Servo Plant Compensation Techniques potx

11
Servo Plant Compensation
Techniques
Servo compensation usually implies that some type of filter network such as
lead/lag circuits or proportional, integral, or differential (PID) algorithms
will be used to stabilize the servo drive. However, there are other types of
compensation that can be used external to the servo drive to compensate for
other things in the servo plant (machine) that can, for example, be structural
resonances or nonlinearities such as lost motion or stiction. These machine
compensation techniques are shown in Figure 1 and are valid for either
hydraulic or electric servo drives.
11.1 DEAD-ZONE NONLINEARITY
Stiction, sometimes referred to as stick-slip, occurring inside a positioning
servo, can result in a servo drive that will null hunt. The definition of a null
hunt is an unstable position loop that has a very low periodic frequency such
as 1 Hz or less with a small (a few thousandths) peak-to-peak amplitude
(limit cycle). The most successful way to avoid stiction problems is to use
antifriction machine way (rollers or hydrostatics) or use a way linear
material that has minimal stiction properties. If stiction-free machine slide
ways cannot be provided, the use of a small dead-zone nonlinearity placed
inside the position loop, preferably at the input to the velocity servo, has
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
had some success in overcoming a null hunt problem. However the dead
zone must be very small (e.g., 0.001 in.); otherwise, the servo drive will have
an instability from too much lost motion. A simple analog dead-zone
nonlinear circuit is shown in Figure 2. The same function can be provided
with a digital algorithm in computer control of machines.
11.2 CHANGE-IN-GAIN NONLINEARITY
In some industrial servo drives it is a requirement to position to a very low
feed rate to obtain a smooth surface finish. This requirement usually occurs
Fig. 1 Servo plant compensation techniques.

Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
in a turning machine application. At feed rates below 0.01 ipm, the
requirement for a smooth surface may not be easily attainable because the
servo drive may have a cogging problem at these low federates. Increasing
the forward loop gain to the velocity drive can overcome the lowfeed
cogging problem but will result in an unstable servo drive. As a compromise,
a change in gain nonlinear circuit can be used to improve the low-feed-rate
smoothness and still have a stable servo drive. The object is to have a high
forward-loop gain in the velocity servo (which is inside a position loop). For
normal operation, the high servo loop gain is reduced by the change-in-gain
circuit at a low velocity to its normal gain, thus maintaining a stable servo
drive. This type of nonlinear circuit has been used successfully for smooth
Fig. 2 Dead-zone nonlinearity.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
feed rates down to 0.001 ipm. The analog version of a change-in-gain
nonlinearity is shown in Figure 3. With digital controls a digital algorithm
can be used.
11.3 STRUCTURAL RESONANCES
Structural resonances or machine dynamics, as it often referred to, is
certainly not a new subject. However, on the morning of November 7, 1940,
the nation awoke to the destruction of the Tacoma Narrows Bridge. A
42 mile-per-hour gale caused the bridge to oscillate thus exciting the
structural resonances of the bridge to a final destruction frequency of about
14 Hz and a peak-to-peak amplitude of 28 ft. The destruction of the bridge
was a wake-up call to the importance of dynamic analysis in structural
Fig. 3 Change-in-gain nonlinearity.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
design in addition to static analysis and design. Some sixty years later the
technology of dynamic analysis is now well known.
To further investigate machine resonances, a typical linear industrial

servo drive can be represented as in Figure 4. The mechanical components
of this servo drive are referred to as the servo system plant. The servo plant
may have a multiplicity of resonant frequencies resulting from a number of
degrees of freedom. In actual practice there will be some resonant
frequencies that are high in frequency and far enough above the servo
drive bandwidth so that they can be ignored. In general there will be a
predominant low resonant frequency that could possibly be close enough to
the servo drive bandwidth to cause a stability problem. Therefore a single
degree of freedom model as shown in Figure 5 can represent the
predominant low-resonant frequency, where:
B
L
¼viscous friction coefficient (lb-in min/rad)
T ¼driving torque, developed by the servo motor (lb-in.)
K ¼mechanical stiffness of the spring mass model (lb-in./rad)
J
M
¼inertia of the motor (lb-in sec
2
)
J
L
¼inertia of the load (lb-in sec
2
)
S ¼laplace operator
From Newton’s second law of motion, the classical equations for this servo
plant (industrial machine system) can be written. In most industrial
machines it can be assumed that the damping B
L

is zero. Therefore the
Fig. 4 Block diagram of a machine feed drive.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
motor equation is:
T
M
¼ J
M
s
2
y
M
þ Kðy
M
À y
L
Þ (11.3-1)
Also the load equation is:
0 ¼ J
Ls
y
L
þ Kðy
L
À y
M
Þ (11.3-2)
Solving for the motor position y
M
and the load position y

L
:
ðJ
M
s
2
þ KÞy
M
¼ T
M
þ y
L
(11.3-3)
y
M
¼
T
M
þ Ky
L
J
M
s
2
þ K
(11.3-4)
and
ðJ
L
s

2
þ KÞy
M
¼ Ky
M
(11.3-5)
y
L
¼
Ky
M
J
L
s
2
þ K
(11.3-6)
Fig. 5 Machine slide free-body diagram.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Further solving for y
M
and y
L
by combining Eq. (11.3-4) and (11.3-6):
y
M
¼
T
M
þ KðKy

M
=J
L
s
2
þ KÞ
J
M
s
2
þ K
(11.3-7)
y
M
¼
T
M
ðJ
L
s
2
þ KÞþK
2
y
M
ðJ
M
s
2
þ KÞðJ

L
s
2
þ KÞ
(11.3-8)
y
M
¼
T
M
ðJ
L
s
2
þ KÞ
ðJ
M
s
2
þ KÞðJ
L
s
2
þ KÞ
þ
K
2
y
M
ðJ

M
s
2
þ KÞðJ
L
s
2
þ KÞ
(11.3-9)
y
M
À
K
2
y
M
ðJ
M
s
2
þ KÞðJ
L
s
2
þ KÞ
¼
T
M
ðJ
L

s
2
þ KÞ
ðJ
M
s
2
þ KÞðJ
L
s
2
þ KÞ
(11.3-10)
ððJ
M
s
2
þ KÞðJ
L
s
2
þ KÞÀK
2
Þy
M
¼ T
M
ðJ
L
s

2
þ KÞ (11.3-11)
y
M
¼
T
M
ðJ
L
s
2
þ KÞ
ðJ
M
s
2
þ KÞðJ
L
s
2
þ KÞÀK
2
(11.3-12)
y
M
¼
T
M
ðJ
L

s
2
þ KÞ
J
M
J
L
s
4
þ KðJ
M
þ J
L
Þs
2
(11.3-13)
y
M
¼
T
M
ðJ
L
s
2
þ KÞ
s
2
ðJ
M

J
L
s
2
þðJ
M
þ J
L
ÞK
(11.3-14)
y
M
¼
T
M
ðJ
L
s
2
þ KÞ
s
2
ðJ
M
þ J
L
ÞððJ
M
J
L

=J
M
þ J
L
Þs
2
þ KÞ
(11.3-15)
Let:
J ¼ J
M
þ J
L
J
P
¼ J
M
J
L
=ðJ
M
þ J
L
Þ
y
M
¼
T
M
ðJ

L
s
2
þ KÞ
s
2
Jðs
2
J
P
þ 1Þ
(11.3-16)
Also:
y
L
¼
Ky
M
J
L
s
2
þ K
(11.3-17)
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
y
L
¼
K
ðJ

L
s
2
þ KÞ
6
T
M
ðJ
L
s
2
þ KÞ
s
2
Jðs
2
J
P
þ KÞ
(11.3-18)
y
L
¼
T
M
K
s
2
Jðs
2

J
p
þ KÞ
(11.3-19)
y
L
T
M
¼
1
ðJ=KÞs
2
ðs
2
ðJ
p
=KÞþ1Þ
(11.3-20)
y
L
T
M
¼
1
ðJ=KÞs
2
ððs
2
=o
2

r
Þþ1Þ
(11.3-21)
o
r
¼ load resonant frequency ¼
ffiffiffiffiffi
K
J
p
s
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
KðJ
M
þ J
L
Þ
J
M
J
L
s
(11.3-22)
From a practical point of view industrial machines and their servo drives
(hydraulic and electric) are to this day still subject to resonant frequency
stability problems. Most industrial servo drives use an inner velocity servo
inside a position servo loop. Hydraulic servo drives have the added variable
of hydraulic fluid resonance, which can be a limiting factor of stability. The
hydraulic resonance o

r
can be observed as a typical second order response
in the Bode frequency response of Figure 6. For hydraulic drives having a
low damping factor d
h
, the resonant peak may be higher than 0 dB gain,
which will result in a resonant oscillation. There are a number of methods to
compensate for this resonant oscillation. First, a small cross-port damping
hole of about 0.002 in. can be used across the motor ports. Secondly, the
velocity loop differential compensation can be varied, which quite often
eliminates the oscillation. Lastly, the velocity loop gain could be lowered,
which can also lower the velocity servo bandwidth. As an index of
performance (I.P.) the hydraulic resonance should by proper sizing be above
200 Hz, and the separation between the velocity servo loop bandwidth o
c
and the hydraulic resonance o
h
should be three to one or greater. Brushless
DC electric drives do not usually have velocity loop resonance problems
unless a more compliant coupling is used internally in the motor to couple a
position transducer to the motor shaft.
Both hydraulic and brushless DC electric drives can have resonance
(stability) problems if the machine is included in the position servo loop.
This is an ongoing problem with industrial machines, in spite of all the
available technology to minimize stability problems. A typical position
servo Bode frequency response is shown in Figure 7. As a figure of merit the
separation between the velocity loop bandwidth o
c
and the position-loop
velocity constant K

v
(gain) should be two to one or greater. The machine
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
resonance o
r
should be at least three times greater than the velocity servo
bandwidth o
c
. However in actual practice the machine resonance inside the
position loop is often quite low (such as 15 Hz). The resonant peak in this
case should be above 0 dB gain, resulting in a resonant oscillation. There are
a number of control techniques that can be applied to compensate for
machine structural resonances that are both low in frequency and inside the
position servo loop. The first control technique is to lower the position-loop
gain K
v
(velocity constant). Depending on how low the machine resonance
is, the position-loop gain may have to be lowered to about 0.5 ipm/mil (8.33/
sec). This solution has been used in numerous industrial positioning servo
drives. However, such a solution degrades servo performance. For very
large machines this may not be acceptable. The I.P. that the servo loop gain
(velocity constant) should be lower than the velocity servo bandwidth by a
factor of two, will be compromised in these circumstances.
A very useful control technique to compensate for a machine
resonance is the use of notch filters. These notch filters are most effective
Fig. 6 Hydraulic velocity servo.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
when placed in cascade with the input to the velocity servo drive. These
notch filters should have a tunable range from approximately 5 Hz to a
couple of decades higher in frequency. The notch filters are effective to

compensate for fixed structural resonances. If the resonance varies due to
such things as load changes, the notch filter will not be effective. Since most
of these unwanted resonant frequencies are analog sinusoidal voltages, a
notch filter can effectively remove these frequencies.
In digital control the algorithm for the notch filter can be used. A
simple analog notch filter is shown in Figure 8 as it appeared in Electronics
Magazine, December 7, 1978. This filter is equivalent to a twin T-notch filter
but it is an active filter versus passive networks, so there are no signal losses.
The frequency of the notch is set by the selection of resistor R. For a
1-microfarad (mF) capacitor (C), the values for R versus the notch frequency
are shown in Figure 9.
The depth of the notch is adjusted by varying the potentiometer P
1
.
Frequency responses of the notch filter for values of R ranging from 40 K
ohms to 200 ohms are shown in Figure 10. A 40-Hz notch filter frequency
response is shown in Figure 11 with a number of potentiometer settings to
Fig. 7 Position loop frequency response.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
show the change in the depth of the notch filter as the potentiometer is
varied.
A few case histories are of interest. In a hydraulic servo valve feed
drive, pump pulsations of 500 Hz traveled through the machine piping to the
servo valve, the hydraulic motor, and finally the feedback tachometer of the
velocity loop. The high sensitivity of the tachometer (100 V/1000 rpm)
sensed the 500-Hz vibration and generated this voltage into the servo drive
electronics, where it was amplified through the entire drive, causing an
undesirable vibration. A 500-Hz notch filter at the tachometer output feed
to the servo amplifier eliminated the vibration problem.
In another case, the switching frequency of a numerical control

(125 Hz) beat with the single-phase, full-wave DC SCR drive frequency
(120 Hz) producing a 5-Hz signal that appeared in the machine servo drive,
causing what appeared to be a 5-Hz instability. Using a notch filter
frequency of the control switching frequency, the 5-Hz beat frequency signal
was eliminated.
In another case history a 45-Hz resonance existed in an air bearing of a
rotary position feedback transducer. Once this resonance was excited it was
amplified through the electronics drive and the machine slide vibrated at
45 Hz.
A 45-Hz notch filter applied at the input to the velocity drive
eliminated the problem. There are numerous possibilities where unwanted
Fig. 8 Notch filter circuit. (Reprinted with permission from Penton Publishing.)
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 9 Notch filter resistance nomogram.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 10 Notch filter characteristics.
Fig. 11 Depth of notch filter characteristics.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
signal frequencies appear in industrial servo drives. This simple notch filter
can readily be used to eliminate all kinds of undesirable signals.
11.4 FREQUENCY SELECTIVE FEEDBACK
Another technique that has been very successful with industrial machines
having low-frequency machine resonances, is known as ‘‘frequency selective
feedback’’. In abbreviated form it requires that the position feedback be
located at the servo motor eliminating the mechanical resonance from the
position servo loop, resulting in a stable servo drive but with significant
position errors. These position errors are compensated for by measuring the
slide position through a low-pass filter; taking the position difference
between the servo motor position and the machine slide position; and
making a correction to the position loop, which is primarily closed at the

servo motor.
Compensator Operation
The complete control circuit has more complexity than comparing two
position transducer outputs. Figure 12 can be used to discuss the actual
operation of the compensating system. This particular compensating scheme
used an instrument servo to perform the compensating function. A software
based frequency selective feedback system will also be discussed. For this
compensating system, the machine-feed servo drive uses a position
measuring feedback resolver (1) connected electrically in series with a
correction resolver (5). Any correction required during positioning is
introduced into the numerical control feedback circuit with the correction
resolver (5).
The compensator circuit includes the positioning servo-motor position
measuring resolver called a compensator feedback resolver (2), a machine
slide linear position measuring transducer (3), and an instrument type
correction servo drive. The difference between the feed servo-drive motor
position and the machine slide position is measured with the compensator
feedback resolver (2) and the linear resolver (3). However, an additional
correction resolver (4) is included in the circuit. Therefore, the instrument
correction servo error is a function of three resolver positions. This is shown
in the block diagram of Figure 13. The resolver positions are shown as angle
y. The total correction error is shown as a function of the three resolver
positions
y
c
¼ y
m
À y
c
À y

s
: (11.4-1)
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 13 Simplified block diagram for frequency selective feedback.
Fig. 12 Analog block diagram for frequency selected feedback.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
As a position error is developed between the feed servo-drive motor
position y
m
and machine slide position y
s
, an error is developed at the
instrument servo drive (Figure 13). The error is also a function of y
c
.
y
e
¼ y
m
À y
c
À y
s
¼ðy
m
À y
s
ÞÀy
c
(11.4-2)

It is significant to note that the bandwidth of the instrument correction
servo must be a low frequency such as 1.5 Hz. This is required to eliminate
the machine structural dynamics from the machine slide feedback position
ðy
s
Þ and is the key to the frequency selective feedback control function.
A correction y
c
is developed at the output of the instrument servo
drive, which is a function of:
y
c
¼ G
1
½ðy
m
À y
s
ÞÀy
c
 (11.4-3)
The correction y
c
is added to the main-feed servo drive by means of resolver
(5). The correction y
c
causes the feed servo drive to move by the amount of
the correction. Therefore both y
m
(at the motor) and y

s
(at the machine slide)
move by the amount of the correction y
c
. The correction can be shown
mathematically to be approximately a function of ðy
m
À y
s
Þ as follows:
Since: y
c
¼ G
l
½ðy
m
À y
s
ÞÀy
c
 from Eq: (11.4-3)
y
c
þ y
c
G
l
¼ðy
m
À y

s
ÞG
l
(11.4-4)
y
c
¼
ðy
m
À y
s
ÞG
1
ð1 þ G
1
Þ
(11.4-5)
y
c
¼ðy
m
À y
c
Þ
1
1=G
1
þ 1
(11.4-6)
1=G

1
5 much smaller than 1
Therefore: y
c
%ðy
m
À y
s
Þ (11.4-7)
The correction can also be shown graphically in Figure 14. For
illustration the relation of the feed servo-drive position y
m
to the machine
slide position y
s
will remain constant during the correction. Figure 14 shows
that the machine slide position y
s
moves by the amount of the correction.
The feed servo-drive motor position moves by the same amount.
From another point of view the instrument servo-drive error must be
reduced to zero after the correction is made. From Eq. (11.4-2) the error is:
y
e
¼ðy
m
À y
s
ÞÀy
c

(11.4-2)
Assuming the initial condition at the start of the correction is such that the
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
machine slide position y
s
is short of the correct position (assume 10 in.) by
0.01 in. and that the location of y
m
and y
s
are as follows:
y
m
¼ a related distance of 10 in.
y
s
¼ a related distance of 9:99 in:
After correction the related positions will be—
y
m
¼ a related distance of 10:01 in :
y
s
¼ a related distance of 10 in:
The relation between y
m
and y
s
is the same but the machine position y
s

has
moved to the correct position.
Fig. 14 Correction diagram for frequency selective feedback.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Meanwhile the correction y
c
(which was equivalent to 0.01 in.) had to
be included in the error of the instrument servo. Since the relation of ðy
m
À
y
s
Þ remains constant, a correction must be made in the instrument servo
loop to reduce the error to zero:
y
c
¼ 0 ¼ðy
m
À y
s
ÞÀy
c
(11.4-8)
The compensator correction resolver (4) serves the purpose to reduce the
instrument servo error to zero.
As the correction process becomes a continual process, as in normal
machine operation, the resolver (4) will continually be in motion to reduce
the error to zero in the correction loop of the instrument servo. At machine
traverse feeds the correction will not be effective, but this is not important
since there will not be any machining operations at the traverse feeds.

Software version of frequency selective feedback
The software version for the correction is shown in Figure 15. This version
can be added to a machine axis that exhibits a structural resonance problem.
The drive is assumed to be a typical commercial electric servo drive with a
current loop and a velocity loop inside a position servo loop.
The difference between the motor position and the machine slide
position is used as an input to a low-pass digital filter. This filter has a very
low bandwidth of about 1.5 Hz. The reason for this is to remove structural
dynamic frequencies from the correction process.
Fig. 15 Software version of frequency selective feedback.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
The machine axis prior to adding frequency selective feedback, is
assumed to have a position transducer on the machine slide. To add
frequency selective feedback, an instrument gearbox must be added to the
rear of the servo motor. A position transducer such as a resolver will be
geared to the motor shaft with a ratio determined by the resolution of the
machine slide feedback transducer.
11.5 FEEDFORWARD CONTROL
The problem of developing an industrial servo drive with high-performance
capabilities for accurate positioning is a subject of much importance. On
multiaxis industrial machine servos using classical type 1 servo control, it is
a requirement that each machine axis have matched position-loop gains to
maintain accuracy in positioning. Quite often this means that all machine
axis servo drives must have their position-loop gains K
v
adjusted to the
poorest performing axis. Consider the basic approach to the design of a
poisoning servo drive illustrated in Figure 16.
This is the classical type 1 servo, which exhibits characteristic errors e
in position that are well known for various inputs y

i
.
Consider a simplified block diagram of the system where G
ðsÞ
is the
servo drive and inner-loop transfer function typically of the following form:
G
ðsÞ
¼
K
v
sF
ðsÞ
(11.5-1)
F
ðsÞ
is a polynomial that represents the dynamics of the servo drive and
servo plant. What is really desired of the servo of Figure 17 is that y
i
¼ y
0
under all conditions. That is, for any y
i
; e ¼ 0. Clearly, this is not possible
for a type 1 servo described by Figure 17.
Fig. 16 Type 1 servo block diagram.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Consider, then, a compensator G
c
that could possibly produce a

system response as desired, placed cascade with the system as shown in
Figure 18.
Since
y
oðsÞ
y
iðsÞ
¼ G
cðsÞ
G
ðsÞ
1 þ G
ðsÞ
¼ 1 (11.5-2)
Thus : G
cðsÞ
¼
1 þ G
ðsÞ
G
ðsÞ
¼
1
G
ðsÞ
þ 1 (11.5-3)
Thus, the desired system may be represented in block diagram form as
Figure 19.
It is convenient to rearrange the diagram of Figure 19 so that the
actual error term e ¼ y

i
À y
o
appears, as Figure 20.
As pointed out by Eq. (11.5-1)
G
ðsÞ
¼
K
v
sF
ðsÞ
for a practical position servo: (11.5-4)
Thus the block diagram of Figure 20 becomes Figure 21.
Note that y
iðsÞ
sF
ðsÞ
K
v
¼ o
iðsÞ
F
ðsÞ
K
v
(11.5-5)
Where o
iðtÞ
¼

dy
iðtÞ
dt
(11.5-6)
Furthermore, a well-designed, high-performance servo G
ðsÞ
should have
dynamics described by F
ðsÞ
sufficiently negligible with respect to the
integration crossover that F
ðsÞ
need not be synthesized in the forward
compensating path. This assumption is important to the simplicity of the
concept, but it is possible to check if it is valid in any specific application.
Fig. 17 Simplified position-loop block diagram.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Fig. 20 Zero-error feedforward block diagram.
Fig. 18 Compensator with position servo.
Fig. 19 Block diagram with feedforward.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Because the inverse of a type 1 system differentiates rather than
integrates the compensating path is simply a velocity signal rather than
a position signal. Consider the final design of the system described in
Figure 22.
From Figure 22
y
oðsÞ
¼ y
iðsÞ

À y
oðsÞ
þ
o
iðsÞ
K
v

K
v
sF
ðsÞ
(11.5-7)
y
oðsÞ
1 þ
K
v
sF
ðsÞ

¼ y
iðsÞ
þ
o
iðsÞ
K
v

K

v
sF
ðsÞ
(11.5-8)
y
oðsÞ
¼
ðK
v
y
iðsÞ
þ o
iðsÞ
Þ
ðsF
ðsÞ
þ K
v
Þ
(11.5-9)
Fig. 21 Complete feedforward block diagram.
Fig. 22 Zero-error position servo.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
Equation (11.5-9) may be used to illustrate how in the ideal case (where
F
ðsÞ
¼ 1) the system is zero error in position regardless of the input.
y
i
¼ Ct therefore y

iðsÞ
¼
C
s
2
(11.5-10)
o
i
¼ C o
iðsÞ
¼
C
s
(11.5-11)
Therefore
y
oðsÞ
¼
K
v
C=s
2
þ C=s
sF
ðsÞ
þ K
v
¼
ðs þ K
v

ÞC
s
2
ðsF
ðsÞ
þ K
v
Þ
(11.5-12)
Or for F
ðsÞ
&1, the lag-lead cancel and
y
oðsÞ
¼
C
s
2
¼ y
iðsÞ
therefore zero error: (11.5-13)
Consider next the steady-state error for the constant velocity case:
e
ðsÞ
¼ y
iðsÞ
À y
oðsÞ
¼
C

s
2
À
C
s
2
s þ K
v
sF
ðsÞ
þ K
v

(11.5-14)
e
ðsÞ
¼
C
s
2
1 À
s þ K
v
sF
ðsÞ
þ K
v

(11.5-15)
Applying the final value theorem, the steady-state position error is:

eð?Þ¼lim se
ðsÞ
¼
lim
s?0
C
s
1 À
s þ K
v
sF
ðsÞ
þ K
v

(11.5-16)
¼
lim C
s?0
1 À
s þ K
v
sF
ðsÞ
þ K
v

1
s
(11.5-17)

¼
lim C
s?0
sF
ðsÞ
þ K
v
À s À K
v
sF
ðsÞ
þ K
v

1
s
(11.5-18)
¼
lim C
s?0
F
ðsÞ
À 1
sF
ðsÞ
þ K
v

(11.5-19)
However, for all possible forms of F

ðsÞ
lim
s?0
F
ðsÞ
¼ 1
Thus: eð?Þ¼0
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved
For constant acceleration input:
y
iðtÞ
¼
1
2
at
2
y
iðsÞ
¼
a
s
3
(11.5-20)
o
iðtÞ
¼ at o
iðsÞ
¼
a
s

2
(11.5-21)
From Eq. 11.5-9
y
iðsÞ
¼
K
v
a=s
3
þ a=s
2
sF
ðsÞ
þ K
v
¼
a
s
3
s þ K
v
sF
ðsÞ
þ K
v
(11.5-22)
And eð?Þ¼
limse
ðsÞ

s?0
¼
a
s
2
sF
ðsÞ
þ K
v
À s À K
v
sF
ðsÞ
þ K
v

(11.5-23)
eð?Þ¼
lim
s?0
a
s
ðF
ðsÞ
À 1Þ
ðsF
ðsÞ
þ K
v
Þ

(11.5-24)
eð?Þ¼
alim
s?0
F
ðsÞ
À 1
s
2
F
ðsÞ
þ K
v
s
¼
alim
s?0
dF
ðsÞ
=ds
ds
2
F
ðsÞ
=ds þ K
v
(11.5-25)
Then
lim
s?0

dF
ðsÞ
ds
?finite value (11.5-26)
While
limit
s?0
ds
2
F
ðsÞ
ds
?0 (11.5-27)
Thus eð?Þ¼
1
K
v
(11.5-28)
The magnitude of the finite position error will depend in this case on the
coefficient of the s term of F(s). If b is this coefficient, then
eð?Þ¼
ab
K
v
(11.5-29)
The compensator technique described holds excellent potential for provid-
ing an outstanding servo drive for the industrial machine. When properly
executed in a well-designed position servo it should virtually eliminate
velocity lag errors and reduce acceleration lag errors to low levels. The
velocity feedforward approach will eliminate position errors for constant

velocity moves on a machine axis if the machine dynamics represented by
F
ðsÞ
in the feedforward term F
ðsÞ
=K
v
exactly respresents the G(s) term of the
forward position loop. Matching of position-loop gains K
v
on all axes will
not be required. For the case of acceleration and deceleration the velocity
feedforward approach will not be effective. For these situations commercial
servo-drive manufacturers will use an additional technique of acceleration
feedforward.
Copyright 2003 by Marcel Dekker, Inc. All Rights Reserved