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Table 2.1
Average values and standard deviation of 10 runs
EKF(c)
EKF
(
c
+
a
)
Online training error
3
.
115126
±
0
.
185
3
.
084547
±
0
.
167
Final models error (training)
5
.
283666
±
0
.
312
5
.
049607
±
0
.
300
Final models error (validation)
3
.
081644
±
0
.
139
3
.
030364
±
0
.
144
RMSE (validation)
4
.
676209
±
0
.
506
4
.
630756
±
0
.
575
2.4.1 Example 1. Nonlinear Static System
Be the nonlinear system:
e
−
0
.
03
x
sin
f
(
x
)
=
(
0
.
1
x
)
(2.36)
with
x
5.
During the validation process the noise is removed to check if the model fit to the
real system without its interference.
The function (
2.36
) will be modeled using two different initial configurations. In
the first case, an initial model composed only of membership functions of Gaussian
type is used, while the second case use a mixture of different membership functions
and uses the
∈[−
150
,
150
]
, towhich is added a noise signal whose covariance is
R
e
=
0
.
matrix to include a restriction on the antecedents of the fuzzy model.
Each case is carried out 10 times, with a 60-40% random split between the training
and validation data subsets. Two fitting algorithms are to be used, the Algorithm2.2 to
set only consequents, EKF(c), and the Algorithm 2.2 followed by the Algorithm 2.3,
EKF
, to adjust both consequents and antecedents of the TS fuzzy model. The
covariance matrices for each of the algorithms will be initialized with
(
c
+
a
)
10
5
and
α
=
10
5
and
β
=
0
.
1 for EKF(ac) algorithm, and
α
=
β
=
1forEKF
(
c
+
a
)
.
2.4.1.1 Case I: Gaussian Membership Functions
In this case, the antecedents of the initial model starts with Gaussian membership
functions uniformly distributed, with all consequents set to zero, and
I
.After
run ten times, the average errors are shown in Table
2.1
. Note that the validation
results are better than training since the algorithm performs online modeling.
For one of the runs, the absolute errors of the final models are shown for training
and validation data in Figs.
2.4
and
2.5
. The evolution of online modeling outputs for
each algorithm shown in Fig.
2.6
, the final response from validation data in Fig.
2.7
,
and the online evolution of absolute errors are shown in Fig.
2.8
. Finally, Fig.
2.9
shows the resulting antecedents of each model, where the changes made by the
EKF
=
(
+
)
algorithm from the original, EKF(c), antecedents can be seen.
Based on the results obtained can be checked that the algorithms modeled prop-
erly online, and obtain final models that adequately represent the original function.
c
a
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