Feedbacks in Hydraulic Servo Systems Rydberg

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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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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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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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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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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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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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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______________________________________________________________________

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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______________________________________________________________________

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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______________________________________________________________________

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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______________________________________________________________________

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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Feedbacks in Hydraulic Servo Systems Rydberg