Research Journal of Applied Sciences, Engineering and Technology 4(4): 289-298, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: August 03, 2011
Accepted: September 25, 2011
Published: February 15, 2012
High Precision Position Control of Electro-Hydraulic Servo System Based
on Feed-Forward Compensation
1
Yao Jian-jun, 1Di Duo-tao, 1Jiang Gui-lin and 2Liu Sheng
1
College of Mechanical and Electrical Engineering,
2
College of Automation, Harbin Engineering University, Harbin 150001, China
Abstract: The study is focused on an electro-hydraulic servo system which is a position control system. It is
a non-minimum phase system when it was discretized with a certain sample time. To improve its tracking
performance and extend its bandwidth, based on invariance principle, feed-forward compensation is developed
by pole-zero placement theory for the system. The task is accomplished by transforming instable zero of the
system into pole of the fitted closed-loop transfer function, forming the zero of feed-forward compensator and
completing the compensation of the instable zero for the closed-loop system. The simulation and experimental
results show the validity of the analytical results and the ability of the proposed algorithm to efficiently improve
the system tracking performance and greatly extend system bandwidth.
Key words: Feed-forward compensation, invariance principle, non-minimum phase system, pole-zero
placement, system bandwidth, tracking performance
not be guaranteed. Adaptive sliding control by Guan and
Pan (2008a) and nonlinear adaptive robust control by
Guan and Pan (2008b) were presented for electrohydraulic system with nonlinear unknown parameters to
improve the tracking performance. Self-tuning fuzzy PID
controller in which the fuzzy controller was used to
update the three parameters of the PID controller was
developed by Wu et al. (2004). Its key issue is how to
design the membership function and it needs to know the
system model. Adaptive control scheme based on Popov
criterion was proposed by Yao et al. (2006). But it has
large computation. Adaline neural network using LMS
adaptive filtering algorithm was proposed by Yao et al.
(2007), whose task was accomplished by adjusting the
weights of the network using LMS algorithm when phase
delay existed between the input and its corresponding
output, and the reference input was weighted in such a
way that it makes the system output track the input
efficiently, thus the weighted input signal was added to
the control system such that the output phase delay was
cancelled leaving the desired signal alone. To cancel
amplitude attenuation and phase delay, Yao et al. (2011b)
developed a network using normalized LMS adaptive
filtering algorithm, whose weights were on-line updated
according to the estimation error between the desired
input and the weighted feedback, the updated weights thus
were copied to the input correction. The two methods can
greatly improve sinusoidal tracking performance.
However, this method is only applicable to fixed-point
sine wave. An adaptive tracking control with reference
INTRODUCTION
Hydraulic control systems are widely used in many
main industrial fields, including aerospace, metallurgy,
transportation, machine, marine technology, modern
scientific experimental device and weapon control (Li,
1990; Yao et al., 2007). Compared with others types of
drives, they have many distinct advantages, such as
developing a comparatively small device with much
larger torque, high precision, higher respond speed, higher
stiffness and higher force-to-weight ratio (Yao et al.,
2008).
As for the control algorithm aspects for electrohydraulic servo system, classical PID controller has been
employed in real applications. The regulation law of the
conventional PID controller is effective for quite a
number of plants, especially for linear time-invariant
system. The control performance is dependent on the
parameters of PID controller. However, its parameters are
set under certain operation condition. Because
nonlinearities and uncertainty usually occur in the electrohydraulic servo system, when the classical PID controller
is used for the system, it only gives mediocre control
performance.
Adaptive position control based on Radial Basis
Function (RBF) neural network and LQ controller was
developed by Knohl and Unbehauen (2000) for a
hydraulic system controlled by a 4/3 way proportional
valve. Its structure is complex and it has a problem of
computation burden, thus its real-time performance can
Corresponding Author: Yao Jian-jun, College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin
150001, Heilongjiang China
289
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
ps
results in a large change of oil flow (John and
Constantime, 1993). The resulting difference in pressure
on the main piston causes motion of the output shaft. The
oil flowing in is supplied by a hydraulic source that
maintains a constant high supply pressure ps, and the oil
on the opposite side of the piston flows into the drain at
low return pressure pb.
The load-induced pressure pL is the difference
between the pressures on each side of the main piston:
pb
xv
qVi
qVo
pi A
po
y
m
Vi
FL
B
Vo
pL =
Fig. 1: Block diagram of electro-hydraulic servo system
FL
= pi − po
A
(1)
where A is the effective area of the cylinder piston, FL the
external force, and pi, p° are the input and output pressure
of the cylinder, respectively.
The pressure drop ªp across the orifice is a function
of the supply pressure ps and the load-induced pressure pL.
Since ps is assumed to be constant, the flow equation for
q is a function of valve displacement xv and pL:
model being 1 was presented by Wang et al. (2005) to
achieve zero phase tracking. The initial value of integral
adaptive law has great influence on the system output.
With the increase of test system requirements, it
requires high performance both in amplitude frequency
bandwidth and in phase frequency bandwidth, especially
for large frequency bandwidth system, for example,
electro-hydraulic servo shaking table, in which its
frequency bandwidth may be required to be bigger than
100 Hz. Though three-variable control can extend system
frequency bandwidth for an electro-hydraulic servo
shaking table (Yao et al., 2010, 2011a), this improvement
is limited by the servo valve frequency bandwidth and the
hydraulic natural frequency. The methods can improve the
system performance, especially the tracking performance,
but it can not widen frequency bandwidth (Yao et al.,
2006, 2007; Knohl and Unbehauen, 2000; Guan and Pan,
2008 a; Guan and Pan, 2008 b; Wu et al., 2004).
In the study, to improve the tracking performance of
electro-hydraulic servo system and extend its bandwidth,
based on invariance principle, feed-forward compensation
is developed by pole-zero placement theory for the
system. High precision position control is accomplished
by transforming instable zero of the system into pole of
the fitted closed-loop transfer function, forming the zero
of feed-forward compensator and completing the
compensation of the instable zero for the closed-loop
system. The proposed control scheme is validated by the
simulation and experimental results.
q = f(xv1, pL)
where, q is the rate of flow of hydraulic fluid through the
valve. The differential rate of flow dq, expressed in terms
of partial derivates, is!
dq =
∂q
∂q
dx +
dp
∂ xv v ∂ pL L
(2)
If q, xv and pL are measured from zero values as reference
points, and if the partial derivatives are constant at the
values they have at zero, the integration of Eq. (2) gives
the flow equation of fluid entering the main cylinder:
⎛ ∂q ⎞ v ⎛ ∂q ⎞
q=⎜
⎟ x o+ ⎜
⎟ PLo
⎝ ∂ xv ⎠ 0
⎝ ∂ pL ⎠ 0
(3)
= K q x v − K c pL
where,
⎛ ∂q ⎞
Kq = ⎜
⎟ is the flow-gain coefficient
⎝ ∂ xv ⎠ o
SYSTEM DESCRIPTION
⎛ ∂q ⎞
Kc = ⎜
⎟ is flow-pressure coefficient
⎝ ∂ pL ⎠ o
The valve-controlled hydraulic actuator is used as a
force amplifier. Very little force is forced to position the
valve, but a large fo xv rce is obtained. The hydraulic unit
is relatively small, which makes its application very
attractive. Figure 1 is the hydraulic system, which is
comprised of a symmetric cylinder, a symmetric servo
valve and a load force. The motion xv of the valve
regulates the flow of oil to either side of the main
cylinder. An input motion of a few thousands of an inch
The oil flow can also be written as:
q=
290
qv1 + qvo
2
(4)
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
where qv1, qvB is the input/output oil flow of the cylinder,
respectively. The continuity equations of the cylinder are
given as:
(q − Ay& )
V
β
=
( Ay& − q )
V
p& i =
β
vi
Gc (s )
r
Feed-forward
compensation
+
−
e
Controller
GF (s )
(5)
u
Plant
y
Gp (s)
i
p& o
vo
o
Fig. 2: Feed-forward compensation
(6)
correction algorithm of drive signals PSD and the
generation of time domain drive signals in the classical
algorithm. The drive signals PSD iteration algorithm with
different step length in different frequency band was
presented to raise the rate of convergence. The method of
generating the drive signal in time domain by filtering a
series of independent white noise with designed FIR
(Finite Impulse Response) filter was presented. The FIR
filter was designed by the Parks-McClellan algorithm
with the information contained in the in the drive signal
PSD. To avoid the time domain randomization procedure,
the execution time of the algorithm was reduced. The
improved PSD iteration algorithm can improve the control
precision, but it has great computation burden, because it
needs to off-line calculate FFT (Fast Fourier Transform)
and IFFT (Inverse Fast Fourier Transform) in time
domain and in frequency domain to generate the drive
signal. What more, for a elastic load, ite convergence
may not be ensured.
For a servo control system, the feed-forward
compensation scheme can extend the system frequency
bandwidth and improve the system tracking performance.
Generally, the feed-forward compensation scheme is
based on the invariance principle (Lin and Tian, 2005). In
other words, the feed-forward compensation loop is
designed to be the inverse of the system closed-loop
transfer function such that the transfer function of the
compensated system is unity. However, when the closedloop system is a non-minimum phase system, this method
will face difficulty. In the invariance principle (Lee et al.,
2001), the unstable zeros of a non-minimum phase system
become the poles of the feed-forward compensation loop,
leading to instability of the feed-forward compensation
part. In real applications, with the increase of the sample
frequency, discrete-time systems discretized from
continuous systems will be non-minimum phase systems.
In this study, for such non-minimum phase systems,
zeros are introduced into the feed-forward compensator to
compensate the unstable zeros of the closed-loop system
(Wang et al., 2005). Its basic theory is the pole-zero
displacement theory.
The feed-forward compensation scheme can be
described by Fig. 2. Gc(s) is the closed loop system
transfer function including the controller, r the input
command. Let Gc(zG1) be a stable discrete-time system
which is discretized from a continuous closed loop
where $ is the bulk modulus of the hydraulic oil and & is
the velocity of the piston, $ is defined as the ratio of
incremental stress to incremental strain. Thus:
ΔpL
β=
ΔV /V
where V is the effective volume of fluid under
compression.
From Eq. (1), (4), (5) and (6), the flow continuity
equation of the cylinder is:
q = Ay& +
Vt
p&
4β L
(7)
where Vt is the total oil volume of the cylinder, Vt = Vi+Vo.
The balance of the force at the sliding carriage is:
ApL = my&& + By& + FL
(8)
where m is the total mass of the piston and the load, B the
equalized viscous damping coefficient, &&y the acceleration
of the piston.
Using Eq. (3), (7), (8) and applying the Laplace
transformation to the resulting third-order differential
equation results (Yao et al., 2007):
Kq
Y=
A
Kc ⎛
s⎞
⎜ 1 + ⎟ FL
ω1 ⎠
A2 ⎝
⎛ s2 2ζ
⎞
s⎜ 2 + h s + 1⎟
⎝ ωh ωh
⎠
Xv −
(9)
where
ωh =
4 βA 2
mVt
βm
ζh =
Kc
A
ω1 =
4βKc
Vt
Vt
+
Vt
B
2 A 4βm
Feed-forward compensation: In order to improve the
control precision of random vibration PSD (Power
Spectral Density) replication, Guan (2007) improved the
291
0
1.5
-10
1.0
Pole
Zero
-20
Imaginary part
Amplitude (dB)
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
-30
-40
-50
-60
0
0.5
0
-0.5
-1.0
Phase (deg)
-1.5
-4 -3.5 -3.0 -2.5 -2.0 -1.5 -1 -0.5
Real part
-90
0
0.5
1
1.5
Fig. 4: Zero- pole plot
-180
and a set-point should be specified. The feedback variable
is the system parameter, which the designer wants to
control, and the set-point is the desired value for the
parameter. The PID controller generates a controller
output, for example, valve position. The controller applies
this output value to the system, which in turn drives the
feedback variable toward the set-point value. The PID
controller calculates the controller action as:
-270
-1
0
10
1
10
10
2
10
Frequency (Hz)
Fig. 3: System closed-loop Bode plot before feed-forward
compensation
system,Gc(s) and Gc(zG1) has s non-minimum phase zeros
with unity static gain. Thus Gc(zG1) can be written as:
Gc ( z −1 ) =
−d
−1
⎡
1
u( t ) = K p ⎢ e( t ) +
TI
⎣
−1
z Nu (z ) Na (z )
D( z −1 )
(10)
1
5
1
e( t ) dt + TD
de( t ) ⎤
⎥
dt ⎦
Simulation results: To validate the proposed feedforward compensation scheme, the system was used for
an electro-hydraulic servo shaking table is adopted as the
plant (Yao et al., 2006). Its transfer function is:
Theorem (Liu, 2003; Xia and Meng, 1995): Let H(zG ) =
Nu(z)Nu(zG1) which meets:
1
(
t
0
where e is the error signal obtained by comparing the setpoint to the feedback variable. Kp is the controller gain.
TI is the integral time, which is also called the reset time.
TD is the derivative time or the rate time.
where, Nu(zG ) = n0+n1zG +...+n5zG , n0 … 0. Nu(zG )
contains zeros of Gc(zG1) on the unit circle and instable
zeros of Gc(zG1) beyond the unit circle, while Na(zG1)
includes all stable zeros of Gc(zG1) within the unit circle.
1
∫
)
∠ H e − jωr = 0
and
(
∠ H e − jωT
)
2
G p ( s) =
[ ( )]
+ Im [ N (e
)] ∀ ω ∈ R
= Re2 N u e − jωT
− jωT
2
where, T is the sample time. Thus the feed-forward
compensator can represented by Liu (2003):
z − s Nu ( z) D( z −1 )
[
N a ( z −1 ) N u (1)
]
2
s s + 47.855s + 24586.24
)
For real electro-hydraulic servo system, P controller is
mostly applied. Before feed-forward compensation is
applied, the system closed-loop bode plot is shown in
Fig. 3 with proportional coefficient Kp = 0.3. From the
figure, it can be seen that there are small -3 dB amplitude
and -90º phase frequency bandwidth, especially for the
amplitude frequency bandwidth. To improve the system
frequency bandwidth and its tracking performance, the
feed-forward compensation in Fig. 2 is applied.
During the simulation, system sample time is 1 ms.
After it is discretized with the sample time, the closedloop transfer function is:
u
G F ( z −1 ) =
(
1803158187
.
2
(11)
Currently, the Proportional-Integral-Derivative (PID)
algorithm is the most common control algorithm used in
industry applications. In PID control, a feedback variable
292
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
Desired signal
Actual output
0
Displacement (y/m)
Amplitude (dB)
0.20
-0.5
-1
-1.5
-2
-2.5
0
0.15
0.10
0.05
0.00
-0.05
-0.10
Phase (deg)
-0.15
-0.20
-45
0.0
0.3
0.6
0.9
Time (t/s)
1.2
1.5
-90
Fig. 7: System sinusoidal response with hybrid frequencies
-135
-1
2
0
1
10
10
Frequency (Hz)
10
Table 1: Experimental parameters
Item
Value
Cylinder stroke
1350
Cylinder piston
K125
K90
Cylinder rod
Oil supply pressure
5
Equivalent external load
5500
Sample time
2
10
Fig. 5: System closed-loop bode plot after feed-forward
compensation
Desired signal
Actual output
Displacement (y/m)
0.10
Table 2: Sinusoidal response analysis
Amplitude
Frequency (Hz)
attenuation (dB)
0.1
0
0.5
-0.5691
1.0
-0.6144
1.7
-0.6600
5.0
-3.7352
0.05
0.00
-0.05
-0.10
0.0
0.1
0.2
0.3
Time (t/s)
0.4
8.898 × 10−5 z 2 + 3513
.
× 10− 4 z + 8.688 × 10−5
z 3 − 2.929 z 2 + 2.883z − 0.9533
Its zero-pole plot is shown in Fig. 4, from which it can be
seen that there is an instable zero, so it is a non-minimum
phase system.
According to Eq. (10), the feed-forward compensator
GF (zG1) is:
( )
GF z −1 =
Experimental results: The experimental setup of an
electro-hydraulic servo system which is an asymmetric
cylinder controlled by a MOOG D792 servo valve is used
to validate the proposed control scheme. The system
configuration is shown in Fig. 8. The hydraulic power
supply provides a high supply pressure ps of 5 MPa as
shown in Table 1, which gives the system parameters. r is
the command input, and y is the feedback displacement.
The servo controller compares the feedback signal from
sensor with the input to produce a command signal to
drive the servo valve, which adjusts the flow of
pressurized oil to move the hydraulic cylinder until the
desired position is attained (Yao et al., 2011a).
1887 z 4 − 5016z 3 + 3940z 2 − 3217
. z − 488.6
z + 0.265
For real control application, desired output ahead of
time can not be known. Since the system sample time is
very small, the value ahead of three sampling period can
be ignored, thus the feed-forward compensator becomes:
( )
GF z −1 =
Phase delay (deg)
-5.616
-17.28
-21.6
-30.6
-64.8
The system closed-loop bode plot after feed-forward
compensation is shown in Fig. 5. It can be seen that the
system amplitude and phase bandwidth are all greatly
enhanced. This is very useful for the applications where
wide frequency bandwidth is required for electrohydraulic servo system.
When the input is 0.1sin(20Bt) m, the system
response is shown in Fig. 6. Figure 7 is the response
corresponding to the command: 0.1sin(16Bt) +
0.05sin(30Bt) + 0.02sin(40Bt) m. From the two figures,
the compensated system has good tracking performance.
0.5
Fig. 6: System sinusoidal response
Gc ( z ) =
Unit
mm
mm
mm
MPa
kg
ms
1887 − 5016z −1 + 3940z − 2 − 3217
. z − 3 − 488.6z − 4
−1
1 + 0.265z
293
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
Hydraulic
power supply
r
Servo
controller
H ydraulic
cylinder
Servo valve
Load
y
Sensor
Fig. 8: System configuration of the electro-hydraulic servo system
Signal
generator
+
0.18 s + 1
0.06 s + 1
K
−
D/A
conditioner
D/A
A/D
conditioner
A/D
Displacement
senso r
Fig. 9: Block diagram of the control system
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
170
175
180
185
190
195
Time t/s
200
205
210
(a) Input frequency is 0.1 Hz
2.1
C ommand
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
245
247
249
251
253
255
Time t/s
(b) Input frequency is 0.5 Hz
294
257
259
261
Servo valve
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
290 291 292 293 294 295 296 297 298 299 300
Time t/s
(c) Input frequency is 1 Hz
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
329
330
331
332
Time t/s
333
334
335
(d) Input frequency is 1.7 Hz
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
375
376
377
Time t/s
(e) Input frequency is 5 Hz
Fig. 10: Sinusoidal response at different frequencies
295
378
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
175
180
185
190
195
200
205
Time t/s
(a) Input frequency is 0.1 Hz
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
290
292
294
296
298
300
Time t/s
(b) Input frequency is 0.5 Hz
2.1
Command
Feedback
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
392
394
396
Time t/s
(c) Input frequency is 1 Hz
296
398
400
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
2.1
C om mand
Feed bac k
Displacement y/mm
1.4
0.7
0.0
-0.7
-1.4
-2.1
388
3 89
39 0
391
3 92
39 3
394
Tim e t/s
(d) Input frequency is 1.7 Hz
2 .1
Co mm a nd
F e ed b ac k
Displacement y/mm
1 .4
0 .7
0 .0
-0 .7
-1 .4
-2 .1
4 4 2 .0
4 4 2 .5
4 43 .0
44 3 .5
4 4 4 .0
T i m e t/ s
(e) Input frequency is 5 Hz
Fig. 11: Sinusoidal response at different frequencies after using the proposed control method
The control system is shown in Fig. 9. The signal
generator is used to generate the command input. The
saturation unit is to make the control signal in the range of
[-10, 10] V, which is also the volt range of D/A (digital to
analog, D/A). The D/A and A/D (analog to digital, A/D)
conditioner are used to condition signals for servo valve
driving signal and for A/D board, respectively.
The sinusoidal responses are shown in Fig. 10, and its
analysis is shown in Table 2, from which it can be seen
clearly that with the increase of input frequency, the
amplitude attenuation and the phase delay become more
serious.
With the frequency response data in Table 2, the
system closed-loop transfer function can be identified by
the MATLAB command, invfreqs, which is useful in
converting magnitude and phase data into transfer
functions. The identified system closed-loop transfer
function is:
y( s)
=
R( s)
The proposed control scheme is then applied to the
electro-hydraulic servo system, and its experimental
results are shown in Fig. 11 plotted with the command
and the feedback signals. From the results, it can be
clearly seen that the feedback signal nearly coincides well
with the command input at all five frequencies, thus the
compensated system has good tracking performance,
improving its frequency characteristics.
CONCLUSION
Though PID controller is widely used in electrohydraulic servo system, because of its limitations, good
tracking can not be achieved in every operation condition.
To enhance system bandwidth and improve the tracking
performance, feed-forward compensator, which is based
on pole-zero placement theory, is developed. When the
electro-hydraulic servo system is discretized, it becomes
a non-minimum phase system. The zeros of the feedforward compensator are obtained by transforming the
system instable zeros to the poles of the fitted transfer
function of the closed-loop system, compensating the
− 1482 s + 7674
s 4 + 0.0703s 3 + 1023s 2 − 23.64 s + 4.78 × 10 4
297
Res. J. Appl. Sci. Eng. Technol., 4(4): 289-298, 2012
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servo shaking table. Proceedings of the 8th World
Congress on Intelligent Control and Automation,
Jinan, China, July 6-9, pp: 2026-2030.
Yao, J.J., W. Fu, S.H. Hu and J.W. Han, 2011a.
Amplitude phase control for electro-hydraulic servo
system based on normalized least-mean-square
adaptive filtering algorithm. J. Central South Univ.
Technol., 18(3): 755-759.
Yao, J.J., S.H. Hu, W. Fu and J.W. Han, 2011b. Impact of
excitation signal upon acceleration harmonic
distortion of an electro-hydraulic shaking table. J.
Vib. Control, 17(7): 1106-1111
instable zeros of the closed-loop system, and high
precision tracking performance is achieved.
For a real system, its amplitude and phase frequency
characteristics can be established by sine sweep tests, thus
the feed-forward compensator can be obtained based on
the fit of the closed-loop system, so the proposed control
scheme can be applied to real application.
ACKNOWLEDGMENT
The author is grateful for the support of the National
Natural Science Foundation of China (No. 50905037),
Specialized Research Fund for the Doctoral Program of
Higher Education of China (No. 20092304120014),
Special Financial Grant from the China Postdoctoral
Science Foundation (No. 201104414), China Postdoctoral
Science Foundation (No.20100471021), Postdoctoral
Science–Research Developmental Foundation of
Heilongjiang Province (No. LBH-Q09134) and the
Foundation of Harbin Engineering University (No.
HEUFT09013).
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