Intelligent Control and Automation, 2011, 2, 251-257
doi:10.4236/ica.2011.23030 Published Online August 2011 (http://www.SciRP.org/journal/ica)
Parameter Identification and Controller Design for the
Velocity Loop in Motion Con tro l Systems
Reimund Neugebauer, Stefan Hofmann, Arvid Hellmich, Holger Schlegel
Institute for Machine Tools and Production Processes, Faculty of Mechanical Engineering,
Chemnitz University of Technology, Chemnitz, Germany
Received June 10, 2011; revised July 1, 2011; accepted July 8, 2011
Abstract
Today the controller commissioning of industrial used servo drives is usually realized in the frequency do-
main with the open-loop frequency response. In contrast to that the cascaded system of position loop, veloc-
ity loop and current loop, which is standard in industrial motion controllers, is described in literature by us-
ing parametric models. Several tuning rules in the time domain are applicable on the basis of these paramet-
ric descriptions. In order to benefit from the variety of tuning rules, an identification method in the time do-
main is required. The paper presents a method for the identification of plant parameters in the time domain.
The approach is based on the auto relay feedback experiment by Åström/Hägglund and a modified tech-
nique of gradual pole compensation. The paper presents the theoretical description as well as the implement-
tation as an automatic application in the motion control system SIMOTION. The identification results as
well as the achievable performance on a test rig with a PI velocity controller will be presented.
Keywords: Identification, Parametric Models, Controller Design, Motion Control
1. Introduction
Nowadays, the identification of the velocity controlled
system is often carried out in the frequency domain.
However, in the area of low frequencies the detection of
mechanical parameters is restricted, if the measure- ment
is performed in the closed loop. There are resulting er-
rors of the magnitude and phase response, which are de-
scribed in [1].
In contrast to the estimation in the frequency domain,
standard identification techniques in the time domain
(e.g. interpreting step responses) have been developed.
These methods demand high measurement accuracy and
are limited, when the expected time constants are in the
range of the sample time.
Derived models for controlled systems of a servo drive
are primarily used for controller tuning. Several tuning
algorithms (e.g. symmetrical optimum) have been pub-
lished, which require exact model parameters as a one
main criterion to be efficient. The model order is another
important criterion for the accuracy of the tuning rules
because standard velocity controller structures are not
able to consider higher order models [2]. According to [3]
the velocity controller (PI-Structure) can be tuned based
on order reduced parametric models even for oscillatory
mechanical systems.
In addition to controller tuning, various limitations can
be defined in servo drives based on identified models [4].
In [5] online monitoring functions like detection of varia-
tion in the moment of inertia or friction moments have
been proposed.
The intention of the paper is to establish a new identi-
fication method in the time domain, which is suitable for
electrical servo drives. It is based on the relay feedback
experiment and the technique of gradual pole compensa-
tion. Based on the identification results, several tuning
rules will be carried out to prove the applicability.
The paper is divided in 5 sections. Subsequent to the
introduction, the identification method is discussed theo-
retically and applied to the velocity loop of an industrial
servo drive. In section 3 several tuning rules will be in-
troduced. Section 4 describes the experimental set-up
and the experimental results are presented. Finally the
conclusions are given.
2. Identification Method
The relay feedback experiment [6] is a well known iden-

R. NEUGEBAUER ET AL. 252
tification approach, which is generally applied to identify
a process Gs(s) on its limit of stability. For this purpose,
the controller Gr(s) is replaced by a relay controller ac-
cording to Figure 1. The original method provides a
nonparametric model with the characteristics ultimate
gain (ku) and ul- timate frequency (ωu). In contrast to that,
the relay feed- back experiment is exclusively used as an
excitation in the presented approach. The controlled
variable x ap- pears as input for a compensator Gc(s). The
aim is to adjust the compensator to the plant parameter
by using the method of gradually compensation of the
dominant pole [7]. This approach combines the advan-
tages of the relay feedback experiment with those of the
parametric model identification (Figure 1).
For the example of a first order transfer function with
dead time (FOPDT) a compensator (Gc) as shown in [8]
is chosen.
 
*1
1
d
sT
sc
k
Gse Gs
Ts s



Ts
(1)
Applying this, the forward path of the control loop
becomes:
 
  

*1
1
c
d
c
yxs c
sT
Xs
Gs GsGs
Ys
Ts
ke sTs




 



(2)
Finally, this equation is expanded in terms of two
components:
 


*
1
1
d
d
sT
c
sT
Xs keYs
s
TT
ke Ys
Ts



 




 



(3)
In the case of T* = T the term reduces to an integral
plus dead time (IPDT) system:

1
d
sT
c
X
ske Ys
s

 

 (4)
As mentioned above, Y(s) is a relay output signal.
With this input sequence, Equation (4) describes an ideal
ramp function x(t). To adjust T*, Equation (3) needs to
be evaluated to satisfy Equation (5):
Figure 1. Identification with gradual pole compensation.

*
*
10
lim 1
d
sT
TT
TT
ke Ts s















(5)
To determine the compensator time constant T*, an
adjustment criterion is of primary importance. Because
of the expected small lag and dead times for electrical
servo drives not all adjustment strategies are suitable.
Consequently the oscillation magnitude of xc(t) is com-
pared with the expected magnitude of the ramp function
xi(t) (Figure 2) and T* is adjusted according to the mag-
nitude ratio in Equation (6).
**
1
ˆ
ˆ
i
nn
c
x
TT
x
 (6)
For the evaluation, only the extreme values of xc(t)
have to be determined, which entails a high toleration of
measurement uncertainty even for lag times which are
smaller than the sample time of the motion controller.
The dead time of the system (Td) can be determined by
the time behaviour of xc(t) and the relay output y(t) [9].
2.1. Velocity Control Loop
The velocity controller (GVC) is typically implemented as
PI controller. The subordinated system consists of the
closed current loop (GCuL) and a mechanical system
(Gmech) which is mainly characterized by the total mo-
ment of inertia J and an oscillatory system. The resulting
structure with an additional nonlinear friction moment
(Mfric) is displayed in Figure 3.
2.2. Identification of the Velocity Controlled
System
The derivation of a parametric model of the velocity
controlled system consists of three steps. In the first step,
the closed current loop is described by using the FOPDT
identification approach, introduced in Section 2. In a
second step, it is shown, how the proposed adjustment
strategy can be applied to a first order plus integral plus
Figure 2. Proposed criterion for compensator adjustment.
Figure 3. Structure of the velocity control loop.

R. NEUGEBAUER ET AL.253
dead time (FOIPDT) system behavior in order to esti-
mate the moment of inertia J. In step three the mechanic-
cal system is classified according to its first resonance
frequency.
Step 1: Identification of the Closed Current Loop
For the closed current loop a first order plus dead time
model with known gain was proposed in [4].
 
 
1
1
d
sT act
CuL
com cur
Ms k
Gse k
MsTs






(7)
The compensator time constant is estimated by
using the structure according to Equation (1) and the
proposed adjustment criterion, applied to the problem of
the closed current loop.

*
cur
T
 
**
1
ˆ
ˆ
i
cur ncur n
c
x
TT
x
 (8)
Step 2: Identification of the Total Moment of Iner-
tia
For the illustrated closed velocity loop the effective
acceleration torque M
acc can be described by the differ-
ence of the actual drive torque Mact and the friction mo-
ment (Figure 3):
accact Fric
MMM (9)
In case the mechanical system is regarded as a single
mass system, the angular momentum can be written as:
acc act
MJ


 (10)
The moment of inertia can not be identified based on
Equation (10), because the acceleration torque Macc can
not be measured. Consequently, an alternative solution
has to be discribed. Therefore, the velocity controller
GVC in Figure 4 has to be substituted with a relay with
hysteresis represented by Equation (13).
The friction moment characteristic in Equation (9) is
nonlinear.

Fric
Mf

 (11)
With the following approach, this nonlinearity has not
to be considered. Instead MFric can be assumed as constant
at selectable operation points for the command velocity
ωcom:

*.

F
riccom op
Mconstfor

  (12)
Considering the gain of the closed current loop in Equa-
tion (7) the position of the summation point in the open
Figure 4. Identification of the moment of inertia.
loop can be moved as shown in Figure 5.
The relay output Mcom in Figure 5 is defined:
*
*
0
0; 2
2
err Hyst
comFricHyst errHyst
F
ricHysterr
MM
M






 



(13)
To use the proposed identification technique for the
moment of inertia, the structure of the velocity loop has
to be expanded by a compensator as well.
Based on the identified closed current loop, the relay
output Mcom can be used for the calculation of the actual
torque Mact. The forward path of the loop in Figure 4 be-
comes a FOIPDT model. Hence, the proposed identifica-
tion method (Figure 1) has to be modified by using a
different compensator structure leading to Figure 6:
 



*
1
1
d
act
CuL mech
com Fric
s
T
cur
s
GsG s MsM
e
Ts Js


 

 
(14)
The influence of the lag time on the actual velocity
ωact can be eliminated by using the modified compensa-
tor


c
Gs

.










**
1
1
d
simpact c
comFric cur
s
T
cur
ssGss
MsMTs e
Ts Js





 
(15)
In the case of an adjusted closed current loop time
constant:





*
*
d
comFric sT
s
impcur cur
MsM
seTT
J
s





(16)
The further derivation is carried out in the time domain.
Especially, the time behaviour of ωsimp is of interest:





*
11 d
comFric sT
simp
MsM
LsL e
Js















(17)
Figure 5. Alternative open loop structure.
Figure 6. Open velocity loop with compensator (grey).

R. NEUGEBAUER ET AL. 254
The dead time does not have to be considered, because
only the magnitude ratio is significant.
 

*
1d
simpcom Fric
tMtMt
J
  
 (18)
The closed loop with relay controller (Figure 4)
achieves oscillation with the time period (TPer). At the
point of time t = 0 the relay output changes to the third
case of Equation (13). Consequently, the oscillation at
the operation point can be expressed as a sum for the
sampled system with the sample time ts.

2**
0
2
1
Per Per
ss
Per
s
TT
tt
s
impFric sFrics
T
kkt
tMktMk
J





 




t





(19)
The structure, shown in Figure 6 is used for the cal-
culation of the moment of inertia. A half-cycle of the
oscillation (19) is sufficient for measuring the magnitude
of ωsimp (simp ). Finally, the resulting formula for calcu-
lation the moment of inertia with
ˆ
ω
ˆ
s
imp

is:
*
2
ˆˆ 22
com
F
ric PerPer
simp simp
M
M
T
J




T
(20)
It has to be pointed out that for this approach the sig-
nificant resonance frequencies have to be damped ade-
quately. For this purpose, notch filters GCF for the current
command value can be used [1]:

2
0
2
00
12 2π2π
12 2π2π
N
CF
D
ss
dff
Gs
ss
dff

 




 



0 
(21)
Step 3: Classification of the mechanical system
The classification of the mechanical system is based
on the location of the first antiresonance frequency fN
compared to the frequency of the 0 dB crossing fd of the
open loop. There are two cases in Table 1 that have to be
considered in order to find a suitable controller setting
for the mechanical system [2].
The location of the resonance frequency and the an-
tiresonance frequency fN can be estimated by using stan-
dard methods of the spectral analysis, such as the Fast
Fourier Transformation [10]. Consequently the motor
moment of inertia JM can be estimated according to
Equation (22).
Table 1. Characterization of the mechanical system.
Coupling Classification Considered inertia for tuning rule
Fixed/Stiff fN >> fd Total moment of inertia (20)
Soft fN << fd Motor moment of inertia (22)
2
0
N
M
f
JJ
f




(22)
After carrying out these three steps, the parametric
model for the velocity controlled system is complete. As
mentioned above, it can be used for the design of the
velocity controller, based on well known tuning rules,
listed in the next section.
3. Tuning Rules
According to [11], “the most direct way to set up con-
troller parameters is the use of tuning rules”. There are
various tuning rules for a wide range of parametric mod-
els put together in [11]. Since the typical structure of a
velocity controller in industrial servo drives is PI, the
choice is reduced to PI controller rules for parametric
models. The structure of the controller is:

1
1
VC P
N
GsKTs






(23)
The extracted tuning rules Table 2 differ in their suit-
ability for electrical servo drives and will be benchmarked
with various criteria in section 4.
4. Experiments and Results
4.1. Experimental Set-Up
The presented approach has been verified on an experi-
mental rig, as shown in Figure 7. It is equipped with the
SIEMENS motion controller SIMOTION D445 and SI-
NAMICS drives. The motion controller is sampled with
500 μs and the drive components with 125 μs.
The experimental set-up contains of a two-mass-sys-
tem. The basic parameters are the total moment of inertia
J, the resonance frequency f0 and antiresonance fre-
quency fN. The preset values of the parameters for the
experimental set-up are shown in Table 3.
Table 2. PI controller tuning rules for FOIPDT model.
Tuning rule KP [Nms/rad] TN [s]
McMillan
2
0.65 2
1.477
1
cur
dcur
d
T
JTT
T

















0.65
3.33 1cur
d
d
T
TT







Sym.
Optimum [4]

2dcur
J
TT

4dcur TT
Samal

π
4dcur
J
TT


3.3 dcur TT

R. NEUGEBAUER ET AL.255
Figure 7. Experiment set-up with motion controller.
Table 3. Parameters of the experimental set-up.
Configuration J [kgm²] f0 [Hz] fN [Hz]
Two-mass-system
(Stiff coupling)
6
1355 10  422 333
4.2. Identification Results
The closed current loop (7) under relay feedback and the
identified model are shown in the following time plot. The
identified model has been calculated in the sample time of
the motion controller. Even for a lag time which is smaller
than the sample time, the reaction curves of the real values
and the modelled values are nearly identical Figure 8.
Hence, the performance of the chosen adjustment strategy
is proven.
According to Equation (13), the hysteresis of the relay
is a free selectable parameter. Therefore, a compromise
between the magnitude of the relay oscillation and the
linearization error for the friction moment (12) has to be
found. The moment of inertia for the mechanical system
at different operation points is plotted with a variable
hysteresis in Figure 9.
The graph demonstrates that the value of the moment
of inertia for the mechanical system can be identified
with sufficient accuracy. A variance of less than 4% can
be achieved for a hysteresis to operation point ratio >0.1.
Comparing the achieved moments of inertia to other in-
vestigations [12,13], the experiments have shown an im-
provement of the accuracy. Simultaneously, a small
value of the velocity (operation point) is sufficient. Hen-
ce, the stress to the mechanical system is smaller. Conse-
Figure 8. Time behavior of closed current loop.
Figure 9. Results for the total moment of inertia.
quently the FOIPDT model (14) can be written with the
identified parameters.
4.3. Controller Design
Using the identified parametric model from Table 4 and
the tuning rules for PI controller from Table 2, the
achievable phase margin and gain margin can be calcu-
lated.
The typical phase margin for set point response is a
range of 70° - 40° and for disturbance response a range
of 50° - 20° [4]. As it is recognizable in Table 5, all
listed tuning rules can be classified as sufficient for a
good disturbance response.
4.4. Achievable Results
The behavior of these tuning rules on disturbance steps is
shown in Figure 10. In addition, the internal tuning rule
of the drive system is listed as “automatic”.
The introduced tuning rules show a better disturbance
response, compared to the automatic tuning of the drive
system. As expected from the open loop stability calcu-
lation Table 5, McMillan and Samal have the smallest
settling times.
Table 4. Identified parameters for controller design.
Model J [kgm²] Td [s] Tcur [s]

1
d
sT
cur
e
J
Ts s


  6
1340 10  3 0.25 10  3 0.4 10


Table 5. Estimated open loop characteristics.
Tuning rulefd [Hz] Gain margin [dB] Phase margin [°]
McMillan 238 5.91 19.2
Sym.
Optimum 129 13.3 35
Samal 188 8.76 25.9

R. NEUGEBAUER ET AL. 256
Figure 10. Disturbance re sponse (ste p function).
The validation of the setpoint response is divided in
two time plots. In a first step the original structure of the
PI-controller is used (Figure 11).
As known from the Symmetrical Optimum, all set-
point responses show an overshoot of up to 80%. Con-
sequently an additional setpoint filter has to be used in
the command value branch. The filter is described by the
following equation:


1
1

F
ilterF N
F
Gs TT
Ts

 (24)
The filter time constant TF was set equal to TN [4] for
all experiments (Figure 12).
By using the proposed filter, the overshoot is reduced
to a range of 0% - 10%. The settling time for all ap-
proaches is about 10ms.
5. Conclusions
In this paper a new identification method of parametric
models for velocity loop parameters in the time domain
has been presented. As an excitation, the auto relay
feedback experiment has been used and has been com-
bined with the method of gradual pole compensation.
Figure 11. Setpoint respon se without filter.
Figure 12. Setpoint response with filter.
The model parameters are identified by applying a cri-
terion, which compares the magnitudes of two signals. A
high accuracy of the model parameters has been achieved.
The advantages of the approach are a less necessary a
priori knowledge compared to other identification meth-
ods, the possibility of a simultaneous identification of
various parameters and a low but sufficient excitation of
the me- chanical system.
The presented algorithm has successfully been imple-
mented as an automatic tool in the motion control system
SIMOTION. Based on the identification results the PI
velocity controller was designed by using various tuning
rules. The achievable results for setpoint and disturbance
responses have been compared.
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R. NEUGEBAUER ET AL. 257
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