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methods of analysis of nonlinear control theory could start to apply to fuzzy
control systems: the Popov's hyperstability criterion (Li and Yonezawa
1991
;Piegat
1997
; Wang and Langari
1994
; Zhao et al.
2001
), the circle criterion (Driankov et al.
1993
; Lin and Wang
1998
), the direct Lyapunov's method (Feng et al.
2002
; Nguyen
et al.
1995
; Siettos and Bafas
2002
; Sun et al.
2003
; Tanaka and Sugeno
1992
; Tong
and Li
2002
), analysis techniques in the phase space (Driankov et al.
1993
; Maeda
and Murakami
1991
), the descriptive function method (Aracil and Gordillo
2000
),
methods based on stability indices and robustness of systems (Aracil and Gordillo
2000
;Araciletal.
1989
), the conicity criterion (Aracil et al.
1991
; Aracil and Gordillo
2000
), methods based on the input/output stability theory (Aracil and Gordillo
2000
;
Lin and Wang
1998
; Malki et al.
1994
) and heuristic methods (Wang et al.
1996
;
Ying
1994
).
More recently, solutions based on
Linear Matrix Inequalities
(LMIs) have been
used to ensure the asymptotic stability of closed loop fuzzy control systems
(Assawinchaichote and Nguang
2004
; Guerra and Vermeiren
2004
; Hong and Nam
2003
; Lam et al.
1999
,
2002
; Tanaka et al.
2001
,
1998
,
2007
). In these methods,
the development of each of the state variables requires the same number of rules,
because the rules are usually expressed in matrix form. The antecedents are well
distributed intentionally, normally meet the requirements of the
Standard Fuzzy Par-
tition
(SFP) (Xiu and Ren
2005
). It is usually considered nil or negligible the affine
term of the consequents. With all these limitations, the design effort and complexity
of the nonlinear system are minimized, reducing it to a set of linear systems more
readily addressed. In general, these methods get good practical results, and allows a
relatively simple design of fuzzy controller, but its scope is restricted by the need for
prior knowledge of the mathematical model of the plant, since otherwise it would
be difficult to comply with all restrictions. In this sense, it is true that in most of the
literature on stability analysis of fuzzy control systems, it start from the prior knowl-
edge of the mathematical model of the plant (Al-Hadithi et al.
2007
), but in these
cases, when the mathematical model of the nonlinear plant is known, there are many
control techniques to ensure its stability with proven results backed by the scientific
community. Therefore, in principle does not seem strictly necessary to use a fuzzy
controller in these cases, this choice although can be supported under the robustness
that usually have these controllers, or the need to obtain a control law linguistically
interpretable. However, when the plant model is unknown, is too complex to obtain
it, or is subject to uncertainties, fuzzy logic is presented as a powerful tool, possibly
one of the best, to deal with the nonlinear controller design. As discussed above, the
use of fuzzy logic allows the controller design is performed formally, but this requires
representing both the model of the plant as the controller, as fuzzy state models, and
get an equivalent mathematical model of such fuzzy models. From these models, it
is possible to obtain the equivalent mathematical model of the closed loop system,
which allows a formal study of the stability and performance of the system in terms
of different quality criteria.
This chapter deals the obtaining of a equivalent mathematical model for a Takagi-
Sugeno (TS) fuzzy system completely general, both for the plant and the con-
troller, then the equivalent mathematical model of the control system closed loop
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