Complexity Reduction in Fuzzy Systems Using Functional Principal Component Analysis - Fuzzy Modeling and Control: Theory and Applications - page 47

Chapter 3
Complexity Reduction in Fuzzy Systems Using
Functional Principal Component Analysis

Juan Manuel Escaño and Carlos Bordons

3.1 Introduction

The ability to build fuzzy logic applications for control problems has been hindered
by well-known problem of combinatorial rules explosion, causing complexity in
modeling. There has been a increased interest in the issues of complexity of fuzzy
systems over recent years. There are a remarkable number of methods aimed at
reducing the complexity of fuzzy systems. Most of them are based on empirical
methods (Gegov 2007 ; Chen and Teng 1996 ;Jin 2000 ; Setnes et al. 1998 ), others with
systematic nature, are practically unenforceable when the number of inputs is large,
(Gegov 2007 ;Ross 2004 ; Yen and Wang 1999 ;Yam 1997 ;Simon 2000 ; Baranyi and
Yam 1997 ; Ciftcioglu 2002 ). In this chapter, a new technique to reduce the number
of rules will be presented. It is based on Functional Principal Component Analysis
(FPCA) , one of the methods of functional analysis. Providing a systematic approach
to the rule reduction, after its application, the new Fuzzy Inference System (FIS) will
have less number of rules. We will see after the application of this technique, there
will be a detriment to the interpretability of the background in the system. The new
fuzzy system will present a non-conventional antecedent set. However, it will not be
a problem for application to control systems.

3.2 Complexity Reduction in Fuzzy Systems

Takagi-Sugeno (TS) Fuzzy systems (Takagi and Sugeno 1985 ) have proven their
effectiveness in control engineering since many years ago. In TS models, the system
may be described by j rules by the following way: