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STABILITY OF TAKAGI-SUGENO FUZZY SYSTEMS
İlker Üstoğlu
Istanbul Technical University, Faculty of Electrical and Electronic Engineering,
Control Engineering Department, Maslak, TR-34469, Istanbul, Turkey
Keywords: Takagi-Sugeno fuzzy systems, Uniform and Exponential stability, Time varying systems.
Abstract: Takagi-Sugeno (T-S) fuzzy models are usually used to describe nonlinear systems by a set of IF-THEN
rules that gives local linear representations of subsystems. The overall model of the system is then formed
as a fuzzy blending of these subsystems. It is important to study their stability or the synthesis of stabilizing
controllers. The stability of TS models has been derived by means of several methods: Lyapunov approach,
switching systems theory, linear system with modeling uncertainties, etc. In this study, the uniform stability,
and uniform exponential stability of a discrete time T-S model is examined. Moreover, a perturbation result
and an instability condition are given. The subsystems of T-S models that is studied here are time varying
and a new exponential stability theorem is given for these types of TS models by examining the existence of
a common matrix sequence.
1 INTRODUCTION
Fuzzy systems can approximate a wide class of
nonlinear systems as accurately as required with
some number of fuzzy IF-THEN rules. They are
known as universal approximators, and their use
offers many advantages (L.X.Wang, 1996). Stability
is the most important concept for analysis and
design of a control system. Stability analysis of
fuzzy systems has been difficult because fuzzy
systems are essentially nonlinear systems (Tanaka
1996, Calcev, 1998, Kim, 2001). The issue of the
stability of fuzzy control systems has been studied
using nonlinear stability frameworks (Tanaka,
1990).
Takagi-Sugeno (T-S) fuzzy models (Takagi,
1985) are nonlinear systems in nature. In this type of
fuzzy model the consequent part of a fuzzy rule is a
mathematical formula, representing local dynamics
in different state space regions (subsystems) as
linear input-output relations (Tanaka, 1996). Thus,
T-S fuzzy systems are considered as a weighted
average of the values in the consequent parts of the
fuzzy rules. The overall model of the system is
consequently a fuzzy blending of these subsystems.
Recently, fuzzy control and modeling is being
used in many practical industrial applications. One
of the first questions to be answered is the stability
of the fuzzy system. Tanaka and Sugeno (Tanaka,
1992), have provided a sufficient condition for the
asymptotic stability of a fuzzy system in the sense of
Lyapunov through the existence of a common
Lyapunov function for all the subsystems.
A system is said to be stable in the sense of
Lyapunov if its trajectories can be made arbitrarily
close to the origin for any initial starting state. When
a system is stable and initial states that are close to
the region of origin converge to the origin, the
system has asymptotic stability. A stable system in
Lyapunov sense does not guarantee asymptotic
stability because asymptotic stability is stricter than
Lyapunov stability.
Additionally, one needs to know how fast the
system converges to the equilibrium point. On the
other hand, exponential stability is used to estimate
how fast the system trajectory approaches and
converges to the equilibrium point as time goes to
infinity. Since exponential stability is stricter than
asymptotic stability it guarantees both Lyapunov
stability and asymptotic stability but not vice versa.
The preliminaries were presented in Section 2.
Section 3 discusses the main results on the uniform
stability, uniform exponential stability and
instability. Moreover, a perturbation result is
presented. Finally, Section 4 contains some
concluding remarks.
213
Üsto
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glu Ä
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r. (2006).
STABILITY OF TAKAGI-SUGENO FUZZY SYSTEMS.
In Proceedings of the Third International Conference on Informatics in Control, Automation and Robotics, pages 213-216
DOI: 10.5220/0001214802130216
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