feedbacks in hydraulic servo systems rydberg
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7/21/2019 Feedbacks in Hydraulic Servo Systems Rydberg
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Linkpings universitet TMHP51IEI / Fluid and Mechanical Engineering Systems____________________________________________________________________________________
Feedbacks in Hydraulic Servo SystemsKarl-Erik Rydberg
2008-10-15
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 1
FEEDBACKS IN ELECTRO-HYDRAULIC SERVO SYSTEMS
1. Linear valve controlled position servo
A linear valve controlled position servo is shown in Figure 1. Leakage flow over thepiston with the flow-pressure coefficient Cp and a viscous friction coefficient Bp are
included in the model. The servo amplifier (controller)is proportional with the gain Ksa.
Figure 1: Valve controlled position servo
The transfer functions (in the frequency domain) of the components in the position
servo are illustrated in Figure 2. Threshold and saturation in the servo valve are
included.
Figure 2: Block-diagram of a linear position servo including valve dynamics and non-linearitys
The transfer function of the valve is
v
v ssG
+
=1
1)( . The hydraulic resonance frequency
and damping is expressed as:tt
pe
hVM
A24 = and
te
t
p
p
t
te
p
ceh
M
V
A
B
V
M
A
K
4+= .
The parameter values of the system are as follows:
Ap= 2,5.10-3m2 e= 1,010
9Pa
Bp= 0 Kf= 25V/m
Kce= 1,010-11m5/Ns Kqi= 0,02 m
3/As
Ksa= 0,1 A/V Mt= 1500 kgVt= 1,010
-3m3 v= 1/v= 0,005 s
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 2
These parameter values gives h= 129 rad/sand h= 0.155.
The open loop gain (Au(s)) of the position servo withKv= hh= 20 1/s(Am= 6 dB) is
shown in Figure 3. Observe that the bandwidth of the valve v= 1/v= 200 rad/sis
higher than the hydraulic resonance frequency h.
Figure 3: Bode-diagram of the open loop gain of the position servo depicted in Figure 2
when the servo valve is assumed to be very fast
Influence of valve dynamics
To really make use of the actuator capability of controlling the load it is very important
that the servo valve is fast enough. Normally the selected valve will have a bandwidth
(v) of at least twice as high as the hydraulic resonance frequency (h). Figure 4shows
the open loop gain of the position servo depicted in Figure 2, with an ordinary valve(v=200 rad/s) and a valve with slow response (v= 20 rad/s).
100
101
102
103
102
100
102
Frequency [rad/s]
Amplitude
100
101
102
103
350
300
250
200
150
100
50
Frequency [rad/s]
Ph
ase
100
101
102
103
102
100
102
Frequency [rad/s]
Amplitude
100
101
102
103
400
350
300
250
200
150
100
50
Frequency [rad/s]
Ph
ase
a) Normal valve bandwidth, v= 200 rad/s b) Valve with low bandwidth, v= 20 rad/s
Figure 4: Bode-diagram of the open loop gain of a position servo with a) fast valve and b) slow valve
From Figure 4 it can be recognised that the open loop gain and thereby the amplitude
margin will be change because of the valve dynamics. For a slow valve (v< h) theopen loop gain can be approximated as
( )ss
KA
v
vu
/1+
, which givesKvmax= vfor a reasonable stability margin.
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 3
Closed loop stiffness
The most important characteristic of the servo system is the closed loop stiffness. The
stiffness of the closed loop system describes the controlled signal deflection Xpdue to
variations in the disturbance force FL. By setting Uc= 0 in the block-diagram in Figure2 the new block-diagram becomes as in Figure 5.
Figure 5: Block-diagram describing the stiffness of a closed loop position servo
The stiffness of the closed loop servo is defined asp
Lc
X
FS
= . If the valve dynamics
and the threshold are neglected the stiffness becomes
+
++
+
+
+++
=
hh
h
h
hv
ce
p
v
cee
t
vhv
h
hv
ce
p
vcs
ss
K
s
K
AK
sK
V
K
ss
KK
s
K
AKS
21
12
1
41
12
2
2
2
2
2
3
2
where the steady state loop gain Kv= KsaKqiKf/Ap. The closed loop stiffness including
valve dynamics is shown in Figure 6. The amplitude curve is normalised as
=ce
p
vs
s
c
K
AKK
K
S2
where,
100 101 102 10310
1
100
101
102
Frequency [rad/s]
Amplitude,
(Sc/Ks)
100
101
102
103
0
50
100
150
200
Frequency [rad/s]
Phase
100 101 102 10310
1
100
101
102
Frequency [rad/s]
Amplitude,
(Sc/Ks)
100
101
102
103
50
0
50
100
150
200
Frequency [rad/s]
Phase
a) Normal valve bandwidth, v= 200 rad/s b) Valve with low bandwidth, v= 20 rad/s
Figure 6: Bode-diagram of the closed loop stiffness with a) fast valve and b) slow valve
In Figure 6b) it can be seen that the valve dynamics reduce the stiffness just at
frequencies around the bandwidth of the valve (v= 20 rad/s).
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The threshold of the servo valve will also cause a position error Xp. If the threshold is
inthe position error isfsa
np
KK
iX
=
, where inis nominal valve input current.
2. Valve controlled position servo with load pressure feedback
The load pressure feedback is used to increase the hydraulic damping in the system. A
negative load pressure signal acts in the same way as a Kc-value (flow-pressure
coefficient) of the servo valve. Load pressure feedback can be of proportional or
dynamic type. Proportional pressure feedbackis shown in Figure 7.
Figure 7: Block-diagram of a linear position servo with proportional pressure feedback (Bp= 0)
Load pressure feedback will mainly increase the hydraulic damping. It works just as a
Kc-value. In the above block diagram the proportional pressure feedback will increase
the effective Kc-value as follows, qivsapfcece KGKKKK +='
. The resulting bode
diagram of the open loop gain (Au(s)) and the closed loop stiffness (Sc(s)) for a
hydraulic damping of h = 0,46 is shown in Figure 8. One negative effect of
proportional pressure feedback is that the steady state stiffness will be reduced.
100
101
102
103
102
100
102
Frequency [rad/s]
Amplitude
100
101
102
103
400
350
300
250
200
150
100
50
Frequency [rad/s]
Phase
100
101
102
103
101
100
101
102
Frequency [rad/s]
Amplitude,
(Sc/Ks)
100
101
102
103
0
50
100
150
200
Frequency [rad/s]
Phase
Figure 8: Open loop gain (to the left) and closed loop stiffness of a position servowith load pressure feedback
Dynamic pressure feedbackis shown in Figure 9. The idea of using dynamic pressure
feedback is that the feedback signal shall reach its maximum value at a frequency,which has to be damped (the hydraulic frequency h). Therefore, the pressure signal
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will be high-pass filtered. At low frequencies the pressure feedback signal is low and
the reduction of the steady state stiffness will be very low compared to proportional
pressure feedback.
Figure 9: Block-diagram of a linear position servo with dynamic pressure feedback (Bp= 0)
3. Valve controlled angular position servo with acc. feedback
Acceleration feedback works in principal as dynamic pressure feedback. When the load
starts oscillate there will be a feedback signal, which increase the hydraulic damping
just at the resonance frequency. The good thing with acceleration feedback is that the
steady state stiffness will not be affected. An angular position servo with acceleration
feedback is shown in Figure 10and the corresponding block-diagram is expressed in
Figure 11.
m
Figure 10: An angular position servo with acceleration feedback (Bm= 0)
From Figure 11 the effect of the acceleration feedback can be expressed as a change in
the second order transfer function of the hydraulic system,
12
1)(
2
2
++
=
ss
sG
h
h
h
h
.
This transfer function will now change to
1)(2
1)(
2
2
+
++
=
ssGD
KKK
ssG
v
m
qi
saac
h
h
h
h
.
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 6
m
m
Figure 11a: Block-diagram of an angular position servo with acceleration feedback (Bm= 0)
Withtt
me
h VJ
D24
= , Gv(s) = 1,0andBm= 0the effective hydraulic damping (including
acceleration feedback) will follow the equation:tt
eqisaac
t
te
m
ceh
JVKKK
V
J
D
K +=* .
Constant acceleration feedback gain (Kac) means that the total damping (*
h ) varies
according to variations in the inertia loadJt, as shown in Figure 11b.
Figure 11b: Damping in an angular position servo with acceleration feedback (Bm= 0)
4. Velocity feedback in position control servos
Pressure and acceleration feedback is used to increase the hydraulic damping and this
makes it possible to increase the steady state loop gain Kvand the closed loop stiffness
will increase. Another way to increase the stiffness of a position servo is to introduce a
velocity feedback. A block-diagram of a linear position servo with velocity feedback is
shown in Figure 12.
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Figure 12: A linear valve controlled position servo with velocity feedback
If the bandwidth of the valve is relatively high and threshold and saturation is neglected
the velocity feedback will give the effect on the hydraulic resonance frequency and
damping as shown in Figure 13.
Figure 13: A linear position servo with velocity feedback included
From Figure 13 the new resonance frequency and damping (hvand hv) caused by thevelocity feedback can be evaluated as
vfv
hhvvfvhhvK
K1
, == , where the velocity loop gain isp
qi
savfvvfvA
KKKK += 1 .
Designing the position control loop for the same amplitude margin as without velocityfeedback gives the following relations:
Steady state loop gain without velocity feedback:f
p
qi
sav
KA
KKK =
Steady state loop gain with velocity feedback: fvfvp
qi
savvv KKA
KKK =
A certain amplitude margin means that hhvK . In this case hvhvhh = , which
implies that vvv KK = and thereby the servo amplifier gain vfvsasav KKK = . With
velocity feedback, the servo amplifier gain (Ksav) can be increased in proportion to the
velocity loop gainKvfvand the servo amplifier gain without velocity feedback,Ksa.
The open loop gain (Au(s)) for a position servo without (Kv = 20) and with velocity
feefback (Kvfv= 10 andKvv=20) is shown in Figure 14.
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100
101
102
103
102
100
102
Frequency [rad/s]
Amplitude
100
101
102
103
300
250
200
150
100
50
Frequency [rad/s]
Phase
Figure 14: Open loop gain for a position servo without and with velocity feefback (Kv=Kvv)
5. Valve controlled velocity servo
If an integrating amplifier is used in a velocity servo the loop gain Au(s) will be in
principle the same as for a position servo with proportional control. Such a velocity
servo is shown in Figure 15.
Figure 15: A linear valve controlled velocity servo
A block diagram of the velocity servo is shown in Figure 16.
Figure 16: Block-diagram of a linear valve controlled velocity servo
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The transfer functions in the above block-diagram are:
1
1)(
+=
v
v ssG
, sK
VsG
cee
t
41)(1 += ,
12
1)(
2
2
++
=
ss
sG
h
h
h
h
An integrating amplifier means that the control error will be integrated and the steady
state control error becomes zero.
6. Proportional valves with integrated position and pressure
transducers
In all fluid power applications a load has to be controlled by an actuator in respect of
speeds and forces. A new dimension of the ways to look upon these control aspects is to
use a control valve (proportional or servo valve), which is capable of controlling both
flow and pressure in the actuator ports (two ports for a double cylinder or motor). Such
a proportional valve has been developed by Ultronics. The principle design of the valve
is shown in Figure 17.
Figure 17: Application with Ultronics proportional valve
From Figure 17 it can be seen the valve has two spools, which make it possible to
control meter-in and meter-out flow of any actuator independently. This facility givesthe opportunity of smooth acceleration and deceleration control of the load by
individual pressure control in each cylinder chamber. The pressure transducers can also
be used for load pressure feedback to increase the hydraulic damping. By measurement
of the pressure drop (p) over a spool the load flow (qL) can be controlled bycalculation of the spool displacement (xv) from the flow equation of the valve, which
gives
pwC
qx
q
Lv
=
2
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 10
7. Electro-hydraulic servo actuators
Today electro-hydraulic actuators are normally manufactured as integrated units. The
servo valve is connected to the actuator (cylinder or motor) and all the transducers
needed for close loop control are integrated in the valve and actuator. An industrial
actuator for linear position control is depicted in Figure 18. The control card for this
actuator includes connectors for all feedback signals and the controller is implemented
in a micro-processor. The input signals to the control card are electric power supply and
a set point signal and than the card deliver a current signal (i) to the servo valve. The
hydraulic part of the actuator system has two connectors, one hydraulic supply line and
one return flow line.
In many industrial applications there is a need for multiple degrees of freedom control
of the load. One application, which requires advanced control, is motion simulator
platforms. This type of platform is often used for dynamic simulation of air-crafts and
cars. A common way to design a platform, which can be moved in a 3D-space, is to use
6 electro-hydraulic linear actuators as shown in Figure 19.
Figure 18: Industrial electro-hydraulic linear position control actuator, MOOG
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 11
Figure 19: Electro-hydraulic motion platform with 6 degrees of freedom, Rexroth
For low power applications (low load weights) the platform shown in Figure 19 is often
realised by using electro-mechanical actuators (electric motor and a ball screw).
A similar control strategy as for the 6 DOF platform can be used for crane(or industrial
robot) tip control. Electro-hydraulic control of a lorry crane is shown in Figure 20.
Z3
X3
h
Figure 20: Crane tip control with optronic sensor for vertical position measurement
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The strategy for 2 DOF crane tip control is shown in Figure 21. A range camera
(optronic sensor) is used to measure the vertical distance (h) between the camera and
the object. Z3 is the vertical co-ordinate from the base line of the crane to the crane tip.
The reference value for the vertical crane tip position is calculated as Z3ref= Z3+hrefh.
The kinematics of the crane structure is calculated by using the signals from position
transducers in the hydraulic cylinders and a geometric description of the crane structure.
However, this will not give the true tip position of the crane tip because of the weakness
in the mechanical structure. By using a range camera it is possible to compensate the
vertical position control according to the mechanical weakness.
Figure 21: Control strategy for crane tip positioning
8. Design examples
ydraulically operated boom with lumped masses
The figure shows a valve controlled cylinder used for operation of a mechanical arm.
The total mass of the moving arm is ML. The distance from the gravity centre of themass to the joint (0) is L. The lever length for the hydraulic cylinder is e, which will
vary according to xp. The piston area is Ap and its pressurised volume is VL and this
volume varies according to the piston position. The effective bulk modulus is e. The
pressure on the piston rod side is assumed as constant,pR= constant. The mass of the
cylinder housing isM0and the mechanical spring coefficient for the connection isKL.
Figure 22: Application with variable mechanical gearing between cylinder and load
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K-E Rydberg Feedbacks in Electro-Hydraulic Servo Systems 13
Equivalent cylinder mass
The equivalent mass loading the piston rod is found from the torque equation for the
joint (0).
eApLMT
LMJ
pLL
Lt
===
..2
2
:Torque
:Inertia
With..
2..
.. pLpL
pX
eLMAp
eX
== .
Introducing the mechanical geare
LU= , the equivalent cylinder mass can be expressed
as,2UMM Lt=
Hydraulic resonance frequency and dampning
Assuming MLas the dominant mass the resonance frequency can be calculated as,
M0> Kh
Cylinder design according to max pressure level
This example is aimed to demonstrate how the cylinder design will influence the
hydraulic frequency and damping. Figure 23 shows a system with a stiff mechanical
structure and the cylinder is assumed to be loaded by one mass (ML).
Figure 23: Cylinder controlled mass with mechanical gear
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As in Fig. 22 the mechanical gear U = L/e. The piston area is selected as,L
Lp
p
gMUA = .
The cylinder volume depends of the load displacement (XL) as,
U
XAV Lp=0 . For the
hydraulic resonance frequency the basic equation is,2
0
2
UMV
A
L
pe
h
= . If the cylinder is
designed for some maximum load pressure (pLmax), withApand V0as described above,
the hydraulic frequency will follow the expression:
maxLL
eh
pX
g
=
.
The hydraulic damping is described as,0
2
2 V
UM
A
K Le
p
ce
h
= or
gX
p
A
UK
L
Le
p
ceh
= max
2
,
where the flow/pressure coefficient (Kce) is assumed to be constant.
The product hhis expressed as,Lt
Leceecehh
XgM
pK
V
K
== max0 22
.
Figure 24shows how the frequency, damping and the product varies according to the
design parameter max load pressure, pLmax.
Figure 24: Hydraulic resonance frequency and damping versus max load pressure
From the equations it can be noticed that the hydraulic damping will be proportional to2/3
maxLp and the product maxLhh p . This indicates that the cylinder-load response willshow less oscillations when the max load pressure is increased. The system response for
different pLmaxis illustrated in Figure 25.
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Figure 25: Response of the cylinder-load dynamics with cylinder design for max load pressure of 100,200 and 300 bar respectively
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K-E Rydberg Controller design 1
______________________________________________________________________
Controller Design for Hydraulic Servo Systems
General structure of the controller
The most general controller of conventional type is the PID-controller. However, evenwith this controller there can still be a need of more dynamic compensations in the
control loop. In a hydraulic system the relative damping is often quite low. A
stabilisation feedback(load pressure or acceleration feedback) can be used to increase
the damping. Depending of the variation of the command signal there will be a delaybetween the derivative of the command signal and the output signal. This delay can be
reduced to a minimum by use of afeed forward gain.
The action of the PID-controller means that the derivative gain increases proportionallyto the frequency. In spite of this behaviour it is important to reduce the gain of the D-
action at high frequencies. Otherwise, the high frequency disturbances on the signalswill be amplified to a level which can mainly influence the function of the system. A
forward loop filteris used to reduce the derivative gain at high frequencies.
From the above discussion the general structure of the controller will be as shown in
Figure 1.
Figure 1: Structure of a PID controller with feed forward gain and stabilisation feedback.
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K-E Rydberg Controller design 2
______________________________________________________________________
Feed forward gain for reduction of velocity error in pos. servo
Assume a linear position servo with a valve controlled piston. In this case a plain
proportional controller is suitable to use and easy to adjust for stability. However, if thecommand signal is changed there will be a phase lag from input to output signal in the
servo. In the position servo the phase lag cause a position error proportional to time
derivative of the command signal (velocity).
If the feed forward gain introduces a derivative of the command signal it will be
possible to more or less eliminate the phase lag. This feed forward gain helps the servocontrol loop (servo valve) to react quickly to a change in the command signal.
Implementation of a feed forward gain in a position servo is shown in the simulink-
model in Figure 2. The feed forward gain is represented by the transfer function
Gff(s) = s/Kv, where Kv is the steady state gain in the control loop from feed forward
input to system output signal. In this case Kv= 20 sec-1
and 1/Kv= 0.05 sec. The feedforward gain also includes a low-pass filter with a break frequency of 1000 rad/s
(compare with the forward loop filter in Figure 1).
Figure 2: Simulink-model of a valve controlled cylinder with position feedback and feed forward gain.
The command signal in Figure 2 is a sine wave. The simulation results in Figure 3
shows that the output signal can follow the command signal with a very small phase lag.
The oscillations at start depends on the relatively low hydraulic damping (h= 0.155) inthe system.
Figure 3: Command and output signal with feed forward gain.
The effect of the feed forward gain can preferable be studied by plotting the outputsignal (Y) versus command signal (X), as illustrated in Figure 4.
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K-E Rydberg Controller design 3
______________________________________________________________________
Figure 4: Output versus command signal without (to the left) and with feed forward gain ina position servo with proportional control.
A notable behaviour of the feed forward gain is that its action is like a pre-filter, which
not affect the control loop gain and the stability margins.
PID Controller
The ProportionalIntegralDerivative controller (PID controller) is a control loopfeedback mechanism widely used in industrial control system. A PID controller
attempts to correct the error between a measured system variable and a desired
command signal by calculating and then outputting a corrective action that can adjust
the process accordingly. A PID controller and its control algorithm are shown in Fig. 5.
Input_U
startTime={0.2}
Output_Y
D_action
DT1
k={0}
Sum
+1
+1
+1
+
k={1}
P_action
I_action
I
k={3}
Saturation
uMax={2}
Saturation
++=t
t
D
I
Pdt
tdUTdU
TtUKtY
0
)()(
1)()(
Figure 5: PID Controller.
Proportional gain
Proportional gain is used for all tuning situations. It introduces a control signal that is
proportional to the error signal. As proportional gain increases, the error decreases and
the feedback signal tracks the command signal more closely. Proportional gain increases
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K-E Rydberg Controller design 4
______________________________________________________________________
system response by boosting the effect of the error signal. However, too much
proportional gain can cause the system to become unstable.
Figure 6: Effects of proportional gain.
Integral gain
With an integral control mode the error signal will be integrated over time, which
improves mean level response during dynamic operation. Integral gain increases system
response during steady state or low-frequency operation and maintain the mean value athigh-frequency operation. The I-gain adjustment determines how much time it takes to
improve the mean level accuracy. Higher integral gain settings increase system
response, but too much gain can cause slow oscillations, as shown in Figure 7.
Figure 7: Effects of integral gain.
The integrator output signal depends upon the I-gain and the input signal level, see
Figure 8. It is very important to set a limit for the output signal, as shown in Figure 8,to prevent the integrator for windup.
Figure 8: Integrator action with different input signals.
An Anti-windup implementation for a PID controller is shown in Figure 9.
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K-E Rydberg Controller design 5
______________________________________________________________________
Figure 9: Anti-windup implementation for I-action in a PID controller.
Derivative gain
With a derivative control mode the feedback signal means it anticipates the rate of
change of the feedback and slows the system response at high rates of change.
Derivative gain provides stability and reduces noise at higher proportional gain settings.The D-gain tends to amplify noise from sensors and to decrease system response when
set is too high. Too much derivative gain can create instability at high frequencies.
Figure 10: Effects of derivative gain
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