New Concepts for the Estimation of Takagi-Sugeno Model Based on Extended Kalman Filter - Fuzzy Modeling and Control: Theory and Applications - page 3

Lian et al. 2006 ; Tanaka andWang 2001 ). This result provides a theoretical foundation
for applying TS fuzzymodels to represent complex nonlinear systems approximately.
Several results have been obtained about the identification of TS fuzzy models
(Cao et al. 1997 ; Johansen et al. 2000 ; Mollov et al. 2004 ). They are based upon
two kinds of approaches, one is to linearize the original nonlinear system in various
operating points when the model of the system is known, and the other is based on
the input-output data collected from the original nonlinear system when its model
is unknown. The authors in Cao et al. ( 1997 ) use a fuzzy clustering method to
identify TS fuzzy models, including identification of the number of fuzzy rules and
parameters of fuzzy membership functions, and identification of parameters of local
linear models by using a least squares method (Skrjanc et al. 2005 ).
In Hong and Lee ( 1996 ), have analyzed that the disadvantages of most fuzzy
systems are that the membership functions and fuzzy rules should be predefined to
map numerical data into linguistic terms and to make fuzzy reasoning work. They
suggested a method based on the fuzzy clustering technique and the decision tables to
derive membership functions and fuzzy rules from numerical data. However, Hong
and Lee's algorithm presented needs to predefine the membership functions of the
input linguistic variables and it simplifies fuzzy rules by a series of merge operations.
As the number of variables becomes larger, the decision table will grow tremendously
and the process of the rule simplification based on the decision tables becomes more
complicated. The authors in Johansen et al. ( 2000 ) suggest a method to identify TS
fuzzy models. Their method aims at improving the local and global approximation
of TS model. However, this complicates the approximation in order to obtain both
targets. It has been shown that constrained and regularized identification methods
may improve interpretability of constituent local models as local linearizations, and
locally weighted least squares method may explicitly address the trade-off between
the local and global accuracy of TS fuzzy models.
In Matía et al. ( 2011 ), the authors proposed to obtain the best features of
Mamdani and TS models by using an affine global model with function approx-
imation capabilities which maintains local interpretation. The suggested model is
composed of variant coefficients which are independently governed by a zeroth
order fuzzy inference system. This model may be interpreted as a generalization of
TS model in which dynamics coefficients have been decoupled. They have shown
that Mamdani and TSmodels can be combined so that local and global interpretations
are preserved.
In Al-Hadithi et al. ( 2011 ), Jiménez et al. ( 2012 ), Al-Hadithi et al. ( 2012 )new
and efficient approaches are presented to improve the local and global estimation
of TS model. The aim is obtaining high function approximation accuracy and fast
convergence. The main problem is that TS identification method can not be applied
when the Membership Functions (MFs) are overlapped by pairs. The first approach
uses the minimum norm method to search for an exact optimum solution at the
expense of increasing complexity and computational cost. The second one is a simple
and less computational method, based on weighting of parameters. This restricts the
use of the TS method because this type of membership function has been widely
used during the last 2decades.