fuzzy control and system modeling (Wang et al. 2015; Yang et al. 2013; Mendel 2016; Zeng et al. 2008). However, previous researches on the approximation of fuzzy system are mainly based on subdivision of refined universe. Total rules of the fuzzy system will achieve exponential growth as the input space dimension increases, which is easy to cause rules explosion. Hence, how to control total rules growth is an important issue that has to be considered. Pedrycz (1989) first put forward the idea of the variable universe of discourse to make an universe is subdivided better near the expected control point, and studied the nonlinear context adaptation questions in the calibration of fuzzy sets (Pedrycz et al. 1997). Li (1997) gave the concept of variable universe based on a group of nonlinear contraction expansion factors, and the design issues of high precision adaptive fuzzy controller are effectively discussed (see Li 1999; Li et al. 2002). In addition, Li et al. (2002) applied the fuzzy controller based on variable universe successfully to the simulation control experiment of level-4 inverted pendulum which was the theoretical basis for spacecraft control and research of chaotic system robustness. Long et al. (2008) proposed a latent genetic algorithm of fuzzy control based on variable universe for two inputs and one output fuzzy controller. The approximation of adaptive fuzzy system based on variable universe was studied by Long et al. (2010). Long et al. (2012) constructed the input output expression of Mamdani fuzzy system based on variable universe using singleton fuzzifier, product inference engine and central average defuzzifier. These useful results demonstrate that fuzzy controller based on variable universe can increase approximation accuracy of nonlinear fuzzy system, which lays foundation for following discussion on fuzzy control and system modeling of complicated fuzzy system. Actually, the nonlinear hybrid fuzzy system based on variable (...truncated)


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Guijun Wang, Yu Li, Xiaoping Li. Approximation performance of the nonlinear hybrid fuzzy system based on variable universe, Psychonomic Bulletin & Review, 2016, pp. 73-84, Volume 2, Issue 2, DOI: 10.1007/s41066-016-0028-z