Delay Time Compensation in the Current Control Loop of Servo
Drives – Higher Bandwidth at no Trade-off
Heiko Schmirgel
Cologne University of Applied Sciences
Betzdorfer Str. 2, 50679 Köln, Germany
Heiko.Schmirgel@FH-Koeln.de, Tel.: +49 221/8275-2480, Fax.: +49 221/8275-2477
Jens Onno Krah
Cologne University of Applied Sciences
Betzdorfer Str. 2, 50679 Köln, Germany
Jens_Onno.Krah@FH-Koeln.de, Tel.: +49 221/8275-2439, Fax.: +49 221/8275-2477
Reiner Berger
Danaher Motion GmbH
Wacholderstr. 40-42, 40489 Düsseldorf, Germany
R.Berger@DanaherMotion.net, Tel.:+49 203/9979-191, Fax.: +49 203/9979-200
Abstract: Even after compensation of nonlinear
phenomena in the power stage, delay and sampling
times remain in the current control loop that can
not be eliminated. These time lags limit the speed
and possible performance of the current controller
and thus of all superposed control algorithms.
An effective method to compensate the time lag in
the current control loop of digital servo amplifiers
is presented. Based on the Smith Predictor
concept, a model predictive control algorithm is
implemented. Usability is ensured by reducing the
algorithm to be dependent on very few
parameters. The presented method provides a fast
and simple solution implemented on the current
control loop level of the cascaded drive control
system.
1. Introduction
High performance servo drives are still a fast growing
market segment. The brushless technology offers
significant advantages in terms of reliability and
motor size. Higher drive complexity was balanced
with additional logic and sophisticated power
electronics. Due to the innovation cycles of the
semiconductor manufacturers the size and cost for
more and more complex drive systems did not
increase. Each new drive generation offers more
performance, more flexibility and higher integration
density.
The challenge is to provide the best possible
performance/cost ratio with the given setup. Better
performance will allow more sophisticated processes
and lower cost makes these technologies attractive to
new applications.
While the hardware setup remains essentially the
same throughout a single generation, better algorithms
have to be developed and constantly improved in
order to make best use of the limited available
computing power. In a cost optimized servo control
environment, the crucial requirement is always to
provide real-time performance. This demands simple
and fast algorithms, still providing the maximum
possible accuracy and dynamics.
Phase lags caused by dead times inside of control
loops reduce the stability margin and limit possible
gains to lower values and thus the dynamics of the
complete system. To reduce the effect of time delays
in the current loop of the drive system the
implementation of an algorithm known as Smith
Predictor [1] inside the current control loop is
proposed. The Smith Predictor offers the possibility to
make use of known system properties in order to
predict the reaction of the controlled system before it
can be measured, which substantially improves
performance of the drive system. It can be entirely
implemented only by changing the current controller
software without the need for additional hardware.
These improvements offer the possibility of designing
a drive system with faster response while avoiding the
well known trade-offs of higher switching losses and
increased EMI (electromagnetic interference), that
require the system to be bigger and more expensive.
The algorithm is implemented on a drive system with
16kHz switching frequency and 32kHz current loop
PCIM EUROPE 2006 • PROCEEDINGS • 541
take an infinitely small
amount of time, almost
PWM
every element inside the
current control loop
Velocity Feedback
i
Current Feedback
contributes to the time
d/dt
delays. Some elements
are even only used at
Position Feedback
Motor
R
certain time instants
which means that they
“wait” until it is “their
Fig. 1: General cascaded drive system
turn” (Fig. 2).
update, and it is shown that a closed loop current
Usually the acquisition of the current feed back signal
control loop bandwidth of 5kHz is achievable.
is such a process. It takes place during one or both of
the zero-sequences (all outlets on low or all outlets on
high potential) that are generated by the PWM2. Control Structure
algorithm. At these instants of time when no
(Circuit, Scheme)
switching takes place noise is low and the current
assumes the average value without
additional
Commonly, the control architecture of drive systems
filtering that would otherwise cause additional delay
consists of three cascaded control loops. The
as well. Together with the time consumed by Analoginnermost loop is the fastest, controlling the current.
to-Digital Conversion this makes up the time delay
Directly superposed to the current loop is the velocity
caused by the feed back system (TFB). Faster A/D
controller. The outer loop is the slowest controlling
hardware can reduce the latter contribution but far
the position (Fig. 1).
more dominant is the sampling process here.
In each loop the command signal is not processed
The time lost due to current control algorithms and
instantaneously, but handed over to the next loop or
calculation of the PWM signals from the voltage
element inside the same loop with a certain time
command values (TPI) can be limited by using
delay. Computation times and other time lags (e.g.
efficient programming techniques. More important is
due to feedback) can only be reduced to a certain
the time delay acquired during the active phase, when
extent. Ultimately a delay time always remains.
the desired current is actually generated by the power
This has a limiting effect on the overall system
stage (TPWM and TPS). This takes naturally a complete
performance, where the most adverse effect on system
interrupt cycle. The voltage drop across the
dynamics has the time lag in the current loop. Since
semiconductor switches is here assumed to be
this is the fastest loop, which is executed with a
compensated by an algorithm equivalent to [2] or
higher frequency than all the other loops.
small enough to be neglected.
Reducing the effect of the time lag in the current loop
In order to reduce any one of the time delays they
and thus increasing its dynamic range will have a
have to be looked at individually giving attention to
positive effect on all superposed loops.
the respective nature and source of the effect. The
compensation with a Smith Predictor is done on a
system level. Here all delay times can be replaced by
3. Sources of Delay Times
one single delay time Td and then be treated together
(Fig. 3).
Since the performance of any computation operation
The sample and hold process contributes a dead time
with a digital processor or microcontroller does not
of one half of a sampling period (Ta/2). As long as the
individual delays of the
computation process are
Td PWM
Td PS Ui
PI-Controller Td PI
L, R
small, which is usually
Power
the case, they can be
PWM
Stage
treated together with the
delay of the active phase,
which adds up to a
Td FB
complete
sampling
A/D
period
(T
).
a
Sampling
Position
Command
Position
Loop
Velocity i*
Loop
Current
Loop
u*
Power
Stage
Conversion
Fig. 2: Delay times inside the current loop
PCIM EUROPE 2006 • PROCEEDINGS • 542
required to be able to predict its
reaction. In this case this is the LR-model of the machine windings
and the applicable dead time of
Td
the complete system Td.
With this information, it is
Sample
Power
ADC
PWM
Controller
possible to make the predictable
+ Hold
Stage
current signal (the effect of the
voltage command on the L-RTa
model) available to the feed back
loop before the delay time. This is
2
Ta
done by adding the predicted
current to the feed back signal
Fig. 3: All delay times can be represented by one single dead time
without a delay. In the predictor
branch of the current control loop,
the L-R first order lag and the
PI-Controller
Td
L,
R
Ui
delay time have to be swapped
a)
(Fig. 4b). The feed back loop then
“sees” the predicted value of the
current at the same time, when it
is actually generated. In digital
motion control systems this is
Td
L, R
easily implemented.
b)
The predicted signal has to be
subtracted from the feed back
ipre
signal again after the delay time
has elapsed, because at that time
Fig. 4: a) Current control loop without Smith Predictor and b) Basic
the regular feed back signal
Smith Predictor concept
contains the information of the
With only one remaining dead time, the
real effect of the voltage command.
representation of the current control loop including all
This leads to the complete current control loop shown
the time delays simplifies to (Fig. 4a).
in (Fig. 5) where the Smith Predictor branch is
The machine windings are modeled as a simple L-R
implemented according to the transfer function (1).
first order lag.
1
(1)
(1 − e − sT )
1 + sTLR
The second part of (1) is the pulse function, described
4. The Prediction Algorithm
in [1].
Active Phase
Information Processing
d
The idea behind the prediction algorithm is to create a
system that can be controlled as if it was its own
minimum phase equivalent (i.e. the same system
without any dead times).
Some knowledge about the controlled system is
Td
PI-Controller
-
Ui
ipre
For the following considerations the current controller
is assumed to be a general regular PI-controller with a
proportional gain factor Kp and
L, R
integration time constant TN(2).
GPI ( s ) =
Td
L, R
5. Effect on System Performance
-
K p (1 + sTN )
(2)
sTN
The system model that consists of
the dead time Td and the L-R first
order lag with its time constant
TLR is added and the control loop
is closed. This leads to an overall
system transfer function before
Fig. 5: Current control loop with Smith Predictor
PCIM EUROPE 2006 • PROCEEDINGS • 543
the Smith Predictor was added (Fig. 4 a)) as in (3)
1
sTN
1 + sTLR
GS =
 K (1 + sTN ) − sTd
1 

e
1 +  p
sTN
1 + sTLR 

(3)
GSm =
sTN
e − sTd
-6 dB
6 dB
-10 dB
10 dB
1
1 + sTLR
(
-1
and (4) can be simplified to (6)
(6)
1
sTN
e − sTd
 K p (1 + sTN )
1  1 + sTLR

1 + 
1 + sTLR 
sTN

(6) represents a system where the dead time is
completely outside of the feed back loop (Fig. 6) [3]
and thus does not affect system dynamics at all. The
relevant part of the system for the PI-controller is in
fact the minimum phase equivalent as desired.
The Nyquist plot of this system illustrates the effect of
the added prediction algorithm. The well tuned
regular PI-controlled system with dead time (Fig. 7,
red curve) shows the typical characteristics of a
process with dead time. The Smith Predictor with
exactly matching parameters (dead time and motor
time constant) eliminates the dead time completely, in
accordance with (6) only the first order time lag
characteristic of the machine is left that affects the
system dynamics (Fig. 7 green curve).
PI-Controller
L, R
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
Real Axis
(5)
K p (1 + sTN )
-20 dB
-1
)
Td = TdS
20 dB
0
-0.5
(4)
If the model is sufficiently accurate, (5) holds
GSm =
-4 dB
0.5
 K (1 + sTN )  − sTd

1
1
1 − e − sTdS  
1 +  p
+
 e
sT
sT
sT
1
1
+
+
N
LR
S



TLR = TS
-2 dB
4 dB
The addition of the Smith Predictor (Fig. 5) with its
own approximations of the dead time TdS and the time
constant of the machine TLRS, changes only the feed
back which leads to (4)
K p (1 + sTN )
0 dB
1
e − sTd
Imaginary Axis
K p (1 + sTN )
Nyquist Diagram
2 dB
Fig. 7: Nyquist plot of the system without Smith
Predictor (red), with Smith Predictor and
ideal parameters and the same gain
(green) and increased gain (blue).
The gain of the PI-controller can now be increased to
much higher levels than before, without impairing the
stability of the system (Fig.7 blue curve). For a true
first order time lag and correctly estimated parameters
the gain could theoretically be infinite. Since the first
order time lag is only an approximation and small
estimation errors in the parameters are not avoidable,
there are certain limits to the increase of the controller
gain. The effect of estimation errors of 15% for the
dead time and 15% for the time constant of the Smith
Predictor is shown in Fig. 8 (red curve) compared to
the ideally tuned Smith Predictor (blue curve). This
system has far less stability margin than the ideally
tuned Smith Predictor but it is still superior to the
system without prediction algorithm since it has a
greatly increased proportional gain and thus higher
dynamics. Additionally the system will also be robust
enough to cope with parameters that are not precisely
tuned which will most probably happen once a
machine heats up during operation.
Td
L, R
Fig. 6:
Resulting system
PCIM EUROPE 2006 • PROCEEDINGS • 544
Nyquist Diagram
7. Automatic Parameter Acquisition
0 dB
2.5
-2 dB
2
1.5
1
Imaginary Axis
0.5
2 dB
-4 dB
4 dB
-6 dB
6 dB
-10 dB
10 dB
20 dB
-20 dB
0
-0.5
-1
-1.5
-2
-2.5
-1
-0.5
0
0.5
1
Real Axis
Fig. 8: Nyquist plot of the system with Smith
Predictor and ideal parameters (blue) and
15% higher dead time and 15% longer
time constant (red).
The data on which the parameterization of the Smith
Predictor is based is taken from machine data bases or
measured manually. Ultimately only few parameters
remain to be tuned that only depend on the connected
motor.
Additionally, all the measurements needed are already
integrated within a normal drive system, and tuning
effort can be limited by introducing certain bounds to
the possible results. Necessary security to instability
problems during the tuning process is provided by the
Smith Predictor gain that can be used as a limiter as
well.
An automated process that acquires the parameters of
the correction function by itself during startup will be
implemented on the same system, to enable the drive
controller to quickly adjust to a number of different
motors.
8. Test Setup
6. Implementation
The proposed algorithm was implemented and tested
on a conventional industrial drive system.
(Servostar 303, 8kHz switching frequency and
16 kHz current loop update and 16 kHz switching
frequency and 32 kHz current loop update with PM
machine 6SM47L3000) and tested against the same
system without prediction algorithm [4].
Due to the simplicity of the new elements additional
calculation time is low enough not to have any effect
on system performance.
In the implementation special attention was given to
usability and simplicity.
Since the dead time depends purely on the inverter
system and its information processing unit it can be
assumed to remain constant, even when the controlled
machine changes. Experimental results show that this
assumption is valid and no adjustment by the user is
necessary.
The parameters left to tune are the resistance and the
inductance of the machine model. These
values should be readily available from
the data of the machine or can be easily
20dB
open loop
measured by the user.
These parameters can only be acquired
10
with a certain accuracy and especially
3
closed loop
0dB
the resistance changes when the
-3
machine
temperature
increases.
100Hz
1kHz
4kHz
Experiments show that according to the
Fig. 8 the algorithm is indeed robust
0°
against minor inaccuracies of these
closed loop
parameters.
open loop
-90°
Additionally, a Smith Predictor gain is
introduced that can adjust the effect of
the Smith Predictor during the tuning
-180°
process.
With these few parameters and the
100Hz
1kHz
4kHz
possibility to limit the smith predictor
gain during tuning procedure, it is
possible to quickly tune the algorithm.
Fig. 9: Bode plots of the current loop at 8 kHz switching
frequency without (red) and with smith predictor
PCIM EUROPE 2006 • PROCEEDINGS • 545
40dB
open loop
20
10
6
0dB
-6
100Hz
closed loop
1kHz
0°
7kHz
It can eliminate the time delay for
tunability of the PI-controller almost
completely, allowing a substantially
increased proportional gain.
Bode plots show the wider dynamic
range of the current loop with Smith
Predictor.
The design of a drive system with
16 kHz switching frequency and 32 kHz
current loop update, that has a closed
loop current control loop bandwidth of
5 kHz was demonstrated.
closed loop
10. Summary / Conclusion
-90°
-180°
100Hz
1kHz
7kHz
Fig. 10: Bode plots of the current loop at 16 kHz switching
frequency without (red) and with smith predictor
Fig. 9 shows the open loop and closed loop bode plots
of the system at 8kHz switching frequency and
16 kHz current loop update without (red curves) and
with the Smith Predictor (blue curves). It can be
clearly seen that the Smith Predictor increases the
dynamic range of the current loop in this low
switching frequency mode by 70% (closed loop
bandwidth at -90° phase angle). Fig. 10 shows the
open loop and closed loop bode plots of the system at
16 kHz switching frequency and 32kHz current loop
update without (red curves) and with the Smith
Predictor (blue curves). In the high switching
frequency mode, the increase of the dynamic range is
still more than 30%. The -90° phase angle bandwidth
can now be increased to over 5 kHz.
Acknowledgement
All implementation and testing of algorithms was
done on a Servostar 300. This servo drive system as
well as the complete funding of this project was
kindly provided by Danaher Motion GmbH,
Düsseldorf, Germany.
References
1.
2.
9. Discussion of the Results
The Smith Predictor, while originally designed for
much slower processes, proves to be able to
compensate even small dead times in current control
loops.
These improvements offer the possibility
of designing a drive system with much
faster response while avoiding the well
known trade-offs of higher switching
losses
and
increased
EMI
(electromagnetic interference), that
require the system to be bigger and more
expensive.
3.
4.
Smith, “A Controller to Overcome Dead Time”,
ISA Journal, vol.6. no.2, pp.28-33, Feb. 1959
Schmirgel, Krah, “Compensation of Nonlinearities
in the IGBT Power Stage of Servo Amplifiers
through Feed Forward Control in the Current Loop”
Proceedings of the PCIM Europe, pp. 94-99,
Nürnberg, June 2005
Veronesi, “Performance Improvement of Smith
Predictor through automatic computation of dead
time”, Yokogawa Technical Report English Edition,
no. 35, pp.25-30, 2003
SR 300 manual, www.DanaherMotion.net
PCIM EUROPE 2006 • PROCEEDINGS • 546