38 3. STATE ESTIMATION OF CYBER-PHYSICAL VEHICLE SYSTEMS
3.5 EXPERIMENT RESULTS AND DISCUSSIONS
In this section, results of the estimated ANN-based brake pressure with LMBP learning al-
gorithm are presented and discussed. e algorithms are implemented in a computer with the
MATLAB 2017a platform. e processor of the computer is an Intel Core i7-4710MQ CPU
which supports 4 cores and 8 threads parallel computing, while the RAM equipped is a 32 G
one. e time consuming for the FFNN training varies with the number of the hidden neu-
rons selected. In this study, since the range of hidden neurons number is from 10–100, thus the
training time for FFNN varies from 0.6–10 s, and the average training time cost for the FFNN
with 70 neurons is 3.4 s.
3.5.1 RESULTS OF THE ANN-BASED BRAKING PRESSURE
ESTIMATION
To quantitatively evaluate the estimation performance, two commonly used indices, namely the
coefficient of determination R
2
and the root-mean-square-error (RMSE), are adopted. e
definitions of the R
2
and RMSE are presented as follows. Suppose the reference data is T D
ft
1
: : : t
N
g, and the predicted value is Y D fy
1
: : : y
N
g. en R
2
can be calculated as:
R
2
D 1
E
res
E
tot
(3.53)
E
res
D
N
X
i
.
t
i
y
i
/
2
(3.54)
E
tot
D
N
X
i
t
i
N
T
2
; (3.55)
where E
res
is the residual sum of square, E
tot
is the total sum of square, and
N
T is the mean value
of the reference data.
e RMSE can be obtained by:
RMSE D
s
P
N
i
.
t
i
y
i
/
2
N
: (3.56)
First, the impact of the neuron number on the brake pressure estimation performance
is analyzed. Considering the complexity of the problem, the estimation performance is tested
under different number of neurons ranging from 10–100. According to Fig. 3.7, as the number
of neurons changes, the estimation accuracy of the FFNN varies slightly. e best prediction
performance is yield by FFNN with the number of neurons at 70.
en, the linear regression performance of the trained model is investigated. Based on the
linear regression result shown in Fig. 3.8, the test regression result R is of 0.96677, indicating
3.5. EXPERIMENT RESULTS AND DISCUSSIONS 39
0.98
0.96
0.94
0.92
0.03
0.025
0.02
0.015
10 20 30 40 50 60 70 80 90 100
10 20 30 40 50
Number of Neurons
60 70 80 90 100
RMSE (MPa) R
2
Figure 3.7: Estimation performance of FFNN with different number of neurons.
1.2
1
0.8
0.6
0.4
0.2
0
Target
Model Output
0 0.2 1 1.20.4 0.6 0.8
Data Point
Fitted Line
Y = T
Figure 3.8: Regression performance of the FFNN model with 70 neurons.
the FFNN model with 70 neurons can accurately estimate the braking pressure through selected
features.
Figure 3.9 shows the brake pressure estimation result in time domain. e x-axis presents
the 1,400 samples of the testing data set, and the y-axis shows the estimation results of the scaled
brake pressure. Since the input and output data for model training is scaled to the range of Œ0; 1,
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