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)