Simulative Investigation of Transfer Function-based Disturbance
Observer for Disturbance Estimation on Electromechanical Axes
Chris Schöberlein
1
, Armin Schleinitz
1
, Holger Schlegel
1
and Matthias Putz
1,2
1
Institute of Machine Tools and Production Processes, Chemnitz University of Technology, Reichenhainer Straße 70,
09126 Chemnitz, Germany
2
Fraunhofer-Institut for Machine Tools and Forming Technologies, Reichenhainer Straße 70, 09126 Chemnitz, Germany
Keywords: Electromechanical Axes, Disturbance Estimation, Process Forces, Transfer Function, Simulation Model,
Frequency Response.
Abstract: In the field of machine tools, applicable solutions for monitoring process forces are becoming increasingly
important. In addition to sensor-based approaches there are also methods which utilize the already available
signals of the machine control. Usually, the motor currents and, when applicable, position values of the feed
axes are considered. By applying reduced order models of the machine axes, non-process components are
subtracted from the measured signals. However, these approaches are often utilizing simplified models or
require additional a-priori knowledge, for example construction data or actual parameter values. The former
in particular has a negative impact on the quality of the estimations. To overcome these disadvantages, this
paper presents a novel observer structure based on the mechanical system transfer function of the feed axis.
One main advantage is achieved by applying scalable and automatically generated models with focus on
distinct frequency ranges. All necessary information is provided by a frequency response of the speed control
plant, as it is typically obtained during the commissioning phase of electromechanical feed axes. By inverting
the system transfer function and considering an additional disturbance transfer function, the quality of the
estimation can be significantly improved compared to previous approaches.
1 INTRODUCTION
The measurement of the forces acting during
manufacturing processes is subject of intensive
research efforts. In industrial application, additional
sensors are commonly utilized and placed close to the
contact point between tool and workpiece. Due to its
proximity to the process, these methods are
characterized by a high sensitivity, but at the same
time increase the complexity of the machine and lead
to additional investment costs (Rizal et al., 2014).
For this reason, there are in particular research-
based approaches which aim to reconstruct the
prevailing process forces from the already available
signals of the installed drive components. The
majority of these approaches focusses on the pure
evaluation of the motor currents of the feed axes
(Stein & Shin, 1986; Altintas, 1992; Sato et al., 2013).
An alternative is provided by so-called disturbance
observers. These consist usually of an order-reduced
model of the mechanical system of the axes and
utilize additional speed and position values. An
overview of the basic methods and its characteristics
as well as a simulative comparison is presented in
(Schöberlein et al., 2020).
All these methods have in common that simplified
models and deviations in the system parameters affect
the accuracy of the estimation. However, the
estimation quality can be improved for individual
approaches. (Yamato et al., 2019) takes the position
dependency of certain system variables into account
and (Yamada et al., 2016) separates individual
vibration components, for example. On the other
hand, these methods are often only applicable in
certain frequency ranges and typically require
detailed system knowledge. Within the scope of this
paper, a novel approach will be presented, which is
based solely on drive-internal measurement
functions. As a consequence of the applied model
generation method, no prior system knowledge is
required.
The paper has the following structure. Section 2
describes the structure of a typical machine tool axis
as well as approaches for modelling its mechanical
Schöberlein, C., Schleinitz, A., Schlegel, H. and Putz, M.
Simulative Investigation of Transfer Function-based Disturbance Observer for Disturbance Estimation on Electromechanical Axes.
DOI: 10.5220/0009858606510658
In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 651-658
ISBN: 978-989-758-442-8
Copyright
c
2020 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
651
Figure 1: Structure of the electromechanical axis and block diagram of the mechanical two-mass system.
transfer behaviour. Moreover, a method is presented
which generates scalable models of the mechanical
system transfer function based on frequency response
measurements of the speed control plant.
In order to enhance the estimation accuracy for
load-side disturbances, an additional disturbance
transfer function is introduced. Section 3 is focussed
on the actual observer structure of the novel transfer
function disturbance observer (TFDOB). For this
purpose, two concepts for inverting the previously
determined transfer functions are presented. One
approach is based on the extension of the transfer
function by an additional P-controller. The other one
focusses on the placement of additional high-
frequency poles. In Section 4, the TFDOB is
implemented in a simulation model in Matlab®
Simulink® and compared with a common observer
structure. The paper closes with a summary and an
outlook on upcoming research goals.
2 AUTOMATIC MODELLING OF
MECHANICAL TRANSFER
FUNCTION
The typical structure of an electromechanical feed
axis is shown in Figure 1. Usually, it can be separated
into an electrical part consisting of an NC control and
a subordinated drive system as well as a mechanical
part. In most applications, the mechanical system is
realized in form of a ball screw drive. As illustrated
in the Figure, process forces usually act on the load
side of the mechanical part only. Position measuring
systems are equipped on motor and load side,
respectively. For reasons of simplification, a rotary
system is assumed for the following investigations.
Hence, the equilibrium of torques is calculated to:
T
m
=I
m
∙K
t
=T
a
+T
l
(1)
T
m
identifies the motor torque provided by the
motor, which is calculated as product of motor current
I
m
and torque constant K
t
. The parameter T
a
denotes
the acceleration torque required for the movement of
the axis and T
l
summarizes all load torques acting on
the motor:
T
l
T
p
T
f
T
g
(2)
As shown in equation (2), besides friction (T
f
) and
potential weight torques (T
g
) the process torques T
p
are included in the load torque and therefore also
affect the actual motor torque.
2.1 Modelling of Electromechanical
Axes
Modelling strategies for electromechanical axes
usually concentrate solely on the mechanical part.
Typically, this part is represented in form of a multi-
ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics
652
mass mechanism. Depending on the complexity and
modelling accuracy, more or less complex models
can be applied. A widely utilized model in control
engineering is the two-mass system (Figure 1, lower
part), which is selected for the following
considerations. Such a system is typically divided
into the following components:
motor-side moment of inertia J
m
,
load-side moment of inertia J
l
,
elastic coupling between J
m
and J
l
with the
torsional rigidity c and damping constant d.
Additional friction torques act on motor (T
f,m
) and
load side (T
f,l
), respectively. Process forces usually
act on the load side only. The total moment of inertia
for a general multi-mass system is calculated as
follows:
J
tot
= J
i
n
i
=
1
(3)
With reference to formula (3), the following
transfer function results for a two-mass system:
G
S,2MS
(s)=
1
J
tot
∙s
J
l
d
∙s
2
+
d
c
∙s+1
J
m
J
l
J
tot
c
∙s
2
+
d
c
∙s+1
(4)
In principle, higher-order models can also be
described with physical parameters. On the other
hand, for systems with model orders higher than three
this is no longer feasible with reasonable effort
(Münster et al., 2014). In addition to the
representation with physical parameters shown in
formula (4), the two-mass system can also be
described with standardized parameters (Hipp et al,
2017):
G
S,2MS
(s)=
1
J
tot
∙s
1
ω
f
2
∙s
2
+
2d
f
ω
f
∙s+1
1
ω
r
2
∙s
2
+
2d
r
ω
r
∙s+1
(5)
Here ω
r
and ω
f
denote the resonance and
antiresonance frequencies, d
r
and d
f
the associated
damping values. As shown in Figure 2, ω
r
and ω
f
can
be taken directly from the frequency response of the
mechanical system. The damping values can only be
identified with great effort. Hence, they are often
determined empirically or utilizing heuristic methods
(Hipp et al., 2017).
Figure 2: Magnitude response of a two-mass system.
2.2 Automatic Model Generation based
on Frequency Measurements
In order to implement the novel TFDOB, the transfer
function of the mechanical system is required. As
illustrated in formula (5), it can be represented by
standardized parameters. A method to generate these
transfer functions automatically is proposed in
(Münster et al., 2014) and (Hipp et al., 2017). This
methodology can be applied to systems of arbitrary
order. The starting point is a measured frequency
response of the mechanical subsystem of the feed
axis. Modern control and drive systems offer
corresponding pre-installed functionalities. The
frequency response is emulated by connecting
individual partial oscillators in series. Each partial
oscillator represents a characteristic part of the
frequency response in the direct and indirect case
(Figure 3). The resulting transfer function of such a
Figure 3: Partial oscillator for indirect (a) and direct (b) controlled systems.
1
20log
1
J
tot
ω
f
ω
r
jω in
rad
s
G
S,2MS
(jω)
in dB
ω
f
ω
r
jω in
rad
s
G
S,PO
indir
(jω)
in dB
G
S,PO
dir
(jω)
in dB
ω
r
jω in
rad
s
b)a)
Simulative Investigation of Transfer Function-based Disturbance Observer for Disturbance Estimation on Electromechanical Axes
653
Figure 4: Block diagram of the transfer function inverter with proportional gain (a) and pole placement (b).
partial oscillator is calculated as:
G
S,PO
(s)=
a∙
1
ω
f
2
∙s
2
+
2d
f
ω
f
∙s+1
1
ω
r
2
∙s
2
+
2d
r
ω
r
∙s+1
(6)
Note that for indirect controlled systems the
parameter a is assigned with 1 and for direct
controlled systems a is 0. The general procedure for a
measured frequency response according to
(Hipp et al., 2017) is:
1. determination of J
tot
by calculating the
gradient at lower frequencies,
2. identification of the resonance and anti-
resonance frequencies,
3. initialization of the damping parameters
with default value of 0.01,
4. empirical adaptation of the damping values.
The fundamental suitability of the proposed method
is demonstrated on an exemplary simulated frequency
response model in Section 4.
3 TRANSFER FUNCTION BASED
DISTURBANCE OBSERVER
The following Section presents a novel type of
disturbance observer based on automatic generated
transfer functions. One main advantage of this
method is that a detailed and scalable model of the
mechanical transfer behaviour is created even without
complex reference measurements, like applied in
(Rudolf, 2014). Since the process forces usually act
on the load side of the feed axis, an additional
disturbance transfer function is introduced.
Therefore, no further measurements or system
knowledge are required.
3.1 Inversion of Mechanical System
Transfer Function
The fundamental idea of model-based approaches for
estimating process forces on feed axes is based on the
assumption that these forces are contained in the
motor current signal. As a consequence, the process
forces can be reconstructed by subtracting all non-
process influences (e.g. friction, acceleration). As a
result of the knowledge of the mechanical transfer
behaviour, the torque required for axis acceleration
can be obtained. Due to the fact that the actual angular
velocity is typically available in modern drive
systems, this acceleration torque is calculated by
inverting the mechanical transfer function G
s
(s):
G
S
(s)=
ω
m
(s)
T
a
(s)
=
N(s)
D(s)
→ G
S
(s)
-1
=
D(s)
N(s)
(7)
As a result of the inversion, the poles of the
original system become the zeros of the inverted
system and vice versa. If this inversion is carried out
using the exemplary transfer function of the two-mass
system from equation (5), the inverted system would
have a differentiating character. This means that the
order of the numerator N(s) is greater than the order
of the denominator D(s). However, such a system
cannot be implemented in reality for reasons of
causality (Schröder, 2015). Hence, two alternative
concepts for inverting the transfer function are
considered (Buchholz, 2007), both illustrated as
block diagram in Figure 4. In the left part of the
illustration (a), the inversion is realized by inserting
the original transfer function into the feedback path
of a P-controller with high controller gain K. This
leads to the following transfer function:
G
S
(s)
-1
=
T
a
(s)
ω
m
(
s
)
=
D(s)
N(
s
)
=
K
1+K∙G
S
(
s
)
(8)
G
S
(s)
-1
=
K
1+K∙
N(s)
D
(
s
)
=
K∙D(s)
D
s
+K∙N(s)
(9)
It becomes clear that the zeros of the inverted
function, depending on the value of K, correspond to
the poles of the original transfer function. The poles
of the inverted function are equally dependent on K,
whereby for large values of K:
lim
K→∞
G
S
(s)
-1
=
D(s)
N(
s
)
(10)
ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics
654
For continuous systems, this method delivers very
good results depending on the value of the controller
gain and an appropriate selection of the solver type
(e.g. ode15s) in Matlab® Simulink®. For practical
systems with discrete input signals, however, the
necessary high gain factors lead to discontinuities and
a truncation of the simulation. In consequence, this
structure is unsuitable for the intended application in
discrete controlled drive systems. Hence, a second
concept is proposed, which is shown in Figure 4b.
The fundamental idea here is to extend the inverted
transfer function with a defined number of high-
frequency poles until the denominator order
corresponds at least to the numerator order:
G
S
(s)
-1
=
T
a
(s)
ω
m
(s)
=G
S
(s)
-1
1
(1+Ts)
n-m
(11)
Main advantage of this inverter structure is its
robustness regarding digital systems and signals.
Hence, this approach will be utilized in the following
and examined in detail in Section 4 for an exemplary
simulated mechanical system.
3.2 Load Disturbance Transfer
Function
As already explained, the process forces usually act
on the load side of the feed axis. Depending on the
type of the mechanical subsystem, the knowledge of
the transfer behaviour between the surgical point of
the disturbance torque T
p
and the reaction torque T
reac
on the motor side may lead to further improvements
of the estimation. The parameters of this transfer
function can be obtained once more from the
mechanical frequency response. In general case, the
transfer function results to:
G
dist
(s)=
T
reac
T
p
=
s
2
+
2d
f,i
ω
f,i
∙s+1
1
ω
f,i
2
∙s
2
+
2d
f,i
ω
f,i
∙s+1
n
i=1
(12)
Again, this transfer function is inverted according
to the previously described strategy by adding high-
frequency poles. The combination of both systems
leads to the observer structure shown in Figure 5.
Note that due to the intended application on digital
systems, the transfer functions are transformed into z-
domain. Furthermore, besides acceleration and
disturbance torques an additional friction model is
required. This paper does not provide a detailed
description of possible friction models. For example,
an approach consisting of four individual friction
parts is discussed in (Schöberlein et al., 2020).
4 SIMULATION RESULTS
In this Section, the TFDOB is verified based on
simulations in Matlab® Simulink®. Hence, a second-
order system with the parameters listed in table 1 is
utilized as representative mechanical model and
implemented as block simulation (see Figure 1). The
cycle time of the observer and its input signals is set
to a typical value of 100 µs. As described in
(Schöberlein et al., 2020), the current control loop
was approximated as a PT2 element. Furthermore, a
current setpoint filter to attenuate the resonance
frequency as well as a low-pass filter with a cut-off
frequency of 1999 Hz were implemented. The speed
controller was parameterized corresponding to the
setting rule of the symmetrical optimum. All
parameters are listed in table 2.
4.1 Inversion of System Transfer
Function
First, the inverted transfer functions of the
mechanical system for acceleration and disturbance
torque are calculated. Hence, the measured frequency
response is analyzed by utilizing the methodology
described in Section 2.2. The results are illustrated in
Figures 6 and 7. Figure 6a shows the result of the
frequency response estimation in the Bode diagram.
All identified parameters are listed in Table 1.
Figure 5: Block diagram of the complete TFDOB.
Simulative Investigation of Transfer Function-based Disturbance Observer for Disturbance Estimation on Electromechanical Axes
655
Table 1: Parameters of the simulated two-mass system.
System parameters Identified parameters
J
tot
(kg∙m
2
)
0.001354
J
tot
(kg∙m
2
)
0.00137
J
m
(kg∙m
2
)
0.000869
ω
f
(Hz)
334.00
J
l
(kg∙m
2
)
0.000485
d
f
0.040
c
m,l
Nm
ra
2150
ω
r
(Hz)
423.19
d
m,l
Nms
ra
0.086
d
r
0.055
It can be stated that despite small deviations for the
estimated value of J
tot
, the approximated model
emulates the block simulation very well. The adjusted
damping values for the green colored model were
determined empirically. Compared to the red colored
model with default damping values, a significant
improved approximation quality of the original model
is achieved. Figure 7a shows the pole-zero map for the
estimated discrete transfer function and its inversion
for different values of the time constant T. Regardless
of the time constant T, the pole and zero positions of
the transfer function are reproduced very precisely.
Furthermore, all poles of the inverted transfer function
are placed within the unit circle around the origin of
coordinates. This means that the inverted system is
stable. The essential effect of smaller time constants is
expressed in the position of the additional real poles,
which influences the bandwidth of the estimation.
4.2 Inversion of Load Disturbance
Transfer Function
In addition to the inverted transfer function of the
mechanical subsystem, the TFDOB includes the
Table 2: Speed controller settings.
Speed controller
K
p
(
Nms
ra
)
0.0802
T
n
(s)
0.00842
T
SSPF
(s)
0.002
Current setpoint filter
ω
FCN
Hz
418.18
ω
FCD
Hz
418.18
1999.00 1999.00
D
FCN
0.0
D
FCD
0.25
0.7 0.7
transfer behaviour for load-side disturbances defined
in Section 3.2. Figure 6b shows the Bode diagram for
the block simulation and the approximated transfer
behaviour for default and adjusted damping values
based on equation (11). Once again, there is good
agreement between simulated and identified transfer
behaviour especially for the adapted damping values.
Considering the pole-zero map in Figure 7b for the
inverted discrete disturbance transfer function for
various time constants, the results of the mechanical
transfer function can be confirmed. This inverted
transfer function is stable, too. Furthermore, the
additional real poles again are shifting in the negative
direction for smaller time constants T.
4.3 Approximation of Disturbance
Torques
After the functionality of the individual subsystems
has been verified, the complete TFDOB is set up
according to Figure 5. To evaluate its performance,
the estimation behaviour in time and frequency
Figure 6: Frequency response for simulated and identified system (a) and disturbance transfer function (b).
a) b)
ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics
656
Figure 7: Pole-zero-map for simulated and inverted system (a) and disturbance transfer function (b).
Figure 8: Simulated step (a) and frequency (b) response for load side disturbance torque.
domain is examined. Hence, a stepwise and
broadband disturbance excitation is applied,
respectively. The disturbance input is connected to
the load side with a magnitude of 2 Nm. To avoid
static friction effects, a speed offset of 50 min
-1
was
specified. The additional time constants of the
inverted transfer functions of the TFDOB were
gradually decreased from 0.1 s to 100 µs. For reasons
of comparison, the conventional disturbance observer
(DOB) is included in the simulation, too. This
observer is also based on measurements of the motor
current and speed. Its only free parameter is the total
moment of inertia, which has been assigned with the
estimated value (see Table 1). Furthermore, a pure
evaluation of the motor current or rather the motor
torque was taken into account. The simulated friction
behaviour corresponds to the parameterization in
(Schöberlein et al., 2020). For the stepwise
disturbance input in time domain (Figure 8a), in
contrast to the DOB, an excitation of the mechanical
natural frequency is avoided for all parameterizations
of the TFDOB. On the other hand, the dimension of
the time constant significantly affects the dynamics of
the estimation. It should be selected as small as
possible, whereby modern drive systems with higher
clock rates may grant further improvements. In
contrast to a pure evaluation of the motor current, the
estimation dynamics of the TFDOB and DOB are
significantly increased.The results can also be
confirmed for a broadband disturbance excitation in
frequency domain, whereby the visible influence of
the current setpoint filter is counteracted for smaller
time constants of the TFDOB. Overall, the estimation
behaviour compared to the DOB or a pure motor
current evaluation is significantly improved over a
wide frequency range. This effect should become
0 0.25 0.5 0.75 1
real axis
(
s
-1
)
-0.4
-0.2
0
0.2
0.4
-0.5 -0.25 0 0.25 0.5 0.75 1
real axis
(
s
-1
)
-0.4
-0.2
0
0.2
0.4
a)
b
)
G
z
-1
, T=0.1s G
z
-1
, T=0.001sG
z
-1
, T=0.01s G
z
-1
, T=0.0001sG
z
T
dist
T
m
DOB TFDOB, T= 0.1s TFDOB, T= 0.01s TFDOB, T= 0.001s TFDOB, T= 0.0001s
a)
b)
Simulative Investigation of Transfer Function-based Disturbance Observer for Disturbance Estimation on Electromechanical Axes
657
significant for more complex mechanical systems, for
example with lower natural frequencies or further
partial oscillators. Due to the scalability of the
presented identification and modelling approach, an
increased estimation quality is expected even for
elastically coupled multi-mass systems.
5 CONCLUSIONS
In this paper, the performance of a novel type of
disturbance observer for electromechanical axes was
examined using a simulation model in Matlab®
Simulink®. The main advantages in contrast to
established structures are the automatic identification
of the observer parameters and their scalability on
systems of multiple order. The determination of the
required transfer functions is based exclusively on
frequency response measurements. By inverting the
determined transfer functions via additional high-
frequency poles the estimation of load-side
disturbances is enabled over a wide frequency range.
The performance compared to an established
structure was demonstrated utilizing an exemplary
simulated electromechanical axis.
Future work should initially analyze the
robustness of the observer structure. For example, this
includes currently not considered influencing factors,
such as signal noise or changing mechanical
parameters and controller settings. A more precise
identification of the damping values of the transfer
functions, for example with heuristic optimization
methods, may lead to further improvements. Finally,
the TFDOB must be subjected to practical
measurements on a real machine tool under process
conditions. Hence, it should be examined if the novel
structure can be supplemented by including signals of
a load-side measuring system.
ACKNOWLEDGEMENTS
Funded by the European Union (European Social
Fund) and the Free State of Saxony.
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