PWARX Model Identification Based on Clustering Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 186

6.2 Experimental Results

The alternative of considering a PWA map is very interesting because the char-
acteristic of the system can be considered as piecewise linear in each operating
phase: the heating phase, the reacting phase and the cooling phase.
Previous works has demonstrated that the adequate estimated orders n a and n b of
each sub-model are equal to two (Talmoudi et al. 2008 ). Thus, we can adopt the
following structure:

<

a 1 ; 1 y ð k 1 Þ a 1 ; 2 y ð k 2 Þþ b 1 ; 1 u ð k 1 Þ

þ b 1 ; 2 u ð k 2 Þ if uð k Þ2 H 1

.
a s ; 1 y

y ð k Þ ¼ ð 39 Þ

:

ð Þþ ð Þþ ð Þ

k 1 a s ; 2 y k 2 b s ; 1 u k 1

þ b s ; 2 u ð k 2 Þ if uð k Þ2 H s

where the regressor vector is de ned by:

T

uð k Þ ¼ ½ y ð k 1 Þ; y ð k 2 Þ; u ð k 1 Þ; u ð k 2 Þ

and the parameter vectors is denoted by:

;

h i ð k Þ ¼ a i ; 1 ; a i ; 2 ; b i ; 1 ; b i ; 2 i ¼ 1 ; ...; s :

We have picked out some input-output measurements from the reactor in order
to identify a model to this process. We have taken two measurement files, one for

the identi cation having a length N = 220 and another one of length N = 160 for the

validation.
The measurement file used in this identi cation is presented in Fig. 14 .

Fig. 14 The real input-output
evolution

2000

1800

1600

1400

1200

1000

0 50 100 150 200 250

k

150

100

50

0

0 50 100 150 200 250