Modelling and Analysis of Step Response Test for

Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
© 2015 Journal of Mechanical Engineering. All rights reserved. DOI:10.5545/sv-jme.2014.2046
Original Scientific Paper
Received for review: 2014-07-08
Received revised form: 2014-10-04
Accepted for publication: 2014-10-27
Modelling and Analysis of Step Response Test
for Hydraulic Automatic Gauge Control
Yi, J.
Yi Jiangang*
1 Jianghan
University, Hubei Key Laboratory of Industrial Fume & Dust Pollution Control, China
The step response for hydraulic automatic gauge control (HAGC) determines the steel rolling speed and the steel sheet thickness in the
process of rolling production. In this paper, the step response test process of HAGC was analysed, and a test approach was proposed for it.
Based on that, the transfer function model of the step response test was established and simulated by using Matlab. In order to reduce the
settling time and the overshoot, an adaptive proportional-integral-derivative (APID) link was presented in order to compensate for the input
signal by using back propagation neural networks (BPNN). The experimental results show that the improved step response test model reaches
the process requirements of HAGC, eliminates the jitter of the HAGC system at the start-up phase, and has better stability as well as faster
response for steel sheet rolling.
Keywords: step response, hydraulic automatic gauge control, proportional-integral-derived controller, artificial neural networks
Highlights
• Proposed the step response test model of HAGC system.
• The working parameters study of the model.
• Presented an APID link for signal compensation.
• Representation of the stability and the flexibility on step response of the HAGC system.
0 INTRODUCTION
Sheet gauge is one of the main quality indicators for
steel sheet in the process of rolling production. To
improve the control precision of sheet gauge, hydraulic
automatic gauge control (HAGC) is currently widely
used. In the process of HAGC, the step response plays
the most important role, because it determines the
steel rolling speed and the steel sheet thickness, and
accordingly influences steel sheet surface quality. The
step response test is a time-domain test method for
system dynamic characteristics. It is used to describe
the dynamic response process of the control system
when the input is a step signal. To achieve uniform
thickness of a steel sheet, the step response parameters
of the HAGC should be adjusted according to the
real-time thickness of steel sheet. However, during
the step response process of HAGC, the step response
parameters are influenced by the interactions of
hydraulic cylinders, servo valves, and various sensors
of the system, and the working time is extremely
short (no more than 1 second). Consequently, it is of
vital importance to model, test, and analyse the step
response of HAGC.
In terms of HAGC system design, Wang et al. and
Taleb et al. developed a real-time simulator for a hotrolling mill based on a digital signal processor, which
can be used for controlling the hydraulic cylinder in
an HAGC system [1] and [2]. Gao et al. proposed a
simulated model of 1100 mm rolling mill HAGC
system by using position-pressure compound control
method [3]. T.S. Tsay presented a command tracking
error square control scheme, and designed feedback
control systems [4]. To achieve good control effect,
many researchers studied the control algorithm of
HAGC. Ang et al. and Mitsantisuk et al. researched
the general design method of control system with
proportional-integral-derived controller (PID) [5] and
[6]. Zhang et al., Dou et al. and Chang et al. analysed
the PID parameters setting problem [7] to [9]. Their
research proved that the PID controller with proper
parameters was efficient, but the setting of the PID
parameters is the main problem. To achieve the desired
strip thickness of the HAGC system, Khosraviet al.
and Song et al. proposed a novel fuzzy adaptive PID
controller [10] and [11]. The simulation results showed
that it was better than traditional PID controller, but
sensitive to parameter variations. Wan et al. and
Kasprzyczak et al. analysed the main parameters of
the hydraulic system and discussed their effects on
system stability [12] to [13].
To solve the problem of multivariable parameters
adjustment of the PID controller, several authors
proposed some intelligent algorithms, such as
evolutionary algorithms, particle swarm optimization
(PSO), artificial neural networks (ANN) and
generalized predictive control method [14] to [18]. The
results indicated the intelligent algorithms improved
the adaptability of the PID controller. However, the
dynamic response process of the controller under step-
*Corr. Author’s Address: Hubei Key Laboratory of Industrial Fume & Dust Pollution Control, Jianghan University, Wuhan, 430056, China, Yjg_wh@yeah.net
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Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
input was not discussed. In the literature, the research
put emphasis on the design, analysis and control of
HAGC, and few papers studied the step response test
of HAGC.
In this paper, the step response test of HAGC is
analysed, a test approach is proposed, and a transfer
function model of the step response test is established
and simulated by using Matlab software. In order
to reduce the settling time and the overshoot, an
adaptive proportional-integral-derivative (APID)
link is presented to compensate for input signal by
using back propagation neural networks (BPNN).
The experimental results show that the improved step
response test model reaches the process requirements
of HAGC, eliminates the jitter of the HAGC system at
the start-up phase, and has better stability as well as a
faster response for steel sheet rolling.
The structure of this paper is organized as
follows. Section 1 introduces the parameters and the
approach of the step response test of HAGC. Section 2
establishes the step response test model with transfer
function. Section 3 simulates the proposed model by
using Matlab, and presents the improved model of the
step response test by adding an APID link based on
BPNN. Section 4 contains the experiments and the
analysis of the improved model. Section 5 is devoted
to the conclusions.
coordinate value represents the displacement of the
piston rod in the HAGC system. Next, the parameters
of the step response test include the rise time tr, the
maximum overshoot Mp, and the settling time ts. The
rise time tr is the time at which the response signal
reaches the first steady-state output, as described in
Eq. (1):
tr = t0.9 − t0.1 , (1)
where t0.9 is the time at which the response signal is
90% of the first steady-state output, and t0.1 is the time
at which the response signal is 10% of the first steadystate output.
The difference between the response signal and
steady-state output functions as the numerator, and the
steady-state output as the denominator, the overshoot
as the ratio of them. Next, the maximum overshoot Mp
can be calculated by Eq. (2):
MP =
xo (∞)
× 100%, (2)
where xo(t) is the displacement of the piston rod at the
time t, and tp is the time at which the response signal
reaches the peak.
In the step response process, the settling time ts
is also called the transition time, which represents the
time at which the HAGC system reaches the steadystate. It is defined as the time at which the value of
xo(t) satisfies Eq. (3):
1 THE STEP RESPONSE TEST OF HAGC
1.1 The Parameters of the Step Response Test
In Fig. 1, the x coordinate value of the response signal
curve represents the step response time, and the y
xo (t )-xo (∞) ≤ 0.05xo (∞). (3)
Fig. 1. The parameters of the step response test
116
xo (t p ) − xo (∞)
Yi, J.
Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
In the parameters of the step response, the settling
time ts reflects the flexibility of the HAGC system,
and the maximum overshoot Mp reflects the stability
of HAGC system. In an HAGC system, it is always
considered that the shorter of ts and Mp, the better of
the control effect.
1.2 The Approach of the Step Response Test
The main components in the step response process
of HAGC are the servo valve, mill cylinder, current
sensors, and displacement sensors. In order to simplify
the test process, the influence of the hydraulic pipe
and hydraulic power components is neglected. Next
the approach of the step response test is shown in Fig.
2, and the main test steps are as follows:
Step 1: The displacement of step signal is given to
the computer test software. It is converted to a voltage
signal by the data acquisition card and is sent to the
current sensor (6).
Step 2: The output signal of the data acquisition
card is converted to current by the current sensor (6),
and then is sent to the servo valve (5) to control the
output flow in valve port A.
Step 3: According to the output flow in the valve
port A, the piston rod (3) of mill cylinder 2 moves updown to control the rolling thickness of steel sheet.
Step 4: The real-time displacement of the rolling
thickness is measured by the displacement sensor
(4), and then is converted to digital signal by the data
acquisition card.
Step 5: The acquired digital signal is sent to the
computer test software, which will be compared with
the input displacement in Step 1 to determine the next
input value.
2 MODELLING OF THE STEP RESPONSE TEST
2.1 The Parameters of the Step Response Test
According to Fig. 2, the step response test scheme is
established, as shown in Fig. 3. The input signal Uv
is the step signal of the expected displacement. The
output signal Yp is the real-time displacement of the
mill cylinder, which is converted to the voltage signal
Up by the displacement sensor and fed back to the
input port of the servo valve. The difference between
Uv and Up, Ue, is converted to the current signal by
the current sensor and is used to drive the servo valve.
The piston rod action of the mill cylinder is controlled
by the output flow of the servo valve.
If the PID link is neglected and the input signals
are sent to drive the servo valve directly, the transfer
function of the servo valve is:
G1 ( s ) =
K sv
, (4)
2ξ sv
s
1
+
+
s
ωsv2 ωsv
2
where Ksv is the output flow gain of the servo valve,
ωsv is the natural frequency of the servo valve, and ξsv
is the damping radio of the servo valve.
The transfer function of the mill cylinder is:
G2 ( s ) =
Ac
KK ce

 s
  s 2 2ξ h
s + 1
 + 1  2 +
 ωr
  ωh ωh

, (5)
where ωr is the transition frequency of the inertia, and
ωh and ξh are the natural frequency and the damping
radio of the mill cylinder. Kce is the overall flowpressure coefficient, K is the load stiffness, and Ac is
the effective area of the piston rod of the mill cylinder.
The transfer function of the current sensor is:
Fig. 2. The step response test of HAGC; 1-Steel sheet, 2-Mill cylinder, 3-Piston rod, 4-Displacement sensor, 5-Servo valve, 6-Current sensor
Modelling and Analysis of Step Response Test for Hydraulic Automatic Gauge Control
117
Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
In terms of Fig. 3 and Eqs. (4) to (8), the overall
transfer function model of the step response test with
conventional PID algorithm can be described as Eq.
(9):
G3 ( s ) = K i , (6)
where Ki is the gain of current.
The transfer function of the displacement sensor
is:
Ac
K sv
KK ce
⋅ (9)
G (s) H (s) = 2
⋅
2ξ sv
s

 s
  s 2 2ξ h
+
+
1
s
+
+
1
s
+
1


 2
ωsv2 ωsv
 ωr
  ωh ωh

1
⋅ Ki K s ⋅ ( K p +
+ Td s ).
Ti s
H ( s ) = K s , (7)
where Ks is the feedback coefficient of displacement.
2.2 Adding PID Link
To reduce the settling time and the maximum overshoot
of HAGC, some researchers proposed compensating
for the input signal by using some algorithms. The
signal compensation is implemented by adding a new
link to improve the system performance. Because the
PID algorithm is flexible, and its parameters can be
easily adjusted, it is widely used in control systems.
Therefore, based on the step response test scheme,
a PID link is added in the step response test scheme
between the input signal Ue and the current sensor, as
shown in Fig. 3.
The PID algorithm includes a proportional part,
an integral part, and a differential part. Consequently,
three coefficients, Kp, Ti and Td, are used in PID
controller for the system control, where Kp is the
proportional coefficient, Ti is the integral coefficient,
and Td is the derivative coefficient. Therefore, the
conventional PID algorithm can be described as:
G4 ( s ) =
Ug
Ue
= Kp +
3 SIMULATION AND IMPROVEMENT
OF THE STEP RESPONSE TEST
3.1 Simulation of the Step Response Test
To analyse the control effect with and without a PID
link in the step response test, the working parameters
are loaded to the established transfer function model
in the HAGC system, and the step response test is
simulated by using the Simulink toolbox in Matlab
software. The simulated model with the working
parameters is shown in Fig. 4. In the simulated model,
a step signal of 1 mm displacement is loaded at the
input point, and the output result is shown as the
blue dot curve in Fig. 5. In Fig. 5, it can be observed
that ts = 140 ms, Mp = 25 %. However, in the HAGC
production process, it is necessary that ts < 100 ms
and Mp < 10 % for steel sheet rolling. Therefore, the
settling time and the maximum overshoot are beyond
the range of the HAGC requirements, which means
the step response test without a PID link cannot be
used to drive the HAGC system directly.
1
+ Td s. (8)
Ti s
Fig. 3. The step response test scheme
Fig. 4. The simulated model with working parameters
118
Yi, J.
Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
rate Kec of Ue as the input values of the ANN, and Kp,
Ti and Td as the output values, the BPNN is used to
calculate the proper PID parameters by training it with
acquired samples.
3.3 Implementation of APID
Fig. 5. The simulated results of the step response test
By adding the PID link in the established model
in Fig. 4, the step response test is simulated with a
conventional PID algorithm, and the output result
is shown as a green solid curve in Fig. 5. It is found
when Kp = 10, Ti = 50, and Td = 0, the settling time
ts = 80 ms, and the maximum overshoot Mp = 9 %,
which meet the process requirements of the HAGC.
Moreover, testing shows that increasing Kp and Td,
and decreasing Ti can further reduce the values of ts
and Mp. However, at the same time, it leads to large
jitters in the rise time of the step response test, which
impairs the stability of the HAGC system.
3.2 Improvement of the Step Response Test
The simulation results of the model with a PID link
indicate that the contradiction between the stability
and flexibility of the HAGC system cannot be solved
by the conventional PID algorithm. This is because
the PID parameters of the conventional PID algorithm
are constant during the process of the step response
test, which cannot be adjusted according to the input
and output signals adaptively. In the actual production
of steel sheet, because of the interactions of the servo
valve, mill cylinder, and sensors in the HAGC system,
the step response is a nonlinear time-varying process.
Fig. 6. The structure of the APID for HAGC system
As a result, an APID algorithm based on BPNN
is proposed. The structure of the APID algorithm is
shown in Fig. 6. The error Ke and the error change
In recent years, many ANN algorithms have become
widely used in both academic research and industrial
development. In all ANN algorithms, BPNN is a multilayer forward-spread network with a minimum mean
square deviation learning method. It has been proved
that BPNN can map all nonlinear functions with single
layers. Therefore, a BPNN is created by using the
Neural Networks toolbox in Matlab to implement the
APID link of HAGC system. The BPNN is composed
of an input layer with 2 neurons (Ke and Kec), a hidden
layer with 4 neurons (set in the neural networks
toolbox) and an output layer with 3 neurons (Kp, Ti and
Td). The training function is TRAINLM, the adaption
learning function is LEARNGDM, and the transfer
function is LOGSIG. The samples are collected from
the steel sheet production of HAGC system. Table 1
lists 10 sets of the normalized data which are used as
the training samples for the built BPNN.
Table 1. The training samples of the BPNN
Number
1
2
3
4
5
6
7
8
9
10
Ke
0.442
0.095
0.119
0.094
0.893
0.792
0.541
0.113
0.867
0.133
Kec
0.193
0.794
0.101
0.099
0.545
0.152
0.085
0.125
0.048
0.020
Kp
0.056
0.150
0.097
0.620
0.212
0.078
0.038
0.255
0.243
0.135
Ti
0.071
0.076
0.070
0.588
0.092
0.063
0.041
0.119
0.008
0.009
Td
0.010
0.093
0.105
0.197
0.081
0.098
0.033
0.100
0.025
0.011
The ability of ANN is generally measured by its
mean-squared error (MSE). With the collected data
in Table 1, the built BPNN is trained, and the change
of the MSE is shown in Fig. 7. It can be seen when
the training MSE goal is 0.01; the training times are
no more than 500, which indicates that the proposed
BPNN is convergent for APID control.
After the BPNN is trained, it can be used to
calculate the APID parameters with current values
of Ke and Kec. The step response test of HAGC with
the APID controller is simulated again by inputting
the same values in the conventional PID controller
(Ke = 0.326, Kec = 0.247), and the result is shown as
the red dashed curve in Fig. 5. In comparing the green
Modelling and Analysis of Step Response Test for Hydraulic Automatic Gauge Control
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Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
solid curve (PID) and the red dash curve (APID), it
is obvious that by using the APID algorithm, the step
response can not only reach the process requirements
of the HAGC, but also eliminate the jitter at the startup phase, which means the HAGC system has better
stability and flexibility with the improved model.
Fig. 8. The HAGC system
Fig. 7. The training result of the BPNN
4 EXPERIMENTAL RESULTS
In order to verify the established step response test
model in Section 2 and Section 3, experiments were
done with the designed HAGC system of the mill
servo cylinder, as shown in Fig. 8. The type of the mill
cylinder is C1450-P20N000, the piston rod diameter
is 1450 mm, and the stroke length is 10 mm. The
embedded computer servo controller receives the
acquired signals from the sensors and the servo valves,
and sends the calculated results to the HAGC system.
The APID algorithm is programmed with Visual C++
and loaded into the HAGC controller. To test step
response ability of the HAGC system, the standard
step input signals were sent through the HAGC
controller, as shown in Fig. 9a. The step response
models without PID link, with conventional PID link
and with APID link were tested, and the experimental
results were shown on the HAGC controller screen,
as shown in Fig. 9b. By comparing Fig. 5 and Fig.
9b, it can be determined that there is good agreement
between the simulated and the experimental results.
By using the APID algorithm and conventional
PID algorithm, the change of the steel sheet thickness
within 100 ms is measured, as shown in Fig. 10. It can
be seen that the thickness change in the step response
test of HAGC with APID is no more than 0.06 mm,
and the surface irregularity has a decreased trend as
120
a)
b)
Fig. 9. The experimental results;
a) Input signal, b) Step response signals
time passed. The thickness change with conventional
PID is about 0.30 mm, far above the value of APID.
Consequently, the improved step response test model
by using APID link can reduce the settling time and
the overshoot, and thus enhance the surface quality
of steel sheet in the HAGC system. This indicates
Yi, J.
Strojniški vestnik - Journal of Mechanical Engineering 61(2015)2, 115-122
the improved step response model is valid, and the
experimental results are consistent with the simulated
results.
Fig. 10. The measured thickness change of steel sheet;
a) with APID, b) with conventional PID
5 CONCLUSIONS
In the process of the step response test of HAGC, it
is difficult to balance the stability and the flexibility
of the system. To improve the control performance
of the system, the approach of adding proper link to
compensate for the input signal is valid. By adding a
PID link, the settling time, and the overshoot can be
reduced. However, the conventional PID algorithm
also leads to jitters of HAGC at start-up phase. In
this paper, based on the established step response test
model, the APID link by using BPNN is proposed
to improve steel sheet quality. The simulated and
experimental results show that the designed step
response model with the APID link is useful for
overcoming jitters, reducing overshoot and settling
time, and accelerating the dynamic response of the
HAGC system. Further research to analyse the step
response influence of the other components in HAGC
systems, such as hydraulic pipes and hydraulic pumps,
should be carried out.
6 ACKNOWLEDGMENT
This work is supported by the National Natural
Science Foundation (Granted No: 51071077), China.
The author also gratefully acknowledges the helpful
comments and suggestions of the reviewers, which
have improved the presentation.
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