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Chapter 5
Polynomial Fuzzy Systems: Stability
and Control
José Luis Pitarch, Antonio Sala and Carlos Vicente Ariño
5.1 Introduction
For a quite general class of nonlinear systems , a systematic modeling methodology is
available to exactly transform them into the so-called Takagi-Sugeno (TS) form and it
is known as sector-nonlinearity fuzzy modeling. This chapter presents an extension
of the technique which builds a family of progressively more precise polynomial
fuzzy models, based on the Taylor series. Takagi-Sugeno models become a particular
case of the proposed technique. Following this methodology, some stability and
stabilization problems can be successfully solved for the resulting polynomial fuzzy
systems (Sala 2009 ; Tanaka et al. 2007a , b , 2009a , b ), by extending themethodologies
in Prajna et al. ( 2004b ) to the fuzzy case.
The presented methodology allows asymptotically exact results for smooth non-
linear systems: if there is a smooth Lyapunov function for it, there will exist a
polynomial Lyapunov function and a polynomial fuzzy model with a finite degree
which will allow to prove stability of the original system (some extra assumptions
apply). Asymptotic exactness applies only in compact regions of interest where
the Taylor series approximation of the nonlinearities, as well as those of a valid
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