The predictions obtained from the model or its blocks can be found by averaging
over the posterior. In our case, the Wiener model parameters of the linear part and
the GP model of the nonlinear part are presented as the first and second moments,
i.e. the mean values and covariances that match the first and second moments of the
Gaussian mixtures of posterior realisations for each of the linear model parameters
and the GP model. The predictions for the models with the parameters and the
GP model obtained with the moment matching can be seen in Figs.
3.4
and
3.5
.
0
100
200
300
400
500
600
700
-1
-0.5
0
0.5
1
k
u
0
100
200
300
400
500
600
700
-1
-0.5
0
0.5
1
k
y
Fig. 3.3
Input and output signals used for the identification of the Wiener model

112
3
Incorporation of Prior Knowledge
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
-1.5
-1
-0.5
0
0.5
1
1.5
ξ
y
±
2
σ
predicted mean
static nonlinearity
Fig. 3.4
The static-nonlinear part of the Wiener model scheme obtained from Gaussian mixtures
for every input point with the matching of moments
10
-2
10
-1
10
0
10
-2
10
0
Frequency response of the linear part
Amplitude
10
-2
10
-1
10
0
-500
-400
-300
-200
-100
0
Phase [degrees]
Frequency [rad/s]
Fig. 3.5
Frequency response of the identified linear part with its mean (
full line
) and a 95%
confidence interval (
grey band
) together with the frequency response of the original linear part
(
dashed line
)
The identified nonlinear part of the Wiener model with a good matching to the
original nonlinearity is shown in Fig.
3.4
.
The Bode plot of the identified linear part, together with the original linear part,
is depicted in Fig.
3.5
. As is clear from Figs.
3.4
and
3.5
, the obtained model shows
a satisfactory performance.

3.2
Wiener and Hammerstein GP Models
113
In the case of a validation with the simulation response of the model with an
input signal that is different from the one used for the identification, we have various
options. We can use the matched mean values and covariances for the linear model
parameters and the matched posterior mean functions and covariance functions of
the GP model to be used in the dynamic system simulation, taking account of the
uncertainties as described in Sect.
2.6
or, alternatively, use the simulation with the
Monte Carlo method.
3.2.2
GP Modelling Used in the Hammerstein Model
The Hammerstein structure consists of a nonlinear static block followed by a linear
dynamic block, as illustrated in Fig.
3.6
. It is a frequently applied, nonlinear dynamic
systems modelling approach. This kind of model can be used where the actuator
dominates the system’s behaviour with its nonlinear static characteristic.
The structure of the Hammerstein model can be linearly parameterised, which
can be reflected in the choice of regressors when modelling with the linear model.
The idea behind this approach is to represent a static nonlinearity with a polynomial
approximation and, in this case, the overall input–output relationship is linear in
the parameters [
18
]. In the case of a GP model identification with a linear covari-
ance function, this approach requires the manual or automated selection of poly-

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- dr. ahmed