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Abstract— As an important safety critical cyber-physical
system (CPS), the braking system is essential to the safe operation
of the electric vehicle. Accurate estimation of the brake pressure is
of great importance for automotive CPS design and control. In
this paper, a novel probabilistic estimation method of brake
pressure is developed for electrified vehicles based on multilayer
Artificial Neural Networks (ANN) with Levenberg-Marquardt
Backpropagation (LMBP) training algorithm. Firstly, the
high-level architecture of the proposed multilayer ANN for brake
pressure estimation is illustrated. Then, the standard
backpropagation (BP) algorithm used for training of the
feed-forward neural network (FFNN) is introduced. Based on the
basic concept of backpropagation, a more efficient training
algorithm of LMBP method is proposed. Next, real vehicle testing
is carried out on a chassis dynamometer under standard driving
cycles. Experimental data of the vehicle and the powertrain
systems are collected, and feature vectors for FFNN training
collection are selected. Finally, the developed multilayer ANN is
trained using the measured vehicle data, and the performance of
the brake pressure estimation is evaluated and compared with
other available learning methods. Experimental results validate
the feasibility and accuracy of the proposed ANN-based method
for braking pressure estimation under real deceleration scenarios.
Index Terms— Cyber-Physical System, Safety Critical System,
Artificial Neural Networks, LMBP, Brake Pressure Estimation,
Electric Vehicle.
I. INTRODUCTION
YBER physical systems, which are distributed, networked
systems that fuse computational processes with the
physical world exhibiting a multidisciplinary nature, have
recently become a research focus [1-4]. As a typical application
of CPS in green transportation, electric vehicles have been
widely studied with different topics by researchers and
engineers from academia, industry and governmental
organizations [5-11]. In an electric vehicle (EV), the cyber
world of control and communication, the physical plant of
electric powertrain, the human driver, and the driving
environment, are tightly coupled and dynamically interacted,
determining the overall system’s performance jointly [12].
These complex subsystems with multi-disciplinary
interactions, strong uncertainties, and hard nonlinearities make
the estimation, control and optimization of electric vehicles
very difficult [13]. Thus, there are still a number of
fundamental issues and critical challenges varying in their
importance from convenience to safety of EV remained open
[14-17].
Among all those concerns in EV CPS, a key one is safety.
Safety critical systems are those ones whose failure or
malfunction may result in serious injury or severe damage to
people, equipment, or environment [18]. As one of the most
important safety critical systems in EV, the correct functioning
of braking system is essential to the safe operation of the
vehicle [19]. There are a variety of safety standards, control
algorithms, and developed devices helping guarantee braking
safety for current EVs. However, with increasing degrees of
electrification, control authority and autonomy of automotive
CPS, safety critical functions of braking system are also
required to evolve to keep pace [20].
In the braking system of a passenger car, the braking torque
is generated by the hydraulic pressure applied in the brake
cylinder. Thus, the accurate measurement of the brake pressure
through a pressure sensor is of great importance for various
braking control functions and chassis stability logics. However,
failures of the brake pressure measurement, which may be
caused by software discrepancies or hardware problems, could
result in vehicle’s critical safety issues. Thus, high-precision
estimation of brake pressure become a hot research area in
automotive CPS design and control. Moreover, in order to
handle the trade-offs between performance and cost,
sensor-less observation is required. This makes the study of
brake pressure estimation highly motivated.
Based on advanced theories and algorithms from the aspect
of control engineering, observation methods of braking
pressure for vehicles have been investigated by researchers
worldwide. In [21], a recursive least square algorithm for
estimation of brake cylinder pressure was proposed based on
the pressure response characteristics of anti-lock braking
Levenberg-Marquardt Backpropagation Training of
Multilayer Neural Networks for State Estimation of
A Safety Critical Cyber-Physical System
Chen Lv, Member, IEEE, Yang Xing, Junzhi Zhang, Xiaoxiang Na, Yutong Li, Teng Liu,
Dongpu Cao, Member, IEEE and Fei-Yue Wang, Fellow, IEEE
C
C. Lv, Y. Xing, and D. Cao are with Advanced Vehicle Engineering Centre,
Cranfield University, Bedford, MK43 0AL, UK (email:{c.lyu,y.xing,
d.cao}@cranfield.ac.uk).
J. Zhang and Y. Li are with the Department of Automotive Engineering,
Tsinghua University, Beijing 100084, P.R. Chin
a
(email:jzhzhang@tsinghua.edu.cn, wilson420813@gmail.com).
X. Na is with the Department of Engineering, University of Cambridge, CB2
1PZ, United Kingdom (e-mail: xnhn2@eng.cam.ac.uk).
T. Liu and F.-Y. Wang are with the State Key Laboratory of Managemen
t
and Control for Complex Systems, Institute of Automation, Chinese Academy
of Sciences, Beijing 100190, P.R. China (email: tengliu17@gmail.com,
feiyue@ieee.org).
(C. Lv and Y. Xing contributed equally to the work. Corresponding authors
are J. Zhang and D. Cao)
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Fig. 1 High-level architecture of the proposed brake pressure estimation algorithm based on multilayer Artificial Neural Networks.
system (ABS). In [22], an extended-kalman-filter-based
estimation algorithm was developed considering hydraulic
model and tyre dynamics. In [23], an algorithm for online
observation of brake pressure was designed through a
developed inverse model, and the algorithm was verified in the
vehicle’s electronic stability program. In [24], the models of
brake pressure increase, decrease and hold are proposed,
respectively, by using the experimental data. And the models
can be used for fast online observation of hydraulic brake
pressure. In [25], a brake pressure estimation algorithm was
proposed for ABS considering the hydraulic fluid
characteristics. In [26], the estimation algorithm was performed
by calculating the volume of fluid flowing through the valve.
The amount of fluid is a function of the pressure differential
across the valve and the actuation time of the valve.
Nevertheless, the existing research on brake pressure
estimation was mainly investigated from the perspective of
control engineering, and an approach with the probabilistic
method, such as machine learning, has rarely been seen.
In this paper, an Artificial-Neural-Network-based estimation
method is studied for accurately observing the brake pressure of
an electric passenger car. The main contribution of this work
lies in the following aspects: 1) an ANN-based machine
learning framework is proposed to quantitatively estimate the
brake pressure of an EV; 2) The proposed approach is
implemented with experimental data obtained via vehicle
testing, and compared with other methods; 3) The proposed
approaches has a great potential to achieve a sensorless design
of the braking control system, removing the brake pressure
sensor existing in the current products and largely reducing the
cost of the system. Moreover, it also provides an additional
redundancy for the safety-critical braking functions.
The rest of this paper is organized as follows. Section II
describes the high-level architecture of the proposed multilayer
ANN for brake pressure estimation. Section III briefly
introduces the standard backpropagation algorithm and
illustrate the notations and basic concepts demanded in the
Levenberg-Marquardt algorithm. Section IV presents details of
the application of the LMBP method to training the
feed-forward neural networks. In Section V, experiment
implementations including feature selection, data collection
and preprocessing are presented. Section VI reports the
experimental results of the proposed brake pressure estimation
algorithm including performance comparison to other
approaches. Finally, conclusions are made in Section VII.
II. M
ULTILAYER
A
RTIFICIAL
N
EURAL
N
ETWORKS
A
RCHITECTURE
In order to achieve the objective of brake pressure
estimation, multilayer artificial neural networks are firstly
constructed with the input of vehicle and powertrain states.
Details of the high-level system architecture and structure of
the component are described in this section.
A. System Architecture
The system architecture with proposed methodology is
shown in Fig. 1. The multilayer artificial neural network
receives state variables of the vehicle and the electric
powertrain system as inputs, and then yields the estimation of
the brake pressure through the activation function. The
Levenberg-Marquardt Backpropagation algorithm is then
operated with the performance function, which is a function of
the ANN-based estimation and the ground truth of brake
pressure. The weight and bias variables are adjusted according
to Levenberg-Marquardt method, and the backpropagation
algorithm is used to calculate the Jacobian matrix of the
performance function with respect to the weight and bias
variables. With updated weights and biases, the ANN further
estimates the brake pressure at the next time step. On the basis
of the above iterative processes, the ANN-based brake pressure
estimation model is well trained. The detailed method and
algorithms are introduced in the following subsection.
B. Multilayer Feed-Forward Neural Network
Fig. 2 Structure of the multilayer feed-forward neural network.
In this work, a multilayer feed-forward neural network is
chosen to estimate brake pressure. A FFNN is composed of one
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input layer, one or more hidden layers and one output layer.
Since a neural network with one hidden layer has the capability
to handle most of the complex functions, in this work the FFNN
with one hidden layer is constructed. Fig. 2 shows the structure
of a multilayer FFNN with one hidden layer.
The basic element of a FFNN is the neuron, which is a
logical-mathematical model that seeks to simulate the behavior
and functions of a biological neuron [27]. Fig. 3 shows the
schematic structure of a neuron. Typically, a neuron has more
than one input. The elements in the input vector
12
[, , , ]
R
pp pp are weighted by elements
12
,,,
j
ww w
of
the weight matrix W respectively.
Fig. 3 Structure of the multi-input neuron.
The neuron has a bias b, which is summed with the weighted
inputs to form the net input n, which can be expressed by
1
R
jj
j
nwpb b
Wp
(1)
Then the net input n passes through an active function f,
which generates the neuron output a.
()afn
(2)
In this study, the log-sigmoid activation function is adopted.
It can be given by the following expression:
1
() 1
x
fx e
(3)
Thus, the multi-input FFNN in Fig. 2 implements the
following equation
22 21 1 12
1, ,
11
(( ))
SR
iijji
ij
af wf wpbb
(4)
where a
2
denotes the output of the overall networks. R is the
number of inputs, S is the number of neurons in the hidden
layer, and
j
p
indicates the jth input.
1
f
and
2
f
are the
activation functions of the hidden layer and output layer,
respectively.
1
i
b
represents the bias of the ith neuron in the
hidden layer, and 2
b
is the bias of the neuron in the output
layer.
1
,ij
w
represents the weight connecting the jth input and
the ith neuron of the hidden layer, and
2
1,i
w
represents the
weight connecting the ith source of the hidden layer to the
output layer neuron.
III. S
TANDARD
B
ACKPROPAGATION
A
LGORITHM
In order to train the established FFNN, the backpropagation
algorithm can be utilized [28]. Considering a multilayer
feedforward neural network, such as the one with three-layer
shown in Fig. 2, its operation can be described using the
following equation:
111 1
()
mmmmm
=
afWab
(5)
where a
m
and a
m+1
are the outputs of the m-th and (m+1)-th
layers of the networks, respectively. b
m+1
is the bias vector of
(m+1)-th layers of the networks.
0,1,..., 1m= M
, where M is
the number of layers of the neural network. The neurons of the
first layer obtain inputs:
0
=ap
(6)
Eq. (6) provides the initial condition for Eq. (5). The outputs
of the neurons in the last layer can be seen as the overall
networks’ outputs:
M
=aa
(7)
The task is to train the network with associations between a
specified set of input-output pairs
11 2 2
{( , ),( , ),...,( , )}
QQ
pt p t p t
,
where p
q
is an input to the network, and t
q
is the corresponding
target output. As each input is applied to the network, the
network output is compared to the target.
The backpropagation algorithm uses mean square error as
the performance index, which is to be minimized by adjusting
the network parameters, as shown in Eq. (8).
( ) [ ] [( ) ( )]
TT
F=E E xee tata
(8)
where x is the vector matrix of network weights and biases.
Using the approximate steepest descent rule, the performance
index F(x) can be approximated by
ˆ( ) (( ) ( )) (( ) ( )) ( )( )
TT
F=k k k k kkxt a t a ee
(9)
where the expectation of the squared error in Eq. (8) has been
replaced by the squared error at iteration step k.
The steepest descent algorithm for the approximate mean
square error is
,,
,
ˆ
(1) ()
mm
ij ij m
ij
F
wk =wk w
(10)
ˆ
(1) ()
mm
ii m
i
F
bk =bk b
(11)
where
is the learning rate.
Based on the chain rule, the derivatives in Eq. (10) and Eq.
(11) can be calculated as:
,,
ˆˆ
m
i
mmm
ij i ij
n
FF
wnw
, ˆˆ
m
i
mmm
iii
n
FF
bnb
(12)
We now define m
i
s
as the sensitivity of
ˆ
F
to changes in the
ith element of the net input at layer m.
ˆ
m
im
i
F
sn
(13)
Using the defined sensitivity, then the derivatives in Eq. (12)
can be simplified as
1
,
ˆ
mm
ij
m
ij
Fsa
w
(14)
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ˆm
i
m
i
F
s
b
(15)
Then the approximate steepest descent algorithm can be
rewritten in matrix form as:
1
(1) () ( )
mmmmT
k= k
WWsa (16)
(1) ()
mmm
k=k
bbs
(17)
where
12
ˆˆˆ ˆ
[ , ,..., ]
m
mT
mmm m
S
FFF F
nn n
sn (18)
To derive the recurrence relationship for the sensitivities, the
following Jacobian matrix is utilized.
11 1
11 1
11 1
12
11 1
22 2
1
12
11 1
12
m
m
mm m
m
mm m
mm m
S
mm m
m
mm m
S
m
mm m
SS S
mm m
S
nn n
nn n
nn n
nn n
nn n
nn n
n
n
(19)
Consider the i, j element in the matrix:
1
11
1
,,
1
()
m
m
j
mmmm
ij ij j
mm
j
a
nwwfn
nn
(20)
Thus, the Jacobian matrix can be rewritten as
1
1()
m
mmm
m
nWFn
n
(21)
where
1
2
() 0 0
0() 0
()
00 ()
m
mm
mm
mm
mm
S
fn
fn
fn
Fn
(22)
Then the recurrence relation for the sensitivity can be
obtained by using the chain rule:
1
1
11
ˆˆ
()( )
T
m
m
mmm
mm mTm
FF
n
snnn
Fn W s
(23)
This recurrence relation is initialized at the final layer as
2
1
()
ˆ(( ) ( ))
2( ) 2( ) ( )
M
S
jj
T
j
M
iMM M
ii i
mm
i
ii ii i
M
i
ta
F
snn n
a
ta tafn
n
ta ta
(24)
Thus the recurrence relation of the sensitivity matrix can be
expressed as
2()( )
MMM
sFnta
(25)
The overall BP learning algorithm is now finalized and can
be summarized as the following steps: 1) firstly, propagate the
input forward through the network; 2) secondly, propagate the
sensitivities backward through the network from the last layer
to the first layer; 3) finally, update the weights and biases using
the approximate steepest descent rule.
IV. LEVENBERG-MARQUARDT BACKPROPAGATION
While backpropagation is a steepest descent algorithm, the
Levenberg-Marquardt algorithm is derived from Newton’s
method that was designed for minimizing functions that are
sums of squares of nonlinear functions [29, 30].
Newton’s method for optimizing a performance index ()Fx
is
1
1kkkk
xxA
g
(26)
2()| k
kF
XX
Ax
(27)
()| k
kF
XX
gx
(28)
where 2()Fx is the Hessian matrix and ( )F
x is the
gradient.
Assume that ()Fx is a sum of squares function:
2
1
() () ()()
N
T
i
i
Fv
xxvxvx
(29)
then the gradient and Hessian matrix are
() 2 ()()
T
FxJxvx (30)
2() 2 ()() 2()
T
F xJxJxSx
(31)
where ()Jx is the Jacobian matrix
11 1
12
22 2
12
12
() () ()
() () ()
()
() () ()
n
n
NN N
n
vv v
xx x
vv v
xx x
vv v
xx x
xx x
xx x
Jx
xx x
(32)
and
2
1
() () ()
N
ii
i
vv
Sx x x (33)
If ( )Sx is assumed to be small then the Hessian matrix can
be approximated as
2() 2 ()()
T
FxJxJx
(34)
Substituting Eq. (30) and Eq. (34) into Eq. (26), we achieve
the Gauss-Newton method as:
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1
[()()] ()()
TT
kkkkk
xJxJxJxvx
(35)
One problem with the Gauss-Newton method is that the
matrix may not be invertible. This can be overcome by using
the following modification to the approximate Hessian matrix:
GH I
(36)
This leads to the Levenberg-Marquardt algorithm [31]:
1
[()() ] ()()
TT
kkkkkk
xJxJx IJxvx
(37)
Using this gradient direction, and recompute the
approximated performance index. If a smaller value is yield,
then the procedure is continued with the k
divided by some
factor 1
. If the value of the performance index is not
reduced, then k
is multiplied by
for the next iteration step.
The key step in this algorithm is the computation of the
Jacobian matrix. The elements of the error vector and the
parameter vector in the Jacobian matrix (32) can be expressed
as
1 2 1,1 2,1 1,2
,1 ,Q
[][ ]
MM
T
NSS
vv v e e e e ev
(38)
11
11 1 1 1 2
12 1,11,2 1 1,1
,
[][ ]
M
T M
NSR S S
x
xxww wbbwbx
(39)
where the subscript N satisfies:
M
NQS (40)
and the subscript n in the Jacobian matrix satisfies:
121 1
(1) ( 1) ( 1)
MM
nSR SS S S
(41)
Making these substitutions into Eq. (32), then the Jacobian
matrix for multilayer network training can be expressed as
1
1
1
1
1,1 1,1 1,1 1,1
11 1 1
1,1 1,2 1
,
2,1 2 ,1 2,1 2 ,1
11 1 1
1,1 1,2 1
,
,1 ,1 ,1 ,1
11 1 1
1,1 1,2 1
,
1,2 1, 2 1,2 1, 2
11 1 1
1,1 1,2 1
,
()
MM M M
SR
SR
SS SS
SR
SR
ee ee
ww w b
ee ee
ww w b
ee ee
ww w b
ee ee
ww w b
Jx
(42)
In standard backpropagation algorithm, the terms in the
Jacobian matrix is calculated as
ˆ() T
qq
ll
F
x
x
ee
x (43)
For the elements of the Jacobian matrix, the terms can be
calculated by
,
,
,
[] kq
h
hl
lij
e
v
x
w
J (44)
Thus in this modified Levenberg-Marquardt algorithm, we
compute the derivatives of the errors, instead of the derivatives
of the squared errors as adopted in standard backpropagation.
Using the concept of sensitivities in the standard
backpropagation process, here we define a new Marquardt
sensitivity as
,
,
,,
kq
mh
ih mm
iq iq
e
v
snn
(45)
where (1)
M
hq S k
.
Using the Marquardt sensitivity with backpropagation
recurrence relationship, the elements of the Jacobian can be
further calculated by
,,, 1
,,,
,,,
[]
m
kq kq iq mm
hl ih jq
mmm
ij iq ij
een
s
a
wnw
J (46)
if xl is a bias,
,,,
,,
,
[]
m
kq kq iq m
hl ih
mmm
iiqi
een
s
bnb
J (47)
The Marquardt sensitivities can be computed using the same
recurrence relations as the one used in the standard BP method,
with one modification at the final layer. The Marquardt
sensitivities at the last layer can be given by
,,
,
,, ,
,
()
()for
0for
M
M
kq kq k,q k,q
M
ih
M
MM
iq iq iq
MM
1q
eta a
snn n
fn ik
ik
(48)
After applying the q
p
to the network and computing the
corresponding output
M
q
a, the LMBP algorithm can be
initialized by
()
MMM
qq
SFn
(49)
Each column of the matrix should be backpropagated
through the network so as to generate one row of the Jacobian
matrix. The columns can also be backpropagated together using
11
()( )
mmmmm
qq q
SFnWS
(50)
The entire Marquardt sensitivity matrices for the overall
layers are then obtained by the following augmentation
12
|||
mmm m
Q
SSS S
(51)
V. EXPERIMENTAL TESTING AND DATA COLLECTION
In order to train the FFNN model with the LMBP algorithm
proposed above and validate its effectiveness for brake pressure
estimation, real vehicle driving data is needed. Thus,
experiments using an electric passenger car are carried out on a
chassis dynamometer. The testing vehicle together with the
testing scenarios, selected feature vectors, data collection and
data pre-processing are described as follows.
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A. Testing Vehicle and Scenario
The experiment is implemented on a chassis dynamometer
with an electric passenger car, as shown in Fig. 4(a). The
utilized electric vehicle is driven by a permanent-magnet
synchronous motor, which is able to work in either driving or
regenerating mode. The battery pack is connected to the electric
motor via D.C. bus, releasing or absorbing power during
driving and regenerative braking processes, respectively. Key
parameters of the test vehicle are presented in Table 1.
(a) (b)
Fig. 4. (a) The testing vehicle operating on a chassis dynamometer; (b) Speed
profile of the NEDC driving cycle.
TABLE 1
KEY PARAMETERS OF THE ELECTRIC VEHICLE.
Parameter Value Unit
Total vehicle mass 1360 kg
Wheel base 2.50 m
Frontal area 2.40 m2
Nominal radius of tyre 0.295 m
Coefficient of air resistance 0.32 —
Motor peak power 45 kW
Motor maximum torque 144 Nm
Motor max imum speed 9000 rpm
Battery voltage 326 V
Battery capacity 66 Ah
To set up the testing scenario on a chassis dynamometer,
standard driving cycles can be utilized. In this study, the New
European Drive Cycle (NEDC) which consists of four repeated
ECE-15 urban driving cycles and one Extra Urban Driving
Cycle (EUDC) is adopted [32]. As Fig. 4(b) shows, the four
successive ECE-15 driving cycles in the first section of the
NEDC represent urban driving with low operating speed while
the second section, i.e. the EUDC driving cycle, indicates a
highway driving scenario with the vehicle speed up to 120
km/h.
B. Data Collection and Processing
Vehicle data and powertrain states on CAN bus are collected
with a sampling frequency of 100 Hz. Finally, experimental
data of 6327 seconds containing six NEDC driving cycles in
total are recorded. The vehicle speed and brake pressure of the
collected testing data during the four successive ECE-15
driving cycles are presented in Fig. 5.
In order to achieve a better training performance of the
FFNN model with machine learning methods, the raw
experimental data are smoothed at first using the following
equation:
1N
ttn
n
dd
N
(52)
where dt is the value of a signal at time t, dtn is the n-th sampled
value of signal d at time step t, and N is the total amount of
samples within each second.
Fig. 5 Collected data of the vehicle speed and corresponding brake pressure.
Then, in order to eliminate the effect brought by different
units of signals utilized, the input signals are scaled to be in the
range of 0 to 1.
C. Feature Selection and Model Training
In this work, the important vehicle and powertrain state
variables are selected for the training of the multilayer ANN
model for brake pressure estimation, while the real value of the
brake pressure is utilized as a ground truth during the training
process. When the electric vehicle is decelerating, the electric
motor operates as a generator, recapturing vehicle’s kinetic
energy. During this period, the values of the motor and battery
current change from positive to negative, indicating that the
battery is charged by regenerative braking energy. Thus, apart
from the vehicle states, the signals of motor speed and torque,
battery current and voltage, state of charge (SoC) are also
chosen as features, i.e. the input vector of the FFNN. The data
of some of the selected feature variables during one driving
cycle are shown in Fig. 6.
Fig. 6 Experimental data of selected features during one driving cycle.
Besides, statistical information, including the mean value,
maximum value, and standard deviation (STD) of some of the
vehicle states in the past few seconds are also adopted in this
0200 400 600 800 1000 1200
0
20
40
60
80
100
120
t [s]
Vehicle Velocity [km/h]
Real
Target
1000 2000 3000 4000 5000 6000
0
1
2
3
4
Time [s]
Braking Pressure
[MPa]
1000 2000 3000 4000 5000 60 00
0
50
100
Vehicle Speed [km/h]
Motor Speed [r/min] Motor Torque [Nm]
Bat. Voltage [V]
Bat. Current [A]
Brake Pressure [MPa]
Bat. SoC [%]
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work. The features used for model training are listed below in
Table 2.
TABLE 2
SELECTED FEATURES FOR FFNN MODEL TRAINING
No. Signal Unit
1 Vehicle Velocity km/h
2 Mean Value of Velocity km/h
3 STD of Velocity km/h
4 Maximum Value of Velocity km/h
5 Vehicle Acceleration m/s2
6 Motor Speed rad/s
7 Motor Torque Nm
8 Battery Current A
9 Battery Voltage V
10 Battery SoC %
11 Gradient of Bat. Voltage V/s
12 Gradient of Bat. Current A/s
After determining the feature vectors, the regression model
of the FFNN is trained. To modulate and evaluate the model
performance, the K-fold cross validation approach is adopted
[33]. In this method, among the K folds divided, (K-1) ones are
utilized to train the model, and the rest one fold is adopted for
testing. Thus, the overall recorded data are divided into two
sets, namely the training set and the testing one. The testing set,
which is used for model validation, contains 1400 samples
chosen randomly from the raw data, and rest of the data are
allocated to the training set. The final evaluation of the model
performance is carried out based on the K test results. In this
work, the value of K is set as 5. Then, with the 5-fold cross
validation, the constructed FFNN is trained using the fast
LMBP algorithm developed in Section IV. Some key parameter
of FFNN are illustrated below.
TABLE 3
KEY PARAMETERS OF THE FFNN MODEL
Parameter Value
Maximum number of epochs to train 1000
Performance goal 0
Maximum validation failures 6
Minimum performance gradient 1e-7
Initial ߤ 0.001
ߤ decrease factor 0.1
ߤ increase factor 10
Maximum ߤ 1e10
Epochs between displays 25
Maximum time to train in seconds Infinite
VI. EXPERIMENT RESULTS AND DISCUSSIONS
In this section, results of the estimated ANN-based brake
pressure with LMBP learning algorithm are presented and
discussed. The algorithms are implemented in a computer with
the MATLAB 2017a platform. The 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 32G
one. The time consuming for the FFNN training varies with the
number of the hidden neurons selected. In this study, since the
range of hidden neurons number is from 10 to 100, thus the
training time for FFNN varies from 0.6s to 10s, and the average
training time cost for the FFNN with 70 neurons is 3.4s.
A. Results of the ANN-based Braking Pressure Estimation
To quantitatively evaluate the estimation performance, two
commonly used indicies, namely the coefficient of
determination R2 and the root-mean-square-error (RMSE), are
adopted. The definitions of the R2 and RMSE are presented as
follows. Suppose the reference data is {}
1N
Ttt, and the
predicted value is { }
1N
Yyy
. Then R2 can be calculated as:
21res
tot
E
RE
(53)
2
()
N
res i i
i
Ety
(54)
2
()
N
tot i
i
EtT
(55)
where res
Eis the residual sum of square, tot
E is the total sum of
square, and Tis the mean value of the reference data.
The RMSE can be obtained by:
2
()
N
ii
i
ty
RMSE N
(56)
Firstly, 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 to 100.
According to Fig. 7, as the number of neurons changes, the
estimation accuracy of the FFNN varies slightly. The best
prediction performance is yield by FFNN with the number of
neurons at 70.
Fig. 7 Estimation performance of FFNN with different number of neurons.
Fig. 8 Regression performance o f the FFNN model with 70 neurons.
10 20 30 40 50 60 70 80 90 100
0.92
0.94
0.96
0.98
R
2
10 20 30 40 50 60 70 80 90 100
0.015
0.02
0.025
0.03
No. of Neurons
RMSE [MPa]
00.2 0.4 0.6 0.8 11.2
0
0.2
0.4
0.6
0.8
1
1.2
Target
Model Output
Data Point
Fitted Line
Y = T
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Then, the linear regression performance of the trained model
is investigated. Based on the linear regression result shown in
Fig. 8, the test regression result R is of 0.96677, indicating the
FFNN model with 70 neurons can accurately estimate the
braking pressure through selected features.
Fig. 9 shows the brake pressure estimation result in time
domain. The x-axis presents the 1400 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 [0, 1], the model testing output
is then falling within the range between 0 and 1 accordingly.
Based on the results, the FFNN model achieves high-precision
regression performance, and the RMSE is around 0.1 MPa,
demonstrating the feasibility and effectiveness of the developed
method.
Fig. 9 ANN-based braking pressure estimation results with 1400 testing data
points.
B. Importance Analysis of the Selected Features
Besides, the utilized feature variables are further investigated
through analyzing the importance of predictors [34]. A larger
value of the predictor importance indicates that the feature
variable has a greater effect on the model output.
Fig. 10 The predictor importance estimation results.
Fig. 10 illustrates the estimation results of the predictor
importance. Based on the results, the most important feature in
the model is the battery current, followed by STD of velocity,
vehicle velocity, and acceleration. Besides, the battery voltage,
the gradients of the battery voltage and current also exert
impacts on the model estimation performance.
C. Comparison of Estimation Results with different Learning
Methods
The developed ANN-based approach is compared with other
machine learning methods, including regression decision tree,
Quadratic support vector machine (SVM), Gaussian process
model, and regression Random Forest. These models are also
trained and tested with the 5-fold cross validation method.
Apart from R2 and RMSE, other two evaluation parameters, i.e.
the training time and the testing time, are also utilized to assess
the performance of different models.
Detailed results of the comparison are shown in Table 4.
According to the results, the single decision tree algorithm
gives much shorter training time and a much faster testing
speed in comparison to the other algorithms. In terms of
real-time application, the regression decision tree could be a
good candidate because of its simplicity and high computation
efficiency. However, with respect to the brake pressure
estimation accuracy (both R2 and RMSE), the developed ANN
algorithm yields the best performance with acceptable training
time and testing speed.
TABLE 4
COMPARISON OF BRAKING PRESSURE ESTIMATION PERFORMANCE
Method R2 RMSE
(MPa)
Training
Time (s)
Testing
Speed(obs/s)
Decision Tree 0.912 0.133 1.092 ~240000
Quadratic SVM 0.867 0.188 141.93 ~46000
Gaussian Process model 0.921 0.125 156.89 ~8100
Random Forest 0.903 0.104 3.79 ~36000
ANN 0.935 0.101 3.42 ~82000
VII. CONCLUSIONS
In this paper, a novel probabilistic estimation method of
brake pressure is developed for a safety critical automotive CPS
based on multilayer ANN with LMBP training algorithm. The
high-level architecture of the proposed multilayer ANN for
brake pressure estimation is illustrated at first. Then, the
standard BP algorithm used for training of FFNN is introduced.
Based on the basic concept of BP, a more efficient algorithm of
LMBP method is developed for model training. The real
vehicle testing is carried out on a chassis dynamometer under
NEDC driving cycles. Experimental data of the vehicle and
powertrain systems is collected, and feature vectors for FFNN
training collection are selected. With the vehicle data obtained,
the developed multilayer ANN is trained. The experimental
results show that the developed ANN model, which is trained
by LMBP, can accurately estimate the brake pressure, and its
performance is advantageous over other learning-based
methods with respect to estimation accuracy, demonstrating the
feasibility and effectiveness of the proposed algorithm.
Further work can be carried out in the following areas: the
proposed algorithm will be further refined with onboard road
testing; intelligent control algorithms of braking system will be
designed based on state estimation.
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Transactions on Industrial Informatics
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Chen Lv is currently a Research Fellow at Advanced
Vehicle Engineering Center, Cranfield University, UK. He
was with the Department of Automotive Engineering,
Tsinghua University, China. He received the Ph.D. degree at
Department of Automotive Engineering, Tsinghua
University, China in 2016. From 2014 to 2015, he was a joint
PhD researcher at EECS Dept., University of California,
Berkeley. His research focuses on cyber-physical system,
hybrid system, advanced vehicle control and intelligence, where he has
contributed over 40 papers and obtained 11 granted China patents. Dr. Lv
serves as a Guest Editor for IEEE/ASME Transactions on Mechatronics, IEEE
Transactions on Industrial Informatics and International Journal of
Powertrains, and an Associate Editor for International Journal of Electric and
Hybrid Vehicles, International Journal of Vehicle Systems Modelling and
Testing, International Journal of Science and Engineering for Smart Vehicles,
and Journal of Advances in Vehicle Engineering. He received the Highly
Commended Paper Award of IMechE UK in 2012, the National Fellowship for
Doctoral Student in 2013, the NSK Outstanding Mechanical Engineering Paper
Award in 2014, the Tsinghua University Graduate Student Academic Rising
Star Nomination Award in 2015, the China SAE Outstanding Paper Award in
2015, the 1st Class Award of China Automotive Industry Scientific and
Technological Invention in 2015, and the Tsinghua University Outstanding
Doctoral Thesis Award in 2016.
Yang Xing received his B.S. in Automatic Control from
Qingdao University of Science and Technology, Shandong,
China, in 2012. He then received his Msc. with distinction in
Control Systems from the department of Automatic Control
and System Engineering, The University of Sheffield, UK, in
2014. Now he is a Ph. D. candidate for Transport Systems,
Cranfield University, UK. His research interests include
driver behaviour modelling, driver-vehicle interaction, and
advance driver assistance systems. His work focuses on the understanding of
driver behaviours and identification of driver intentions using machine-learning
methods for intelligent and automated vehicles.
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Xiaoxiang Na received the B.Sc. and M.Sc. degrees in
automotive engineering from the College of Automotive
Engineering, Jilin University, Changchun, China, in 2007
and 2009, respectively, and the Ph.D. degree in
driver-vehicle dynamics from the Department of
Engineering, University of Cambridge (CUED),
Cambridge, U.K., in 2014. He is currently a Research
Associate with CUED. His main research interests include
driver–vehicle dynamics and vehicle in-service monitoring.
Yutong Li is a Ph.D. candidate at Tsinghua Univ., China.
He received the bachelor’s degree in internal combustion
engine engineering at the Jilin University in 2008. He is
currently working in the Vehicle Dynamics & Control Lab
at Univ. of California, Berkeley, under supervision of
Prof. J. Karl Hedrick. His research interest covers
nonlinear control, receding horizon control and its
application to vehicle engineering. He received Tsinghua
first class scholarship for doctoral student in 2013 and
2015, and the China SAE Outstanding Paper Award in 2015
.
Teng Liu received the B.S. degree in mathematics from
Beijing Institute of Technology, Beijing, China, 2011. He
received his Ph.D. degree in the vehicle engineering from
Beijing Institute of Technology (BIT), Beijing, in 2017.
His Ph.D. dissertation, under the supervision of Dr.
Fengchun Sun, was entitled “Reinforcement
learning-based energy management for hybrid electric
vehicles.” Dr. Liu is now a postdoctoral research fellow at
Institute of Automation, Chinese Academy of Sciences, China.
Dr. Liu has more than 6 years’ research and working experience in new
energy vehicle and control. His current research focuses on parallel driving,
parallel reinforcement learning, automated driving, and energy management of
electrified vehicles. He has published over 20 papers in these areas.
Dongpu Cao received the Ph.D. degree from Concordia
University, Canada, in 2008. He is currently a Senior
Lecturer at Advanced Vehicle Engineering Center,
Cranfield University, UK. His research focuses on vehicle
dynamics and control, automated driving and parallel
driving, where he has contributed more than 100
publications and 1 US patent. He received the ASME
AVTT’2010 Best Paper Award and 2012 SAE Arch T.
Colwell Merit Award. Dr. Cao serves as an Associate Editor for IEEE
TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS,
IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, IEEE
TRANSACTIONS ON INDUSTRIAL ELECTRONICS, IEEE/ASME
TRANSACTIONS ON MECHATRONICS and ASME JOURNAL OF
DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. He has been a
Guest Editor for VEHICLE SYSTEM DYNAMICS, and IEEE
TRANSACTIONS ON HUMAN-MACHINE SYSTEMS. He serves on the
SAE International Vehicle Dynamics Standards Committee and a few ASME,
SAE, IEEE technical committees.
Fei-Yue Wang (S'87-M'89-SM'94-F'03) received
his Ph.D. in Computer and Systems Engineering
from Rensselaer Polytechnic Institute, Troy, New
York in 1990. He joined the University of Arizona
in 1990 and became a Professor and Director of
the Robotics and Automation Lab (RAL) and
Program in Advanced Research for Complex
Systems (PARCS). In 1999, he founded the
Intelligent Control and Systems Engineering
Center at the Institute of Automation, Chinese Academy of Sciences (CAS),
Beijing, China, under the support of the Outstanding Overseas Chinese Talents
Program from the State Planning Council and ''100Talent Program'' from CAS,
and in 2002, was appointed as the Director of the Key Lab of Complex Systems
and Intelligence Science, CAS. From 2006 to 2010, he was Vice President for
Research, Education, and Academic Exchanges at the Institute of Automation,
CAS. In 2011, he became the State Specially Appointed Expert and the
Director of the State Key Laboratory of Management and Control for Complex
Systems.
Dr. Wang's current research focuses on methods and applications for parallel
systems, social computing, and knowledge automation. He was the Founding
Editor-in-Chief of the International Journal of Intelligent Control and Systems
(1995-2000), Founding EiC of IEEE ITS Magazine (2006-2007), EiC of IEEE
Intelligent Systems (2009-2012), and EiC of IEEE Transactions on ITS
(2009-2016). Currently he is EiC of IEEE Transactions on Computational
Social Systems, Founding EiC of IEEE/CAA Journal of Automatica Sinica,
and Chinese Journal of Command and Control. Since 1997, he has served as
General or Program Chair of more than 20 IEEE, INFORMS, ACM, and
ASME conferences. He was the President of IEEE ITS Society (2005-2007),
Chinese Association for Science and Technology (CAST, USA) in 2005, the
American Zhu Kezhen Education Foundation (2007-2008), and the Vice
President of the ACM China Council (2010-2011). Since 2008, he has been the
Vice President and Secretary General of Chinese Association of Automation.
Dr. Wang has been elected as Fellow of IEEE, INCOSE, IFAC, ASME, and
AAAS. In 2007, he received the National Prize in Natural Sciences of China
and was awarded the Outstanding Scientist by ACM for his research
contributions in intelligent control and social computing. He received IEEE ITS
Outstanding Application and Research Awards in 2009, 2011 and 2015, and
IEEE SMC Norbert Wiener Award in 2014.
