PWARX Model Identification Based on Clustering Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 164

point. For each operation point, we associate a simple sub-model (linear or af ne)

with it. Indeed, a complex system can be modeled as a hybrid system switching
between simple sub-models.
This chapter addresses the problem of identi

cation of hybrid systems repre-
sented by piecewise autoregressive models with exogenous input (PWARX). This
problem consists in building mathematical models of hybrid systems from observed
input-output data. The PWARX models have attracted a considerable attention in
recent years, since they provide an ef

cient solution for modeling a wide range of
engineering applications (Roll et al. 2004 ; Nakada et al. 2005 ; Wen et al. 2007 ;Xu
et al. 2012 ). In addition, these models are able to approximate any nonlinear system
with arbitrary accuracy (Lin and Unbehauen 1992 ). Moreover, the PWA model can
be considered as a generic representation for other hybrid models such as jump
linear models (JL models) (Vidal et al. 2002 ), Markov jump linear models (MJL
models) (Doucet et al. 2001 ), mixed logic dynamical models (MLD models)
(Bemporad et al. 2000 ), max-min-plus-scaling systems (MMPS models)
(De Schutter and Van den Boom 2000 ), linear complementarity models (LC
models) (Vander-Schaft and Schumacher 1998 ), extended linear complementarity
models (ELC models) (De Schutter and De Moor 1999 ). In fact, the transfer of the
results of PWARX models to other classes of hybrid systems is insured thanks to
the properties of equivalence of PWARX models (Heemels et al. 2001 ). The
PWARX models are obtained by decomposing the regression domain into a finite
number of non-overlapping convex polyhedral regions and by associating a simple
linear model with each region. Consequently, two main problems must be con-
sidered for the identi cation of PWARX models: one is the estimation of the

parameters of the sub-models and two is the determination of the hyperplanes
de

ning the partitions of the state-input regression. Consequently, the identi cation

of PWARX models is one of the most dif cult problems that represent an area of

research where considerable work has been done in the last decade. In fact,
numerous solutions have been proposed in the literature for the identi cation of the

PWARX models such as the clustering-based solution (Ferrari-Trecate et al. 2003 ),
the Bayesian solution (Juloski et al. 2005 ), the bounded-error solution (Bemporad
et al. 2005 ), the greeting solution (Bemporad et al. 2003 ), the sparse optimization
solution (Bako 2011 ; Bako and Lecoeuche 2013 ), and so on. The sparse solutions
do not smooth out the effect of the measurement noise. Then, they often fail in real
time applications since the measurement data are usually contaminated by an
unknown additional noise. The greedy algorithms are very time consuming since
they involve the solution of NP-hard problems. In addition, it can cause a loss of
information because it sometimes fails to associate data to the appropriate regres-
sors. The Bayesian approach assumes that the probability density functions of the
unknown parameters of the system are known a priori. Otherwise, it requires an
additional sequential processing to improve the identi

cation results. The clustering
solution is based on a simple and instructive procedure. It does not require a priori
knowledge of the system. Therefore, only the clustering approach is considered in
this chapter. This solution consists of three main steps, which are data classi cation,

parameter estimation and region reconstruction. It is easy to remark that the