New Concepts for the Estimation of Takagi-Sugeno Model Based on Extended Kalman Filter - Fuzzy Modeling and Control: Theory and Applications - page 4

In Skrjanc et al. ( 2005 ) a method of interval fuzzy model identification was devel-
oped. Themethod combines a fuzzy identificationmethodologywith some ideas from
linear programming theory. The idea is then extended to modeling the optimal lower
and upper bound functions that define the band which contains all the measurement
values. This results in lower and upper fuzzy models or a fuzzy model with a set of
lower and upper parameters. This approach can also be used to compress information
in the case of large amount of data and in the case of robust system identification.
The method can be efficiently used in the case of the approximation of the nonlinear
functions family.
In Kumar et al. ( 2006 ), a study has outlined a new min-max approach to the
fuzzy clustering, estimation, and identification with uncertain data. The proposed
approach minimizes the worst-case effect of data uncertainties and modeling errors
on estimation performance without making any statistical assumption and requiring a
priori knowledge of uncertainties. Simulation studies have been provided to show the
better performance of the proposed method in comparison to the standard techniques.
The developed fuzzy estimation theory was applied to a real world application of
physical fitness classification and modeling.
A fuzzy system containing a dynamic rule base is proposed in Chen and Saif
( 2005 ). The characteristic of the proposed system is in the dynamic nature of its rule
base which has a fixed number of rules and allows the fuzzy sets to dynamically
change or move with the inputs. The number of the rules in the proposed system can
be small, and chosen by the designer. The proposed system is capable of approxi-
mating any continuous function on an arbitrarily large compact domain. Moreover, it
can even approximate any uniformly continuous function on infinite domains. This
paper addresses existence conditions, and as well provides constructive sufficient
conditions so that the new fuzzy system can approximate any continuous function
with bounded partial derivatives.
Several methods are used to deal with the problem of optimizing membership
functions, which are either derivative-based or derivative-free methods. The deriva-
tive free approaches are desirable because they are more robust than derivative-based
methods with respect to finding global minimum and with respect to a wide range
of objective function and MFs types. The main drawback is that they converge more
slowly than derivative-based techniques (Tao and Taur 1999 ). On the other hand,
derivative-based methods tend to converge to local minimums. In addition, they are
limited to specific objective functions and types of inference and MFs. The most
common approaches are: gradient descent (Simon 2000a ), least squares (Skrjanc
et al. 2005 ), back propagation (Wangand and Mendel 1992 ) and Kalman filtering
(Simon 2002b , c ).
The relation of Kalman filter techniques with fuzzy models have been widely
shown in different applications. In Matía et al. ( 2006 ) a fuzzy Kalman filter was
introduced to use a possibilistic instead of probabilistic representation of uncertainty.
But the use of the probabilistic Kalman filter training to optimize the MFs of a fuzzy
system was introduced by Simon ( 2002c ) for motor winding current estimation.
The used MFs were assumed as symmetric triangular forms. The Kalman filter train-
ing was extended to asymmetric triangles in Simon ( 2002b ), and a matrix was defined