Suboptimal Recursive Methodology for Takagi-Sugeno Fuzzy Models Identification - Fuzzy Modeling and Control: Theory and Applications - page 25

can be critical. On the other hand, is relatively easy to convert them into nonlin-
ear state models (Andújar and Barragán 2005 ; Andújar et al. 2009 ), which support
formal analysis to use in control engineering. Up to date, many fuzzy modeling algo-
rithms based on input-output data have been proposed (Babuška 1995 ; Denaï et al.
2007 ; Horikawa et al. 1992 ; Jang 1993 ). Moreover, in many cases it is required that
modeling algorithm works online with the system, and do it properly in the presence
of noise.
In order to design a TS fuzzy modeling algorithm based on input-output data,
which can work online with the system, properly in presence of noise and can be very
efficient computationally, the extendedKalman filter (EKF) (Kalman 1960 ; Maybeck
1979 ) is used. The Kalman filter is the minimum-variance state estimator for linear
dynamic systems withwhite noise with zero-mean value. This is an efficient recursive
filter that estimates the internal states of a linear dynamic system froma series of noisy
measurement. It is used in a wide range of engineering and econometric applications,
from radar and computer vision to estimation of structural macroeconomic models,
and it is an important topic in control theory and control systems engineering. The
EKF (Maybeck 1979 ) is a modification of the Kalman filter that can be used to
estimate the state in nonlinear systems. EKF linearizes the system around the current
parameters. These algorithms update the parameters been consistent with previous
data, and usually converges in a few iterations.
The Kalman filter has been used with fuzzy logic in various applications, such
as the extraction of rules from a given rule base (Liang and John 1999 ), parameters
optimization of defuzzification mechanisms that are based on both Gaussian and
polynomial distributions (Jiang and Li 1996 ) or in optimization of consequents of
Takagi-Sugeno models (Ramaswamy et al. 1993 ). In 2002, Simon introduced the
use of Kalman filter for adjusting the parameters of a TS fuzzy model (Simon 2002 ,
2003 ), assuming that antecedents were membership functions of triangular type, and
using its center of gravity to perform the adaptation process. Later, other proposals
have been made (Al-Hadithi et al. 2012 ; Chafaa et al. 2007 ).
Motivated by the successful use of Kalman filter in the works presented above,
and considering that this algorithm can be applied in real time (Jiménez et al. 2008 ), a
general methodology for use EKF to estimate the adaptive parameters of a general TS
fuzzy model, was presented in (Barragán et al. 2011a , b , 2013 ). This methodology
uses the excellent features of Kalman filter to obtain fuzzy models of unknown
systems from input-output data. This chapter focuses on this latter approach, because
it adjusts both the antecedents such as the consequents, it does not limit the size of
input/output vectors, neither the type or distribution of themembership functions used
in the definition of fuzzy sets of the model. For this, firstly presents the formulation
of the problem and the EKF. Subsequently, the problem of using the EKF to fuzzy
modeling is treated, and several algorithms for EKF fuzzy modeling are presented.
Finally, to illustrate the procedure, several examples are performed and compared
the results with several methodologies accepted by the scientific community.