Linköpings universitet
TMHP51
IEI / Fluid and Mechanical Engineering Systems
____________________________________________________________________________________
Feedbacks in Hydraulic Servo Systems
Karl-Erik Rydberg
2008-10-15
K-E Rydberg
1
Feedbacks in Electro-Hydraulic Servo Systems
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 the
piston 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.
Fix reference
Cp
be
Ap
P1 V1
xp
Bp
be
Ap
Mt
V2 P2
FL
qL2
qL1
Position
transducer
xv
Servo
uc + amplifier i
Ksa
u f
Kf
Ps = const.
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.
FL
ö
Vt
Kce æç
÷
+
s
1
ç
Ap2 è 4 be K ce ÷ø
Threshold
uc
+
-
Ksa
Saturation
iv imax
ir
ein
Kqi
1
Ap 1 + s +
wv
.
-
Au(s)
1
xp 1 xp
s 2 + 2 dh s +
s
1
wh2 wh
Kf
Figure 2: Block-diagram of a linear position servo including valve dynamics and non-linearity’s
The transfer function of the valve is G ( s) =
v
and damping is expressed as: ω h =
4 β e A p2
1
1+
s
. The hydraulic resonance frequency
ωv
and δ h =
K ce
Ap
βeMt
M tVt
Vt
The parameter values of the system are as follows:
Ap = 2,5.10-3 m2
βe = 1,0·109 Pa
Kf = 25 V/m
Bp = 0
-11
5
Kqi = 0,02 m3/As
Kce = 1,0·10 m /Ns
Mt = 1500 kg
Ksa = 0,1 A/V
-3
3
τv = 1/ωv = 0,005 s
Vt = 1,0·10 m
+
Bp
4 Ap
Vt
.
βeM t
K-E Rydberg
2
Feedbacks in Electro-Hydraulic Servo Systems
These parameter values gives ωh = 129 rad/s and δh = 0.155.
The open loop gain (Au(s)) of the position servo with Kv = δhωh = 20 1/s (Am = 6 dB) is
shown in Figure 3. Observe that the bandwidth of the valve ωv = 1/τv = 200 rad/s is
higher than the hydraulic resonance frequency ωh.
Phase shift [degrees]
Amplitude [dB]
Kv
= dh = 0,155
wh
Am
Frequency [w/wh]
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
(ωv) of at least twice as high as the hydraulic resonance frequency (ωh). Figure 4 shows
the open loop gain of the position servo depicted in Figure 2, with an ordinary valve
(ωv=200 rad/s) and a valve with slow response (ωv = 20 rad/s).
2
2
10
Amplitude
Amplitude
10
0
10
−2
0
10
−2
10
10
0
10
1
2
10
10
3
0
10
10
1
−50
−50
−100
3
10
−150
Phase
−150
Phase
10
Frequency [rad/s]
−100
−200
−250
−200
−250
−300
−300
−350
0
10
2
10
Frequency [rad/s]
−350
1
2
10
10
Frequency [rad/s]
a) Normal valve bandwidth, ωv = 200 rad/s
3
10
−400
0
10
1
2
10
10
3
10
Frequency [rad/s]
b) Valve with low bandwidth, ωv = 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 (ωv < ωh) the
open loop gain can be approximated as
Kv
Au ≈
, which gives Kvmax = ωv for a reasonable stability margin.
(1 + s / ω v )s
K-E Rydberg
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Feedbacks in Electro-Hydraulic Servo Systems
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 ∆Xp due to
variations in the disturbance force ∆FL. By setting Uc = 0 in the block-diagram in Figure
2 the new block-diagram becomes as in Figure 5.
ö
Vt
DFL Kce æç
÷
ç1 + 4 b K s÷ 2
Ap è
e ce ø
.
1
s 2 + 2 dh s +
1
wh2 wh
-
Saturation
Kqi
1
Ap 1 + s
wv
xp
1
s
DXp
Threshold
imax
ein
Ksa
Kf
Figure 5: Block-diagram describing the stiffness of a closed loop position servo
The stiffness of the closed loop servo is defined as S c =
− ∆FL
. If the valve dynamics
∆X p
and the threshold are neglected the stiffness becomes

 s
  s 2 2δ h
2δ h 2
s




+
1
⋅
+
s
+
1
+
s
+
+
1
 ω2 ω
A p2 K vω h2 K vω h
A p2  K v
Kv
h
  h

Sc = K v
⋅
≈ Kv
⋅
V
K ce
K ce

s 
t
1+
s
1 +

4 β e K ce
 2δ hω h 
s3
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
A p2
Sc
, where K s = K v
⋅
Ks
K ce
2
2
10
Amplitude, (Sc/Ks)
Amplitude, (Sc/Ks)
10
1
10
0
10
−1
10
1
10
0
10
−1
0
10
1
2
10
10
10
3
10
0
10
1
3
10
200
150
Phase
150
Phase
10
Frequency [rad/s]
200
100
50
0
0
10
2
10
Frequency [rad/s]
100
50
0
1
2
10
10
Frequency [rad/s]
a) Normal valve bandwidth, ωv = 200 rad/s
3
10
−50
0
10
1
2
10
10
3
10
Frequency [rad/s]
b) Valve with low bandwidth, ωv = 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 (ωv = 20 rad/s).
K-E Rydberg
4
Feedbacks in Electro-Hydraulic Servo Systems
The threshold of the servo valve will also cause a position error ∆Xpε. If the threshold is
ε ⋅ in
ε⋅in the position error is ∆X pε =
, where in is nominal valve input current.
K sa K f
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 feedback is shown in Figure 7.
Proportional pressure feedback
Kpf
-
uc +
-
Ksa
i
Gv Kqi
+
-
1
Kce + Vt s / 4be
PL
+
Ap
FL
-
1
Mt s
.
xp 1_ xp
s
Ap
Kf
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, K ce' = K ce + K pf K sa Gv K qi . The resulting bode
diagram of the open loop gain (Au(s)) and the closed loop stiffness (Sc(s)) for a
hydraulic damping of δh = 0,46 is shown in Figure 8. One negative effect of
proportional pressure feedback is that the steady state stiffness will be reduced.
2
2
10
Amplitude, (Sc/Ks)
Amplitude
10
0
10
−2
10
1
10
0
10
−1
0
10
1
2
10
10
10
3
10
0
10
1
2
10
Frequency [rad/s]
10
3
10
Frequency [rad/s]
−50
200
−100
150
−200
Phase
Phase
−150
−250
−300
100
50
−350
−400
0
10
1
2
10
10
Frequency [rad/s]
3
10
0
0
10
1
2
10
10
3
10
Frequency [rad/s]
Figure 8: Open loop gain (to the left) and closed loop stiffness of a position servo
with load pressure feedback
Dynamic pressure feedback is 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 ωh). Therefore, the pressure signal
K-E Rydberg
5
Feedbacks in Electro-Hydraulic Servo Systems
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.
Dynamic pressure feedback
s/wf
s/wf + 1
uc
+
-
Ksa
-
i
Kpf
Gv Kqi +
-
1
Kce +
Vt
s
4be
PL
Ap
FL
-
1
Mt s
+
.
1
_ xp
s
xp
Ap
Kf
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 10 and the corresponding block-diagram is expressed in
Figure 11.
Kf
Uc
+
+
Kac
Greg
V1
ps
..
qm
qm
Jt
pL
TL
V2
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
1
the second order transfer function of the hydraulic system, Gh ( s ) = 2
.
2δ h
s
+
s +1
2
ωh
This transfer function will now change to Gh ( s ) =
ωh
1
K qi
 2δ h


+
+
K
K
Gv ( s )  s + 1
ac
sa
2

Dm
ωh  ωh

s2
.
K-E Rydberg
6
Feedbacks in Electro-Hydraulic Servo Systems
TL
ö
Vt
Kce æç
÷
+
s
1
Dm2 çè 4 be K ce ÷ø
uc
+
-
+
-
Ksa
iv
Kqi
1
Dm 1 + s +
wv
-
.
1
qm
s 2 + 2 dh s +
1
wh2 wh
Kac.s
Acceleration feedback
Kf
Position feedback
1 qm
s
Figure 11a: Block-diagram of an angular position servo with acceleration feedback (Bm = 0)
With ωh =
4 β e Dm2
, Gv(s) = 1,0 and Bm = 0 the effective hydraulic damping (including
J tVt
acceleration feedback) will follow the equation: δ h* =
K ce
Dm
βeJ t
Vt
+ K ac K sa K qi
βe
Vt J t
.
Constant acceleration feedback gain (Kac) means that the total damping ( δ h* ) varies
according to variations in the inertia load Jt, 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 Kv and 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.
K-E Rydberg
7
Feedbacks in Electro-Hydraulic Servo Systems
FL
ö
Vt
Kce æç
÷
+
s
1
ç
Ap2 è 4 be K ce ÷ø
Threshold
uc
+
-
+
Ksav
-
Saturation
Kqi
1
Ap 1 + s +
wv
iv imax
ir
ein
.
-
1
xp 1 xp
s 2 + 2 dh s +
s
1
wh2 wh
Kfv
Velocity feedback
Kf
Position feedback
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.
FL
ö
Vt
Kce æç
÷
ç1 + 4 b K s÷
2
Ap è
e ce ø
uc
+
-
Ksav
Kqi
Ap
iv
+
Kf
Kvfv = 1 + Kfv Ksav
Kqi
Ap
.
1/ Kvfv
xp
2
2
s
d
h
+
s +1
Kvfv wh2 Kvfv wh
1 xp
s
Position feedback
Figure 13: A linear position servo with velocity feedback included
From Figure 13 the new resonance frequency and damping (ωhv and δhv) caused by the
velocity feedback can be evaluated as
ω hv = ω h K vfv , δ hv = δ h
1
K vfv
, where the velocity loop gain is K vfv = 1 + K fv K sav
K qi
Ap
.
Designing the position control loop for the same amplitude margin as without velocity
feedback gives the following relations:
Steady state loop gain without velocity feedback: K v = K sa
Steady state loop gain with velocity feedback: K vv = K sav
K qi
Ap
Kf
K qi
A p K vfv
Kf
A certain amplitude margin means that K v ∝ ω hδ h . In this case ω hδ h = ω hvδ hv , which
implies that K v = K vv and thereby the servo amplifier gain K sav = K sa K vfv . With
velocity feedback, the servo amplifier gain (Ksav) can be increased in proportion to the
velocity loop gain Kvfv and 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 and Kvv =20) is shown in Figure 14.
K-E Rydberg
8
Feedbacks in Electro-Hydraulic Servo Systems
2
Amplitude
10
0
10
−2
10
0
1
10
2
10
3
10
10
Frequency [rad/s]
−50
Phase
−100
−150
−200
−250
−300
0
10
1
2
10
3
10
10
Frequency [rad/s]
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.
Ap
vp
Ap
Mt
FL
Velocity
transducer
uc +
Integrating
servo ampl.
uf -
Ksa
____
s
Kf
i
Ps = const.
Figure 15: A linear valve controlled velocity servo
A block diagram of the velocity servo is shown in Figure 16.
FL
Integrating
amplifier
uc
+
-
Ksa
1 ir
s
Threshold
Kce
Gl
Ap2
-
Saturation
i imax
Kqi
G
Ap v +
ein
.
Gh
Au(s)
Kf
Figure 16: Block-diagram of a linear valve controlled velocity servo
xp
K-E Rydberg
9
Feedbacks in Electro-Hydraulic Servo Systems
The transfer functions in the above block-diagram are:
Vt
1
1
Gv ( s ) =
, G1 ( s ) = 1 +
s , Gh ( s) = 2
s
2δ h
4 β e K ce
s
+1
+
s +1
2
ωv
ωh
ωh
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.
U
X
Load
Uc
+
-
Pressure
and
position
signals
+
Valve
controller
X
U
X
U
Output signals
Supply pressure
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 gives
the 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 by
calculation of the spool displacement (xv) from the flow equation of the valve, which
gives
qL
xv =
2
Cq w
∆p
ρ
K-E Rydberg
7.
Feedbacks in Electro-Hydraulic Servo Systems
10
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
K-E Rydberg
11
Feedbacks in Electro-Hydraulic Servo Systems
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.
Mechanical System
Z3
Optronic
Sensor
h
Hydraulic
System
Computer Based
Measurement and
Control System
X3
Figure 20: Crane tip control with optronic sensor for vertical position measurement
K-E Rydberg
12
Feedbacks in Electro-Hydraulic Servo Systems
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+href–h.
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.
X3ref
href
+
Xp1r
+
S
Inverse
Kinematics Xp2r
Z3ref
+
+
-
Leadfilter
-
PI-
xv1r
PI-
xv2r
S
contr
S
contr
-
Leadfilter
Gcyl1
Xp1
Kinematics
Xp2
Gcyl2
X3
Z3
q3
h Range
Camera
Figure 21: Control strategy for crane tip positioning
8.
Design examples
Hydraulically 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 the
mass 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 βe. The
pressure on the piston rod side is assumed as constant, pR = constant. The mass of the
cylinder housing is M0 and the mechanical spring coefficient for the connection is KL.
L
0
ML
q
e
xp
Ap
V L pL
KL
pR = constant
M0
xv
Figure 22: Application with variable mechanical gearing between cylinder and load
K-E Rydberg
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Feedbacks in Electro-Hydraulic Servo Systems
Equivalent cylinder mass
The equivalent mass loading the piston rod is found from the torque equation for the
joint (0).
..
Inertia : J t = M L L2
..
Torque : Tθ = M L L2 θ = p L A p e
2
Xp
 L  ..
With θ =
⇒ p L Ap = M L   X p .
e
e
..
Introducing the mechanical gear U =
L
, the equivalent cylinder mass can be expressed
e
as, M t = M LU 2
Hydraulic resonance frequency and dampning
Assuming ML as the dominant mass the resonance frequency can be calculated as,
M0 << ML gives ωh =
Ke
1
=
Mt U
2
Ke
where 1 = V L 2 + 1 ⇒ K e = K L β e A p 2
ML
K e β e Ap K L
K LV L + β e A p
Low mechanical friction gives the hydraulic damping: δ h =
βe M L
K ce
U
2 Ap
VL
According to the gear U it can be observed that an increase of the gear gives a reduced
ωh but an increased δh. Studying the product δh·ωh the expression is,
2
βe
K
K ce
β e M L 1 β e Ap
U
δ h ωh ≈
⇒ δ hω h ≈ ce
, for KL >> Kh
VL U VL M L
2 VL0 + Ap x p
2 Ap
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).
L
ML
e
xp
Ap
V0 pL
p
Figure 23: Cylinder controlled mass with mechanical gear
xL
K-E Rydberg
14
Feedbacks in Electro-Hydraulic Servo Systems
As in Fig. 22 the mechanical gear U = L/e. The piston area is selected as, Ap = U
The cylinder volume depends of the load displacement (XL) as, V0 = Ap
hydraulic resonance frequency the basic equation is, ωh =
M Lg
.
pL
XL
. For the
U
β e Ap2
. If the cylinder is
V0 M LU 2
designed for some maximum load pressure (pLmax), with Ap and V0 as described above,
the hydraulic frequency will follow the expression:
ωh =
βe ⋅ g
X L ⋅ p L max
The hydraulic damping is described as, δ h = K ce
2 Ap
.
β e M LU 2 or δ = K ce ⋅ U
h
V0
β e ⋅ p L max
2 Ap
XL ⋅g
,
where the flow/pressure coefficient (Kce) is assumed to be constant.
The product δh·ωh is expressed as, δ hωh =
K ce β e K ce β e ⋅ p L max
=
.
2 V0
2 Mt ⋅ g ⋅ XL
Figure 24 shows 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 to
2
p L3 /max
and the product δ hωh ∝ p L max . This indicates that the cylinder-load response will
show less oscillations when the max load pressure is increased. The system response for
different pLmax is illustrated in Figure 25.
K-E Rydberg
Feedbacks in Electro-Hydraulic Servo Systems
15
Figure 25: Response of the cylinder-load dynamics with cylinder design for max load pressure of 100,
200 and 300 bar respectively
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, even
with 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 delay
between the derivative of the command signal and the output signal. This delay can be
reduced to a minimum by use of a feed forward gain.
The action of the PID-controller means that the derivative gain increases proportionally
to the frequency. In spite of this behaviour it is important to reduce the gain of the Daction at high frequencies. Otherwise, the high frequency disturbances on the signals
will be amplified to a level which can mainly influence the function of the system. A
forward loop filter is 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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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 the
command 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 servo
control 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 “simulinkmodel” 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 feed
forward 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 (δh = 0.155) in
the 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 output
signal (Y) versus command signal (X), as illustrated in Figure 4.
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Figure 4: Output versus command signal without (to the left) and with feed forward gain in
a 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 Proportional–Integral–Derivative controller (PID controller) is a control loop
feedback 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.
P_action
k={1}
Output_Y
Input_U
I_action
I
startTime={0.2}
Sum
+1
+1
Saturation
+
+1
k={3}
uMax={2}
D_action
DT1
k={0}
t
Y (t ) = K P ⋅ U (t ) +
1
dU (t )
U (τ )dτ + TD
∫
TI t0
dt
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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system response by boosting the effect of the error signal. However, too much
proportional gain can cause the system to become unstable.
Command signal
Too low gain
Output signal
Optimal gain
Too high gain
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 at
high-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.
Command signal
Output signal
Optimal gain
High gain
Too high gain
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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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.
Overshoot
Ringing
Low rate
Optimum rate
Figure 10: Effects of derivative gain
Too much rate