Simulative Investigation of Transfer Function-based Disturbance

Observer for Disturbance Estimation on Electromechanical Axes

Chris Schöberlein

, Armin Schleinitz

, Holger Schlegel

and Matthias Putz

1,2

Institute of Machine Tools and Production Processes, Chemnitz University of Technology, Reichenhainer Straße 70,

09126 Chemnitz, Germany

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

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:

∙K

(1)

identifies the motor torque provided by the

motor, which is calculated as product of motor current

and torque constant K

. The parameter T

denotes

the acceleration torque required for the movement of

the axis and T

summarizes all load torques acting on

the motor:

 T

T

(2)

As shown in equation (2), besides friction (T

) and

potential weight torques (T

) the process torques T

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-

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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

 load-side moment of inertia J

 elastic coupling between J

and J

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:

tot

= J

(3)

With reference to formula (3), the following

transfer function results for a two-mass system:

S,2MS

(s)=

tot

∙s

∙

∙s

∙s+1

tot

∙s

∙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):

S,2MS

(s)=

tot

∙s

∙





∙s

∙s+1





∙s

∙s+1

(5)

Here ω

and ω

denote the resonance and

antiresonance frequencies, d

and d

the associated

damping values. As shown in Figure 2, ω

and ω

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.

20log

tot

jω in

rad

S,2MS

(jω)

in dB

jω in

rad

S,PO

indir

(jω)

in dB

S,PO

dir

(jω)

in dB

jω in

rad

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:

S,PO

(s)=

a∙





∙s

∙s+1





∙s

∙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)=

(s)

N(s)

D(s)

→ G

(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:

(s)

-1

(s)

(

)

D(s)

)

1+K∙G

(

)

(8)

(s)

-1

1+K∙

N(s)

(

)

K∙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→∞

(s)

-1

D(s)

)

(10)

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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:

(s)

-1

(s)

-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

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:

dist

(s)=

reac

= 

f,i

∙s+1



f,i



∙s

f,i

∙s+1

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

tot

(kg∙m

)

0.001354

tot

(kg∙m

)

0.00137

(kg∙m

)

0.000869

(Hz)

334.00

(kg∙m

)

0.000485

0.040

m,l





2150

(Hz)

423.19

m,l



Nms



0.086

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

(

Nms

)

0.0802

(s)

0.00842

SSPF

(s)

0.002

Current setpoint filter

FCN





418.18

FCD





418.18

1999.00 1999.00

FCN

0.0

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)

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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

(

-1

)

-0.4

-0.2

0.2

0.4

-0.5 -0.25 0 0.25 0.5 0.75 1

real axis

(

-1

)

-0.4

-0.2

0.2

0.4

)

-1

, T=0.1s G

-1

, T=0.001sG

-1

, T=0.01s G

-1

, T=0.0001sG

dist

DOB TFDOB, T= 0.1s TFDOB, T= 0.01s TFDOB, T= 0.001s TFDOB, T= 0.0001s

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.

REFERENCES

Rizal, M., Ghani, J., Nuawi, M., Haron, C., 2014.

Measurement of cutting forces in CNC turning centers:

a review. In International Journal of Mechanical

Engineering, 7(10), 2083-2097

Stein, J., Shin, K. 1986. Current monitoring of controlled

DC spindle drives, In Journal of Dynamic Systems,

Measurement and Control, 108, 189-295

Altintas, Y., 1992. Prediction of cutting forces and tool

breakage in milling from feed drive current

measurements. In Journal of Engineering for Industry,

114, 386-392

Sato, R., Hasegawa, M., Shirase, K., 2013. Cutting force

monitoring based on the frequency analysis of feed

motor torques. In Journal of ME Japan

Schöberlein, C., Norberger, M., Schlegel, H., Putz, M.,

2020. Simulation and disturbance estimation of speed-

controlled mechatronic drive systems. In MATEC Web

of Conferences, 306

Yamato, S., Sugiyama, A., Suzuki, N., Irino, N., Imabeppu,

Y., 2019. Enhancement of cutting force observer by

identification of position and force-amplitude

dependent model parameters. In International Journal

of Advanced Manufacturing Technology, 104, 3589-

3605

Yamada, S., Kakinuma, Y., 2016. Mode decoupled cutting

force monitoring by applying multi encoder-based

disturbance observer. In International Journal on

Advanced Manufacturing Technologies, 92, 4081-4093

Münster, R., Walther, M., Schlegel, H., Drossel, W., 2014.

Experimental and simulation-based investigation of a

velocity controller extension on a ball screw system. In

Mechatronics 2014 – the 14

Mechatronics Forum

International Conference, 226-234

Hipp, K., Schöberlein, C., Schlegel, H., Neugebauer, R.,

2017. Simulation based optimization for controller

parametrization of machine tool axes – advanced

application. In Journal of Machine Engineering, 17(1),

57-68

Rudolf, T. 2014. Adaptierbare Parametrierung von

Diagnosesystemen durch Verwendung digitaler

Antriebssignale in der Prozessüberwachung. In

Ergebnisse aus der Produktionstechnik

Schröder, D. 2015. Elektrische Antriebe – Regelung von

Antriebssystemen, Springer Vieweg. Heidelberg, 4

edition.

Buchholz, J. 2007. Inversion impossible? GRIN. Munich.

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