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8
<
a 1 ; 1 y
ð
k
1
Þþ
b 1 ; 1 u
ð
k
1
Þþ
b 1 ; 2 u
ð
k
2
Þ
if
k
Þ2
H 1
.
a s ; 1 y ð k
y
ð
k
Þ ¼
ð
37
Þ
:
1
Þþ b s ; 1 u ð k
1
Þþ b s ; 2 u ð k
2
Þ
if k Þ2 H s
where the regressor vector is de
ned by:
T
k
Þ ¼
½
y
ð
k
1
Þ;
u
ð
k
1
Þ;
u
ð
k
2
Þ
and the parameter vectors are denoted by:
;
h i ð
Þ ¼
a i ; 1 ;
b i ; 1 ;
¼
; ...;
:
k
b i ; 2
i
1
s
5.4.3 Input Design
The input design is an important aspect to be considered when implementing
nonlinear system identi
cation experiments. In fact, two main properties must be
verified by this input in order to generate representative data measurements to be
used in identi
cation purpose. First, the input must be able to excitep the totality of
dynamics range present in the system. Second, the used input signal must illustrate
the response of the system to a range of amplitude changes since these models have
nonlinear gains. For these reasons, we have considered the Multi-Sine sequence as
input sequences to identify the ph neutralization process since it satis
es the above
two conditions. It presents several frequencies and exhibits different amplitude
changes. The dynamic of this input is de
ned according to the dominant time
constant range of the process. The amplitudes are selected to cover the totality
operating region around the nominal value of the base
fl
ow rate q 3 ¼
15
:
6ml/s.
5.4.4 Results
The nonlinear model of the pH process de
ned by Eqs. ( 32 ), ( 34 ), ( 35 ) and ( 36 ) and
the parameters of Table 6 is used to generate the output using a Multi-Sine exci-
tation Sequence. The system output is corrupted by a Gaussian white noise with
zero mean and standard deviation
001 in order to simulate industrial situa-
tions where the obtained measurements are often noisy. The obtained input-output
data illustrated in Fig. 8 are then divided into two parts. The
r ¼
0
:
first part is used for the
identi
cation and the second is considered for the validation purpose.
The number of neighboring is chosen n q ¼
85 for the two methods. The
DBSCAN approach uses the following synthesis parameters
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