PWARX Model Identification Based on Clustering Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 173

The relative error expressed in percentage (%) is given by:

￿

j yk ðÞ^ yk ðÞ j

ðÞ ¼ ð Þ

e r k 100 21

j y ðÞ j

where y ^ ð k Þ and y(k) are the estimated and the real outputs at time k.

The identi ed model is considered acceptable if e r is close to 0 %.

5.2 Identi cation Results of a PWARX Model

Consider the following PWARX model (Boukharouba 2011 ):

8
<
:

½ 0 : 40 : 50 : 3 uð k Þþ e ð k Þ if uð k Þ2 H 1 ;

y ð k Þ ¼ ½ 0 : 70 : 6 0 : 5 uð k Þþ e ð k Þ if uð k Þ2 H 2 ; ð 22 Þ

½ 0 : 4 0 : 2 0 : 2 uð k Þþ e ð k Þ if uð k Þ2 H 3 ;

2

H 1 ¼ u 2< : ½ 10 : 30 u 0 and 0 ½ 0 : 50 u [ 0

2

H 2 ¼ u 2< : ½ 10 : 30 u 0 and 1 ½ 0 : 30 u \ 0 ð 23 Þ

2

H 3 ¼ u 2< : ½ 1 0 : 30 u 0 and 0 ½ 0 : 50 u 0

T is the regressor vector.
System ( 22 ) is simulated using an input signal u(k) and a noise signal e(k) which
are normal distributions with variances respectively 0.5 and 0.05. The output y(k)is
presented in Fig. 1 .
Table 1 presents the estimated parameter vectors obtained with the proposed
methods and the k-means one.

where s =3,n a =1,n b = 1, and uð k Þ ¼ ½ y ð k 1 Þ u ð k 1 Þ

Fig. 1 The real output of the
system (squares: output of
sub-model 1, triangles: output
of sub-model 2 and dots:
output of sub-model 3)

1.5

1

0.5

0

−0.5

−1

2

2

1

1

0

0