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A class of hierarchical fuzzy systems with constraints on the fuzzy rules
194 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 13, NO. 2, APRIL 2005
A Class of Hierarchical Fuzzy Systems With
Constraints on the Fuzzy Rules
Moon G. Joo and Jin S. Lee
Abstract—This paper presents a class of hierarchical fuzzy sys-
tems where previous layer outputs are used not in IF-parts, but only
in THEN-parts of the fuzzy rules of the current layer. The proposed
scheme is shown to be a universal approximator to any real contin-
uous function on a compact set if complete fuzzy sets are used in
the IF-parts of the fuzzy rules with singleton fuzzifier and center
average defuzzifier. From the example of ball-and-beam control
system simulation, it is demonstrated that the proposed scheme ap-
proximates with high accuracy a model nonlinear controller with
fewer fuzzy rules than a centralized fuzzy system, and its control
performance is comparable to that of a nonlinear controller.
Index Terms—Fuzzy control, hierarchical fuzzy logic system,
Stone–Weierstrass theorem, universal approximation.
I. INTRODUCTION
ONE OF THE important issues in fuzzy logic systems ishow to reduce the number of involved fuzzy rules and
their corresponding computation requirements. In fact, the
number of fuzzy rules grows exponentially with the number
of input variables. Specifically, a single-output fuzzy logic
system with input variables and fuzzy sets defined for
each input variable requires number of fuzzy rules. To
overcome the problem, the idea of using hierarchical structure
in designing a fuzzy system has been reported by Raju and
Zhou [1], [2], where input variables are put into a collection of
low-dimensional fuzzy logic units (FLUs) and the outputs of
the FLUs are used as input variables for the FLU in the next
layer. According to them, the number of fuzzy rules employed
in a hierarchical fuzzy system (HFS) is proportional to the
number of input variables.
In HFS, however, the retrieval of physical meanings from the
outputs of the FLUs in the previous layer with or without sup-
pleme
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