Bode Plot based Auto-Tuning –
Enhanced Solution for High Performance Servo Drives
J. O. Krah
Danaher Motion GmbH
Wachholder Str. 40-42 40489 Düsseldorf Germany
Email: j.krah@danaher-motion.de Tel. +49 203 9979 133 Fax. +49 203 9979 200
Abstract
This paper describes a new self-tuning method using
an automatic bode plot based filter-configuration
algorithm. Most existing auto-tuning algorithms are
based on simple load inertia estimation. They work
fine, if the user more or less does not need them, but
they struggle in more complex systems. The described
approach is also targeting complex and difficult
problems, which usually appear in low damped direct
drive machines with significant motor inertia to load
inertia mismatches [1].
Introduction
High performance servo drives are still a fast growing
market segment. The brush-less technology offers
significant advantages in terms of reliability and
motor size. The higher drive complexity is covered
with additional logic and new power electronic
components. Due to the innovation cycles of the
semiconductor suppliers the size and the cost of the
more and more complex drives did not increase. Each
new drive generation offers more performance, more
flexibility and higher integration density [2,3].
The servo drive component cost will continually
decrease in the near future. On the other side the
mechanical design is more and more cost and weight
optimized, which results in less stiff constructions.
Therefore the installation and set-up time (= cost) is a
steady growing issue. A reliable self-tuning algorithm
can help to decrease the set-up time.
Control structure
A very common control architecture in the motion
control is the cascaded control structure. The most
inner loop is also the fastest loop, the current control.
The next outer loop is the velocity control. The
slowest and most outer loop is the position control.
Due to feed forward and other extensions like
observers the performance of the system can be
significant improved [4]. At least the feed forward
does not change control loop stability. Therefore the
cascaded control structure is the base for the
developed self-tuning algorithms.
382 • PCIM EUROPE 2004 • PROCEEDINGS
The current control loop depends mainly on the drive
internal delay times and on the motor winding
inductance and resistance. It is very common that
motor and drive are selected from one supplier. This
vendor should also provide the current control
parameter set [5].
Due to the fact that the position is just the integration
of the speed the position loop gain depends mainly on
the achieved velocity loop bandwidth.
Tuning the velocity loop is the real task. Here we find
the motor inertia and the load inertia connected via
compliant mechanics. The system contains also
friction and backlash. This can result in complex
resonance issues. Friction for example can help to
damp the behavior. In systems with low mechanical
damping the drive has to provide electronic damping
by using filters in the current command path.
Compliantly coupled load
The load and the motor are two independent inertias
connected by non-rigid components. The equivalent
spring constant of the entire transmission is c,
illustrated in Figure 1. It describes the torque
produced by a position difference between motor and
load. The viscous damping term d describes the
torque, which is produced by the velocity difference
of motor and load.
An ideal servo system would follow the
position/velocity command without delay and
following error. This would require an extreme high
loop gain. In real systems the gain is limited. A too
high gain produces signal overshoot or excessive
ringing. A very easy and common way to set-up and
tM
c, d
JM
JL
Figure 1: Compliantly coupled motor and load
check the gain is reviewing a velocity loop step
response scope plot, figure 2. This is perfect to check
the result, but it is more or less tuning by trial and
error. If more then one parameter needs to be
determined only a real servo expert has a chance to
get an optimal result. There is no direct way to see the
actual valid stability margin to system rigging.
Reviewing the velocity command to measured
velocity Bode plot gives easier to use information.
This is called closed loop Bode plot and is shown in
figure 3. We can read directly the peaking (output
amplitude is higher than input amplitude) and the
bandwidth (90° or –3dB) of the velocity loop. These
are key performance indices of the velocity control
loop. The closed loop Bode plot allows an easier
interpretation of the information than the pure step
response, but a straightforward gain setting is still not
possible. The tuning process is still more or less trial
and error.
Using the velocity error to measured velocity Bode
plot offers more information, figure 4. This is called
open loop bode plot. Using this graph the gain margin
and the phase margin are directly visible. Knowing
the selected margins we can directly calculate how
much the gain should be changed (x dB) to get the
desired tuning. This is very practical with simple
single inertia systems. Most real systems are not that
simple. Often we find compliant coupled two or more
mass systems, sometimes with significant friction
and/or some backslash.
This is easy to see in the open loop bode plot. The
frequency related gain reduction is requested, some
phase lag comes with it. It is rather often that in a real
system placing a filter in front of the current
controller can help to increase the gain margin. Sizing
that filter is not an easy job. In general there are three
common ways to get the current set-point filter
parameters:
Figure 2: Velocity loop: step response scope plot
AKM 42 with ServoStar 300 – no load
Filters in motion control systems
There are two main reasons why current set-point
filters are used in motion control systems:
• Reduce noise from a low resolution feedback
system
• Increase gain and stiffness of compliant load
Low-resolution encoder and resolver-based systems
with high velocity loop gains generates audible noise
in the motor due to the noise on top of the current
command. Simple low pass filters can reduce this
noise. A more advanced solution is a Luenberger
observer to estimate a velocity signal with less noise
[4].
To suppress oscillations of compliant coupled load
filters are used to increase gain in the lower
frequency area for more stiffness without decreasing
the gain margin at the first phase crossover. To
control a motor without load an additional filter is
usually not necessary. The behavior of a good
servomotor is like a simple integration (torque
(~current) -> speed). Due to the compliant coupling
with an additional inertia we can get a system with
torsional oscillations. The ideal filter would be the
inverse transfer function of the compliant coupling
[6].
Figure 3: Velocity loop: closed loop bode plot
AKM 42 with ServoStar 300 – no load
Figure 4: Velocity loop: open loop bode plot
AKM 42 with ServoStar 300 – no load
(controller integral part is switched off)
PCIM EUROPE 2004 • PROCEEDINGS • 383
1.
2.
3.
Set the filter parameters by experience of a servo
expert
Set the filter parameters by more or less trial and
error
Use the measured or calculated inverse transfer
function to set the filter
Most applications are not just compliant two mass
systems. We find often compliant systems with
distributed mass or multiple mass configurations.
Here is the calculation or the estimation of the inverse
transfer function with an acceptable order very
difficult or not possible. For these applications setting
the filters is the real task. To get an understanding of
the behavior of these systems we can look at an ideal
two mass system. It is always assumed that the
feedback system is mounted on the motor side as
usual in the industry.
Detailed analyses of a two mass system in the
frequency domain
In the following chapter the motor inertia is always
constant. The load inertia and the spring constant are
the characterizing variables. In the industry it is well
known that a good approach is matching motor inertia
with load inertia (1:1). The coupling should be as stiff
as possible.
Figure 5 shows a Bode plot of a low damped non
inertia-matched system with JL ~ 9JM. Three different
sections characterize the plot:
• In the frequency range well below the antiresonant frequency the plant acts like a scaled
(low frequency) inertia with J = J L + J M . The
frequency of the anti-resonance (zero) is defined
by:
c
1
(1)
f AR =
Hz
2π J L
•
•
In the frequency range well above the low
damped resonant frequency the plat acts like a
scaled inertia with only the value of the motor
(high frequency) inertia J = J M . The frequency
of the resonance (pole) is defined by:
c
1
(2)
fR =
2π JJLL+JJMM
The inertia ratio, the spring constant and the
system damping are defining the plot in the
transition area of the resonant frequencies.
The transfer function can be formed in a two-part
notation [6]:
V (s )  1  J M

=
T (S )  J M s  J L + J M

J s 2 + ds + c 
s 2 + ds + c 
L
JLJM
JL +JM
(3)
The term on the left is just the motor behavior and the
term on the right is the effect of the compliant
384 • PCIM EUROPE 2004 • PROCEEDINGS
Figure 5: Bode plot of motor / load plant with
coupled load [6]
black: motor only
red: motor with ideal coupled load
blue: motor with compliantly coupled
load (low damped)
coupled load. An ideal filter would be similar or equal
to the inverse function of the right part.
The high frequency gain ( s = ∞ ) of the right part is
one and the low frequency gain (s = 0) is
JM
(4)
G (s = 0) =
JL + JM
Figure 6 shows the low frequency gain drop as a
function of the load to inertia ratio. This gain drop
results also in a bandwidth drop. Figure 7 shows the
resonance frequencies as a function of the load to
motor inertia ratio. The anti-resonance is strictly
moving with the square root of the inertia ratio. The
resonance frequency is only slightly moving.
fR
=
f AR
JL + JM
JM
(5)
In a servo system with matching inertia ( J M = J L ) the
resonance frequency is 2 higher than the antiresonance frequency. If the load inertia is significant
higher than the motor inertia, the anti resonance
frequency will drop down. To achieve an appropriate
bandwidth a stiffer coupling c is required (for
example a direct drive solution).
By comparing the anti-resonance (nominator from the
second part of equation 3) with the standard
oscillation equation:
+10
+10
0
0
F (ω )
∆F (ω << ω 0 )
-20
- 17 dB
-20
265 Hz
-30 dB
-30 dB
38 Hz
-40
-40
0.01
0.1
1
JL/JM
10
100
Figure 6: Low frequency “gain drop” as function of
the load to inertia ratio
ϕ
0
-90
-180
-270°
1
10
100
f
1 k Hz
10 k
Figure 9: Bode plot of a first order and second order
lag filter fz = 38 Hz, fp = 265 Hz
and DAR=DR=0.5
resonance
100
- 34 dB
10
JL/JM
anti-resonance
1
0.1
0.1
0.3
1
f / f0
3
coupled two mass system are:
• Low gain at “low” frequencies < fAR
• High gain at “high” frequencies >fR
• Phase lead between fAR and fR (figure 5)
• Significantly influenced by the inertia ratio
10
Figure 7: resonance / anti-resonance frequency as
function of the load to motor inertia ratio
ω 02Z + 2 DZ ω 0 Z s + s 2 = 0
we get for the anti-resonance damping:
(6)
DZ = d 2 J L c
(7)
(
)
The desired filter to compensate a two inertia
compliantly coupled system should own high gain at
low frequencies and low gain at high frequencies.
The phase lag between the corner frequencies should
be less than the phase lead of the system in this area.
If the gain drop at low frequencies is low (< 15 dB)
and the system is well damped a two parameter
( ω Z , ω P ) first order lag filter is a good approach:
s + ωZ ωP
(10)
F1 (s ) =
; ω = 2π f
s + ωP ωZ
In case of higher inertia mismatches – which results in
a higher gain drop – or a low damped system a four
parameter ( ω Z , ω P , DZ , D P ) second order lag filter
(bi quad) could be better fitting:
2
s 2 + 2 DZ ω Z s + ω Z2 ω p
(11)
F2 (s ) = 2
s + 2 DPω P s + ω P2 ω Z2
Figure 9 shows two Bode plots of these filters.
Setting these four parameters without
experience or guideline just by trial and
error is nearly impossible.
An optimization criterion is to
maximize the closed loop bandwidth
with a stable well-damped behavior.
The Nyquist stability criteria can help
to determine that parameter set.
and the resonance damping:


JLJM
(8)
D P = d  2
c 
J
+
J
L
M


Interesting is the relation between these equations:
DP
=
DZ
JL + JM
JM
(9)
The equation shows that in an ideal two mass system
the resonance frequency (pole) is always better
damped than the anti-resonance (zero).
Figure 8 shows several idealized Bode plots
(damping not shown) with different load inertia
configurations. Parameter for the plots is the inertia
ratio.
Wrapped up the key behaviors of the compliantly
+20
+10
0
F (ω )
10
-20 dB
-30
3
1
0.3
0.1
JL = 100 JM
-40
0.1
0.3
1
3
10
Figure 8: several idealized bode plots with different load inertia
configurations – damping is not shown – see figure 5
30
Bode plot based velocity loop autotuning
The task is to determine the two
parameters of the velocity loop PIcontroller and - if necessary - to select
and parameterize the first or second
order lag filter to compensate the
PCIM EUROPE 2004 • PROCEEDINGS • 385
•
the velocity observer is well adapted [4].
Figure 11 shows such a measured open loop Bode
plot (green) of an industrial linear motion system,
figure 10. The red marked plot shows the gain
increasing effect of the compliant coupled load.
Figure 10: Belt driven linear motion system produced
by Montech – www.montech.ch
described compliant coupled load behavior. This is
executed in three major steps:
1. Measure system transfer function
2. Select and parameterize anti resonance lag filter
3. Set PI-gains according to Nyquist stability
criteria
Measure system transfer function
In the shown plots we can see that the system has
mainly the behavior of a PI-controlled single
integrator with delay. Due to the integral part of the
velocity loop controller the phase shift in the low
frequency area is close to 180°. The behavior in the
high frequency area is significantly influenced by the
current loop performance and the velocity estimation
method. By using an observer to eliminate the digital
differentiation delay this observer has to be adapted to
the system. The digital drive ServoStar 300 can run a
simple – first order - inertia estimation at a defined
frequency (usually 300-500Hz). This frequency
should be set above of the system resonance
frequency to adapt the observer to the high frequency
(motor only) inertia.
According to this constrains the system transfer
function can be best measured when:
• all internal filters are switched off,
• the current loop is well tuned [5],
• the velocity PI-integral part is off or very low and
Select and parameterize anti resonance lag filter
Figure 12 shows more detailed the gain increase
effect of the compliant coupled load (red). The blue
plot shows the gain approximation of a 2nd order lead
filter. The parameters ( f Z = 74 Hz , f P = 146 Hz and
dc-gain = -1.5dB) are fitted by using a simple least
square adoption to the measured plot. The fitting
works considerably good as shown in the Bode plot.
The highlight is that here a compliant coupled threeinertia system (belt gear box + linear motion belt) is
compensated quite well.
The requested current set-point filter is the inverse
transfer function of the approximated 2nd order filter
( f P = 74 Hz , f Z = 146 Hz ). The dc-gain can be ignored
and the first approximation of the damping is set to:
DZ = D P =
1
2
The frequency with the highest phase lag is here:
f lag = f P f Z = 105 Hz
This is fitting perfectly with the phase lead (at 105
Hz) of the measured open loop bode plot, figure 13.
According to the Nichols diagram here is a minimum
gain requested. Due to digital delays and system time
constants the phase shift crosses 180° in an area of
several 100 Hz. Here is a maximum gain requested.
The frequency dependent minimum or maximum
value is defined by the corresponding open loop phase
shift.
Nyquist stability criteria and the Bode plot
The easiest interpretation of the Nyquist stability
criteria is to check of the gain margin (at 180° phase)
and the phase margin (at 0dB gain) in the open loop
Bode plot, Figure 13.
But this is a verification at only two points. What is
beside these two points? We can see the detailed
behavior much better in the Nichols diagram, figure
14. Here we see the phase-gain relation for each
frequency.
14
12
10
dB
8
6
4
2
0
-2
-4
20 - 600 Hz
Figure 11: Open loop Bode plot (green) and
the gain increase effect of the
compliant coupled load (red)
Montech linear motion system
386 • PCIM EUROPE 2004 • PROCEEDINGS
Figure 12: Measured gain increase effect of the
compliant coupled load (red)
2nd order filter approximation (blue)
Montech linear motion system
+20
F (ω )
I-Part
10 dB gain margin
+10
+40
0 dB Gain
0
-20 dB
3 dB
1 dB
+10
0
-90
-180
-270
-360°
10
0.5 dB
+20
-30
ϕ
f
+30
-10
6 dB
0
30
100
50° phase margin
f
300 Hz
1k
F (ω )
12 dB
180° phase
Figure 13: Open loop Bode plot with 50° phase
margin and 10 dB gain margin
Accepting 1dB peaking (red oval) we can see that the
gain is limited if the phase shift is between 115° to
245°. For low frequencies – due to the phase shift - a
minimum gain is requested, for high frequencies the
maximum is shown. The gain margin would be 5 dB;
phase margin would be 65°.
Set PI-gains according to Nyquist stability
criteria
By switching the integral part off we get the doted
line in figure 14. Now the gain can be increased until
the plot touches the lower side of choused peaking
oval (the plot is moving up). In the next step the
integral part can be increased until the plot touches
the upper side of the oval (the plot is moving left).
Figure 15 shows the Bode plot of the tuned velocity
loop.
Summary
Instead of tuning a servo drive system by highly
educated servo experts the Bode plot based auto-tune
can provide reliable results in a short time frame. The
result is documented with an open and closed loop
Bode plot that shows key performance indices like
bandwidth and stability margin.
The key advantages of using a Bode plot based autotuning are:
• Ease of use
• Very good results in a short time frame
• Less set up cost
This results in advantages for the designed product:
• Higher reliability and lower fault risk
• Higher productivity
• Fast development time
References
[1] V. Wesselak, M. Köhler and G. Schäfer, „Robust
Speed Control Based on the Identification of
Mechanical Parameters“, PCIM, Germany 2003.
[2] J. O. Krah and K. Neumayer, “Motorsteuerung –
kompakt und flexibel”, Elektronik Heft 3/2004,
Weka Verlag.
-20 dB
-30
velocity observer
& digital delay
-40
360°
270
180
ϕ
90
0
Figure 14: Nichols plot - always open loop –
here with 0.5 dB peaking
doted line: without integral part
Figure 15: Open loop (green) and
closed loop (red) bode plot of the
Montech linear motion system with 2nd
order lag filter and switched on integral
[3] J. O. Krah, S. Geiger and G. Jaskowski, “Free
Programmable Signal Processing inside a High
Performance Servo Amplifier”, PCIM, Germany,
1998.
[4] J. O. Krah, “Software Resolver-to-Digital
Converter for High Performance Servo Drives”,
PCIM, Germany 1999.
[5] J. O. Krah, J. Holtz, “High Performance Current
Regulation for Low Inductance Servo Motors”, IEEE
Industry Appl. Soc. Annual Meeting, St Louis, Oct.
1998.
[6] G. Ellis, Cures for Mechanical Resonance in
Industrial Servo Systems, PCIM Germany 2001.
PCIM EUROPE 2004 • PROCEEDINGS • 387