PWARX Model Identification Based on Clustering Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 169

Equation ( 9 ) can be exploited to eliminate the outliers. As this equation attribute
to the outliers a low potential, we can under which the local

parameters are not accepted and then removed from the data set. This threshold is
described by the following equation:

fix a threshold c

c ¼ min ð P Þþb ð max ð P Þ min ð P Þ Þ: ð 11 Þ

where P is the vector containing the potentials P i such that P ¼ P 1 ; ...; P N ½ and b

is a parameter chosen as 0 1.

The elimination of outliers reduces the parameter vectors to (

\ b \

N 0 ) h i ; i ¼ 1 ; ...;

(N 0 \ N). Then, from this new data set, we select the data point with the highest

potential value as the

first cluster center.

h 1 be this

rst center and P 1 be its potential. The other potentials P j ,

Let ð j ¼

1 ; ...; N 0 Þ are then updated using this expression:

2

kk

4
r 2

b h i h 1

P 1 e P i ( P i : ð 12 Þ

Expression ( 13 ) allows to associate lower potentials to the local parameters close
to the first center. Consequently, this choice guaranties that these parameters are not

selected as cluster centers in the next step. The parameter r b is a positive constant
that must be chosen larger than r a to avoid obtaining cluster centers which are too
close to each other. The constant r b is computed using this formula:

r b ¼ N X

N

:

max
j¼1 : n q

h i h j ð 13 Þ

i¼1

In general after obtaining the kth cluster center, the potential of every local
parameter is updated by the following formula:

2

kk

b h i h k

P k e

4
r 2

P i ( P i : ð 14 Þ

h k where P k and are respectively the potential and the center of the kth local

parameter.
The number of sub-models s is a parameter that we would like to determine.
Therefore, we have developed some criteria for accepting or rejecting the cluster
centers as it is explained in the algorithm of the next section.
To search the elements belonging to each cluster, we compute the distance
between the estimated output and the real one and classify uð k Þ within the cluster

which has the minimum distance.

;

T
i u k

arg min h y k i ¼ 1 ; ...; s : ð 15 Þ