Content uploaded by Max Mulder
All content in this area was uploaded by Max Mulder on Jun 18, 2015
Content may be subject to copyright.
Identiﬁcation of the Feedforward Component in
Manual Control with Predictable Target Signals
F. M. Drop, Student member, IEEE, D. M. Pool, Student member, IEEE, H. J. Damveld,
M. M. van Paassen, Member, IEEE, and M. Mulder
Abstract—In the manual control of a dynamic system, the
human controller often follows a visible and predictable reference
path. Compared to a purely feedback control strategy, perfor-
mance can be improved by making use of this knowledge of the
reference. The operator could effectively introduce feedforward
control in conjunction with a feedback path to compensate
for errors, as hypothesized in literature. However, feedforward
behavior has never been identiﬁed from experimental data,
nor have the hypothesized models been validated. This paper
investigates human control behavior in pursuit tracking of a
predictable reference signal while being perturbed by a quasi-
random multi-sine disturbance signal. An experiment was done in
which the relative strength of the target and disturbance signals
were varied systematically. The anticipated changes in control
behavior were studied by means of an ARX model analysis and by
ﬁtting three parametric human controller models: two different
feedback models and a combined feedforward and feedback
model. The ARX analysis shows that the experiment participants
employed control action on both the error and the target signal.
The control action on the target was similar to the inverse of
the system dynamics. Model ﬁts show that this behavior can be
modeled best by the combined feedforward and feedback model.
I. INTRO DUC TI O N
MANUAL control of a dynamic system requires the
Human Controller (HC) to efﬁciently steer the system
along a certain target path while being perturbed by distur-
bances. An example is driving along a winding road while the
car’s motion is perturbed by wind gusts. The HC uses various
sources of information, like visual information of the outside
world and vestibular or somatosensory information on the
current state of the system. To study manual control, real-life
situations are often simpliﬁed to tracking tasks. The example
above can be represented as a combined target following and
disturbance rejection task.
Previous research on manual control behavior has mostly fo-
cussed on compensatory behavior, in response to unpredictable
target signals. The resulting control task is, however, not di-
rectly representative for realistic ﬂight and driving maneuvers.
Therefore, in this paper we consider behavior in response to
more realistic and predictable target signals.
The various control strategies the HC can use during track-
ing tasks have been grounded in the Successive Organization
of Perception (SOP) scheme of Krendel and McRuer , .
F. M. Drop is with the Max Planck Institute for Biological Cybernetics,
T¨ubingen (Germany), firstname.lastname@example.org.
All other authors are with the Control and Simulation section, Faculty of
Aerospace engineering, Delft University of Technology (Netherlands).
This scheme distinguishes three levels of control, i.e., compen-
satory –, pursuit –, and precognitive control –
, through which the HC might proceed when learning a
new control task.
The compensatory strategy consists of controlling solely on
the ‘error’ between the system output and the target signal,
in a closed-loop feedback fashion. It is used by the HC when
little experience with the control task is available or when
confronted with unpredictable target signals presented on a
compensatory display, that shows only the tracking error.
Wasicko et al.  investigated the hypothesized pursuit
strategy for unpredictable target signals by comparing the
compensatory display to the pursuit display, which explicitly
presents the target, the system output and the error. HC
behavior was measured to be different and pursuit display
performance was better, suggesting that the HC was using
a combination of feedforward control on the target signal
and feedback on the remaining error. Feedforward control is
deﬁned as all control actions based on the target signal: either
from perceiving the target on the display or from memorized
or inferred knowledge on the target signal properties.
The highest level of the SOP, precognitive control, is deﬁned
as an open-loop feedforward mode in which the HC executes
a learned control input with little to no feedback.
Magdaleno et al.  hypothesized that the HC might
reach the pursuit and precognitive control stages faster with
predictable target signals. A signal is considered predictable
when the remaining course of the signal can be predicted
after the onset of a signal segment is recognized by the
HC. Refs.  and  found evidence for feedforward
behavior in performance metrics for predictable single sine
target signals. The tracking lags of 50 ms in response to
double sine target signals, as reported by Yamashita ,
are 150-200ms lower than typical lags found in compensatory
feedback-only tracking, Ref. , and can only be explained
by a signiﬁcant feedforward component in the HC control
behaviour. Despite this empirical evidence supporting the feed-
forward hypothesis, feedforward behavior was never found by
system identiﬁcation techniques nor were feedforward models
developed and validated by experimental data.
It is the aim of the present paper to identify the ex-
pected feedforward behavior in response to predictable target
signals from experimental data and to develop a model of
this feedforward component. Identifying the compensatory
and feedforward components simultaneously requires both a
target and a disturbance signal of considerable magnitude
. The addition of a disturbance signal might negatively
F. M. Drop, D. M. Pool, H. J. Damveld, M. M. Van Paassen, and M. Mulder, “Feedforward
Behavior in Manual Control Tasks with Predictable Target Signals,” IEEE Trans. on
Cybernetics, vol. 43, no. 6, pp. 1936–1949, 2013.
inﬂuence the ability of the HC to exert feedforward action
on the target, however. Ref.  only found evidence for
a feedforward operation in response to a predictable target
signal when the quasi-random disturbance was not present.
Hence, the identiﬁcation requirement to insert an additional
disturbance signal might harm the feedforward operation we
intend to identify. For this purpose, the relative strength of
the predictable target signal and the unpredictable disturbance
signal will be systematically varied over a broad range. Our
main hypothesis is that the feedforward path can be identiﬁed,
and that it is similar to the inverse of the system dynamics.
The experimental data will be collected from a realistic
control task that resembles aircraft pitch attitude tracking.
The target signal is composed of predictable ramp segments,
whereas the unpredictable disturbance signal is a sum-of-sines
signal. The controlled element is a single integrator, a highly
simpliﬁed model of aircraft elevator to pitch angle dynamics.
The proposed identiﬁcation of feedforward behavior will
be done through two independent system identiﬁcation tech-
niques. First, an ARX model analysis is used which does not
enforce a particular model structure to ﬁt the data. Second,
three different parametric human control model structures are
ﬁt using a time-domain maximum likelihood method: a basic
error feedback model, an extended error feedback model, and
a model combining feedback on the error with an explicit
feedforward operation on the target.
The paper is structured as follows. Section II further intro-
duces the SOP, and compensatory and feedforward control.
Section III describes the models we will use to study the
observed behavior and makes a prediction of what control
strategy can be expected for what situation. Section IV de-
scribes the details of a human-in-the-loop tracking experiment.
The results of this experiment are presented in Section V. The
paper ends with a discussion and conclusions.
A. The Successive Organisation of Perception
1) The control task: This paper focuses on human control
behavior in a combined target tracking and disturbance re-
jection task, shown in Fig. 1. The HC controls the dynamic
system Ycsuch that the error, deﬁned as the target minus the
system output, or e=ft−θ, remains as small as possible.
Meanwhile, the system is perturbed by disturbance fd.
+ + +
Fig. 1. Control scheme studied here. The HC can use ft, the system output
θand the error eto generate the control signal u.
2) Task variables and learning: The SOP theory postulates
three levels of control through which the HC might proceed
while learning a particular control task , . That is, the
achieved level of control in the HC is a function of the task
variables and his obtained experience with the task. In the
ﬁrst stage (compensatory), the human only responds to the
error and control behavior can be modeled as pure feedback
control. In the second stage (pursuit), the HC uses perceived
information on the target signal ftand the system output
θin addition to compensatory action on the error signal e,
to improve performance, see Fig. 2. The signal nindicates
remnant, accounting for non-linearities present in the HC,
and is the residual of the control signal that is not modeled
by the linear model. In the third stage (precognitive), the
HC recognizes a pattern in the target signal and selects an
appropriate learned response to be used in open-loop fashion.
Fig. 2. Linear model of multi-loop pursuit behavior.
Whether or not sufﬁcient learning will lead to the achieve-
ment of a particular level of control depends on the task
variables. Relevant task variables are 1) the tracking display,
Fig. 3, either the compensatory or the pursuit display ,
, 2) the system dynamics , 3) the properties of the
target and disturbance signals (forcing functions), and 4) the
presence of additional cues (e.g., vestibular).
(a) Compensatory display
(b) Pursuit display
Fig. 3. Compensatory and pursuit displays for pitch control. Both displays
only show the current values of the signals. No post or preview information
3) Predictable target signals: An important property of
the target signal is whether or not it is predictable. The two
main dimensions of target signal predictability, as identiﬁed by
Ref. 13, are signal coherence and waveform shape complexity.
On the predictable end of the spectrum are single sine waves,
which are very coherent and have a simple waveform. On
the other end of the spectrum are the unpredictable signals,
such as ﬁltered white noise and multi-sine signals with many
Realistic ﬂight and driving maneuvers are similar to discrete
patterns with a high coherence, as described in Ref. , such
as steps and ramps. Such discrete patterns are predictable,
because once the onset of the pattern is recognized, the
remainder is also known. Two examples of predictable discrete
patterns are given in Fig. 4.
McRuer at al.  hypothesized that three phases can be
distinguished in the response to a step target signal. After a
short delay phase (I), the HC perceives an unusually large
error and recognizes the step in the target signal, for which an
appropriate response is available. During the rapid-response
phase that follows (II), the HC is hypothesized to switch to an
open-loop control strategy in order to quickly reduce the error,
and then switch back to compensatory control to suppress
remaining errors (III).
(a) Response to step target
(b) Response to ramp target
Fig. 4. Typical responses, and deﬁnition of the response phases to two
predictable discrete patterns.
This paper is concerned with target signals composed of
ramps, not steps. We therefore propose a similar subdivision
of the response to a ramp target signal, Fig. 4(b). Due to the
delay phase (I), during which the human controller is unaware
of the onset of the ramp, one is suddenly confronted with
an error whose magnitude depends on the ramp velocity. We
hypothesize that in response to this error, the HC might also
switch to an open-loop control strategy (phase IIa).
During phase IIb, designated the “ramp-tracking phase”, the
HC has to match the velocity of the system to the velocity of
the target. During this phase, the HC has likely recognized
the signal as a ramp and can make use of its predictability
property for the remainder of the ramp. We therefore expect
to see feedforward behavior during phase IIb and will focus
our further analyses on this phase.
B. Compensatory control models
In every practical control situation, there will be some
unpredictability, causing errors that can only be corrected for
in a closed-loop feedback fashion, see Fig. 5.
+ + +++
Fig. 5. Single-loop compensatory model. The HC only perceives the error e
or is assumed to respond only to the error, even if other signals are available.
McRuer et al.  captured the fundamentals of compen-
satory manual control with the Crossover model, stating that
the human dynamics adapt to the system dynamics , .
The Extended Crossover model approximates linear controller
dynamics in the crossover region for unpredictable forcing
functions and a compensatory display to:
Ype(s) = Kp
TIs+ 1 e−sτe,(1)
with the equalization in parentheses and an equivalent time
delay τe. It is only valid near the crossover frequency ωc.
The Simpliﬁed Precision model, Ref. , describes human
dynamics for a wider frequency range. For frequencies be-
yond crossover, neuromuscular dynamics were added, studied
extensively in Refs. –. The combined manipulator and
human neuromuscular dynamics are commonly modelled as:
Ynms(s) = ω2
s2+ 2ζnmsωnms s+ω2
with natural frequency ωnms and damping ζnms .
C. Feedforward control models
For compensatory control it has been shown that increased
tracking performance requires a higher crossover frequency
. However, due to the presence of time delays in the closed-
loop system (originating from the HC), there is a limit to which
ωccan be increased without sacriﬁcing closed-loop stability.
In case of a known, predictable and perceivable target signal
in addition to the error e, the HC may apply a feedforward
control action. This could increase performance without sac-
rifycing stability. An ideal feedforward control law would be
equal to the inverse system dynamics , :
Yc(s)⇒u(s) = 1
The system output θis then found to be:
θ(s) = Yc(s)·u(s) = Yc(s)·1
Yc(s)·ft(s) = ft(s),(4)
i.e., exactly equal to the target ft, yielding zero tracking error.
III. CONTRO L BE HAVI OR MOD EL S A ND SIMU LATIO NS
A. Characteristic forcing function properties
We aim to model and identify feedforward strategies in
a control task with a target signal composed of predictable
ramp-segments and an unpredictable multi-sine disturbance
signal. Ref. 17 performed an experiment with two variations
in the target (short and fast versus long and slow ramps) and
two variations in the disturbance (no or a strong disturbance),
also for single integrator dynamics. Evidence of feedforward
behavior was found only in the condition with long and
slow ramps without disturbance, but not when a disturbance
was present. A disturbance signal of a certain magnitude is
however needed for identiﬁcation purposes. In this study, we
will therefore systematically vary the steepness of the ramps
and the magnitude of the disturbance.
The different variations of the target and disturbance signals
all relate to two baseline signals (f∗
d) by a simple gain:
The baseline target signal f∗
tis composed of a series of ramps
with a steepness of 1 deg/s, such that multiplication by q
results in ramps with a steepness of qdeg/s. The duration of
the ramp segments is constant, such that qalso affects the ﬁnal
amplitude of the target. Both forcing functions are discussed
in more detail in Section IV-A3.
B. Ramp-tracking performance metrics
Before introducing the three models to be studied, an
analytic performance metric is derived that will allow us
to compare the models. The analytic performance metric
is the error during the ramp-tracking phase as deﬁned in
The following was ﬁrst derived by Wasicko et al. 
and is based on the scheme of Fig. 2. When modeling the
HC as a linear controller, either Ypt,Ypeor Ypθcan be
omitted, because of the linear relationship between e,ftand θ.
Thus, the different responses cannot be identiﬁed separately.
We decided to omit Ypθbecause we expect a response on
the predictable target signal ftand the model would thus
have a higher resemblance to the HC. Also, the reduced
control scheme is then similar to feedforward control schemes
employed in common automatic controllers. The closed loop
transfer function of error edue to target ftthen equals:
1 + Yβ(s),(6)
with the ‘equivalent open-loop’ describing function Yβ:
Yβ(s) = Yc(s) (Ypt(s) + Ype(s))
The steady-state error of the controller to a certain target ft
can be calculated using the Final Value Theorem :
ess = lim
t→∞ e(t) = lim
A ramp input signal, starting at t= 0, of inﬁnite duration, and
with steepness qis given, in the Laplace domain, as ft(s) =
q/s2. Substituting this relation and Eq. 6 into Eq. 8, the ramp-
tracking error of a generic controller in control of Yc=Kc/s
to this ramp input with steepness qequals:
eramp = lim
1 + Yβ(s)= lim
1 + Yβ(s).(9)
This metric will be used to evaluate the ramp-tracking perfor-
mance of the three models to be introduced next.
C. Compensatory control models
1) Basic Compensatory Model: Two compensatory mod-
els will be postulated. For the manual control of integrator
dynamics, humans can be modeled as a feedback controller
with gain equalization only . This compensatory model is
referred to here as the Basic Compensatory Model (BCM):
pe(s) = Kpee−sτpeYnms(s).(10)
Substituting Eq. 10 in Eq. 9 and setting Ypt= 0, the ramp-
tracking error of the BCM is found:
ramp = lim
1 + Yβ(s)=q
Tracking performance improves for higher controller gain
Kpe, at the cost, however, of closed loop stability.
2) Full Compensatory Model: The HC might adopt a
compensatory control strategy that better suits ramp-tracking,
with equal stability. To obtain insight in what this strategy
might be, we return to the equalization terms in the Simpliﬁed
Precision model, in its most generic form :
Ype(s) = K′
TIs+ 1 e−sτpeYnms (s),(12)
which can be rewritten to the equivalent:
pe(s) = Kpe
referred to here as the Full Compensatory Model (FCM).
Rewriting Eq. 12 into 13 clariﬁes the trade-off between
stability and performance in the following, as ωIand ωLhave
an effect on both the static gain (and thus system stability)
and the ramp-tracking performance. TLand TIhave the same
effect, but are less convenient.
The FCM acts as an integrator between ωIand ωL, for
ωI< ωL. Substituting Eq. 13 into Eq. 9 yields:
ramp = lim
1 + Yβ(s)=q
Hence, in addition to the gain Kpe, the lag and lead corner
frequencies ωIand ωLalso affect ramp-tracking error. It can
be improved, as compared to the BCM, by keeping ωI< ωL
and taking ωIas low as possible. A Bode plot of the resulting
transfer function, shown in Fig. 6 for two values of ωI(Ype1
and Ype2), illustrates the effect of ωIon the controller phase
margin. Controller Ype1, with a smaller ωI, yields smaller
ramp-tracking errors than controller Ype2, at the cost of a
smaller phase margin. Clearly, with a compensatory control
strategy a trade off between open loop gain at low frequencies,
and thus error reduction there, and stability, is inevitable.
10-2 10-1 100101
10-2 10-1 100101
Fig. 6. Bode plot of the FCM for two values of ωI.∆Phase margin indicates
the difference in phase margin between the two controllers.
D. Feedforward control model
The proposed Feedforward Model (FFM) consists of a
compensatory control loop augmented with a feedforward path
acting directly on ft, see Fig. 7.
The feedforward path YFFM
ptconsists of an equalization
term, inverse system dynamics, a time delay and the neuro-
muscular dynamics of Eq. 2:
pt(s) = Kpt
TIs+ 1 ·1
Fig. 7. The FFM, a combined feedforward and compensatory HC model.
It will dominate the control contribution necessary to track
ft. Hence, the error ewill be caused primarily by the dis-
turbance signal fd. We limit our study to disturbance signals
with frequency content around the crossover frequency and
assume that compensatory control action in the presence of
feedforward action can be modeled by means of the Simpliﬁed
Precision model tuned for single integrator dynamics .
Thus, the compensatory path of the FFM is equal to the BCM:
pe(s) = YBC M
pe(s) = Kpee−sτpeYnms(s).(16)
Substituting Eqs. 15 and 10 in Eq. 9, the ramp-tracking error
of the FFM is found to be:
ramp = lim
1 + Yβ(s)=q
which can be brought to zero by choosing Kptequal to 1,
without affecting closed loop stability.
Fig. 8 shows a typical response of the FFM. For single
integrator system dynamics, ramp tracking requires a constant
non-zero input and a distinctive “plateau” in the control signal
emerges. During a “hold segment” the control signal is only
due to the disturbance fdand the feedforward path Ypthas
zero output. During a “ramp segment” both feedforward and
compensatory paths contribute to the control signal.
Hold segmentRamp segment
0 5 10 15 20
Fig. 8. Typical simulated response of the FFM to a ramp target and
disturbance signal (q= 3 deg/s, Kd= 1, remnant is zero).
The feedforward controller, Eq. 15, is identical to the one
proposed in Ref. . The lag equalization TIonly affects the
rise time of the feedforward control (upt, Fig. 8) just following
the ramp onset. Because this rise time is very short, it will be
difﬁcult to estimate TIfrom noisy data. Therefore, based on
the results of Ref. , it is ﬁxed here to 0.2 seconds.
The feedforward model of McRuer et al. [2, p. 59] is similar
to the FFM, but contains explicit ‘switches’ to model changes
in behavior during the control task. For ramp target following,
this might occur during phase IIa (see Fig. 4(b)). Explicitly
modeling these switches is beyond the scope of this study.
The target signal ftwas designed such that only a few ‘ramp-
onsets’ were included, making the possible effects of switching
behavior on the analysis as small as possible.
E. Simulations of control behavior
To predict the effect of the two task variables on control
strategy selection and to design the experiment conditions we
will ﬁrst make use of simulations by means of linear models.
Since the models are linear, the forcing functions are scaled
linearly (no change in frequency content) and only relative
metrics are considered, the analyses will not be affected by
the absolute values of qand Kd. Therefore, the analyses are
performed as a function of the ratio of qand Kd, referred to
here as the Steepness Disturbance Ratio (SDR):
SDR = q
The steepness qis expressed in degrees per second and the
disturbance gain Kdis a unitless quantity.
All simulations apply the exact forcing functions used in
the experiment, described in Section IV. The model parameter
values are listed in Table I. To allow for a fair comparison, the
model parameters are chosen such that the phase margin of the
compensatory path of the controller is equal for all models (55
deg). The choice of the value for the phase margin is based
on the available measurement data of Ref. .
MOD EL PARA M ET ER VAL UE S U SE D I N S I MU LATI O NS .
Model KptτptKpeωLωIτpeωnms ζnms ωc
s rad/s rad/s s rad/s rad/s
BCM - - 2.5 - - 0.2 12 0.2 2.6
FCM - - 2.0 0.3 0.05 0.2 12 0.2 2.1
FFM 1 0.2 2.5 - - 0.2 12 0.2 2.6
1) Prediction of control strategy selection: The metric
used to predict the control strategy selection is the relative
performance improvement of the FCM and FFM over the
BCM. It is deﬁned as the variance of the error of the BCM
divided by the variance of the error of the FCM or FFM:
Performance increase = σ2
The simulated performance increase is shown in Fig. 9 for a
large range of SDR. It shows that, as expected, the alternative
control models predict a better performance for higher values
of SDR. However, the FFM has a performance increase that
is considerably larger than the FCM.
For a low SDR, the FCM performs worse than the BCM.
The low frequency lag degrades performance if the distur-
bances are large relative to the ramps. Hence, utilization of low
frequency lag does not only involve a trade-off between perfor-
mance and stability, but also a trade-off between disturbance
attenuation performance and ramp-tracking performance.
The FFM is, in contrast to the FCM, able to achieve a better
or equal performance compared to the BCM, for the complete
range of the SDR. Apparently, the feedforward control strategy
does neither involve a trade-off between performance and
Basic compensatory (BCM)
Full compensatory (FCM)
10-2 10-1 100101102103
Fig. 9. Relative performance increase of the FCM and FFM models with
respect to the BCM (baseline).
stability, nor a trade-off between disturbance attenuation and
Concluding, it is likely that for SDR values larger than 1
the HC will change behavior from adopting a BCM strategy
to a low-frequency lag strategy, FCM, or, with much better
performance, a feedforward strategy, FFM.
2) Choice of conditions: From Fig. 9 it can be concluded
that the most interesting range in SDR is between 1 and
10, since the differences in performance of the three control
strategies vary greatly. The practical constraints on the exper-
imental display (screen size, resolution and the visibility of
the disturbances due to fd) pose an upper limit on the ramp
steepness qin the order of 4 deg/s. To attain an SDR of 10, the
disturbance gain Kdshould be equal to 0.4 or lower. By means
of a small experiment it was veriﬁed whether Kd= 0.4would
cause the HC behavior to change signiﬁcantly. This was not the
case as will be shown in the results of the main experiment.
IV. EXP E RI MEN T
To validate the proposed models and the theoretical concepts
concerning the identiﬁcation of changes in control behavior, a
human-in-the-loop experiment has been conducted.
1) Apparatus: The tracking task was presented on a central
visual display in a ‘pursuit’ conﬁguration, see Fig. 3(b). The
display update rate was 60 Hz and the time delay of the image
generation in the order of 20-25 ms . The display measured
22 by 22 cm with 800 by 800 pixels resolution, and was placed
at a distance of 90 cm from the subject’s eyes. No outside
visuals and no motion cues were available.
Subjects used the fore/aft axis of an electro-hydraulic side-
stick to give their control inputs, u. The stick had no break-out
force and a maximum deﬂection of ±13 deg. Its stiffness was
set to 1.0 N/deg over the full deﬂection range, and its inertia
to 0.01 kg ·m2; the damping coefﬁcient was 0.2. The lateral
axis of the sidestick was locked.
2) Controlled element dynamics: Single integrator dynam-
ics were considered: Yc=Kc/s. The gain constant Kcwas
chosen such that subjects: 1) would never reach the maximum
deﬂection limits of the stick at ±13 deg, and 2) were still able
to give ﬁne, accurate control inputs. A gain of Kc= 1.0 met
both requirements, as will be shown by the results.
3) Independent variables and forcing functions: The inde-
pendent variables were the ramp steepness qand disturbance
signal gain Kd. There were four levels of qand three levels of
Kd, listed in Table II. As mentioned, there is an upper limit
on qof 4 deg/s and a lower limit on Kdof 0.4. To be able
to compare the experimental data of this study to the data of
Ref. , qand Kdfor the condition with SDR = 1 were
chosen equal to those in condition S3 of Ref. . That is,
q= 1 and Kd= 1. The other values were chosen in-between
the maximum and minimum values of qand Kd. Fig. 9 shows
the resulting SDR values of the nine ramp-tracking conditions.
Note that for the R0 target signal ft= 0, reducing the
task to compensatory disturbance rejection only. The three R0
conditions were added to observe the compensatory behavior
for the three levels of Kd, necessary to correctly interpret the
results of the conditions with a ramp target signal.
Each subject performed each combination of ramp steepness
and disturbance signal gain, resulting in a total of twelve
conditions. A condition will be referred to in this paper with
the syntax “RxDy”, where x indicates the Ramp steepness and
y the Disturbance gain as a percentage. For example, R1D40
designates the condition with q= 1 deg/s and Kd= 0.4.
IND EP EN DENT VAR I AB LE S.
Target signal, ftDisturbance signal, fd
Name q, deg/s Name Kd
R0 0 D40 0.4
R1 1 D70 0.7
R2 2 D100 1.0
The forcing functions ftand fdare shown, for all levels of
qand Kd, in Fig. 10. The ramp-tracking signal ftconsisted of
two short (5 s) and two long (10 s) ramps of equal steepness.
0 10 20 30 40 50 60 70 80
(a) Predictable target signals ftconsisting of ramp segments.
0 10 20 30 40 50 60 70 80
(b) Unpredictable quasi-random multi-sine disturbance signals fd.
Fig. 10. Target and disturbance signals used in the experiment.
The disturbance signal fdwas a multi-sine signal, consisting
of ten sets of adjacent frequency components. The phases
of the sinusoids were chosen such that the signal appeared
random. It was identical to the signal used in Ref. .
4) Subjects and instructions: Six subjects, all males with
extensive tracking experience, aged 24-49 years (29 years
avg.), were instructed to minimize pitch tracking error e
presented on the display. After each run the tracking score
was given: the root mean square (RMS) of the error e.
5) Procedure: Subjects performed the 12 conditions in two
sessions with six conditions. The two sessions were performed
on separate days. Conditions were randomized over subjects
using a balanced Latin square design.
The individual tracking runs of the experiment lasted 90
seconds, of which the last 81.92 seconds were used as the
measurement data. Tracking performance was monitored by
the experimenter. When subject proﬁciency in performing the
tracking task had reached an asymptote, ﬁve repetitions at this
constant level of tracking performance were collected as the
measurement data. On average, each session took 3 hours.
During the experiment, the time traces of the error signal
e, the control signal uand the pitch attitude θwere recorded
for ﬁve repetitions of each experimental condition. From u
the control signal rate of change ˙uwas reconstructed. The
ﬁve time traces were averaged to reduce effects of remnant,
resulting in one time trace for each subject for each condition.
Using these averaged time traces all dependent measures were
calculated, followed by the ARX model analysis, and the
parameter identiﬁcation of the three control models.
B. Dependent measures
1) Non-parametric measures: The error variance σ2
the tracking performance metric. The variance of the control
signal rate of change σ2
˙u, was the control activity metric. Note
that the control signal itself contains a ‘plateau’ during ramp-
tracking segments, see Fig. 8. Using its variance would suggest
that control activity increased, whereas subjects did not put
more effort in their control inputs.
To examine whether behavior is linear and constant in the
range 0.4≤Kd≤1.0, it is observed in the frequency domain,
using crossover frequencies and phase margins metrics.
2) Parametric measures: Two linear time-invariant ARX
models were estimated from the measured control signal u.
The single channel model, ARX(e), contains a response on the
error only and thus assumes a pure feedback control structure,
Fig. 11(a). Its frequency response is comparable to BCM and
FCM results. The multichannel model, ARX(ft, e), also allows
for a response on the target ftdirectly, as would be the case for
feedforward behavior, Fig. 11(b). Its frequency response can
be compared to the FFM results. ARX models are commonly
used for system identiﬁcation of a large variety of dynamic
systems , including multichannel LTI models for modeling
human behavior with two partially correlated inputs .
The ARX models are ﬁt to the measured control signal u,
yielding estimates ˆ
Ypt(s) = Bt/A and ˆ
Ype(s) = Be/A. The
ARX method does not enforce a particular model form with
parameters having physical meaning; it describes the set of LTI
models with an accuracy depending on the polynomial orders.
The latter were chosen such that the resulting model was stable
and the Akaike Final Prediction Error was minimal .
(a) Single channel model ARX(e)
(b) Multichannel model ARX(ft, e)
Fig. 11. The ARX model structures.
In addition, the three models proposed in Sections III-C
and III-D were ﬁt to the data through a time-domain Maximum
Likelihood Estimation (MLE) method .
The quality of the ARX and MLE ﬁts is expressed by the
Variance Accounted For (VAF):
VAF = 1−PN
with ˆuthe modeled and uthe measured control signal.
The analysis in section III-E revealed that a transition
from compensatory to feedforward control is most likely for
an SDR >1, based on the observation that the relative
performance improvement of the hypothesized FFM over the
BCM increases dramatically.
For the conditions without a ramp target signal (R0), it was
hypothesized that subjects’ control behavior would be invariant
and described best by the BCM (H.I).
For the conditions with ramp targets, it was hypothesized
that the subjects’ strategy depended on the SDR. For low
values of the SDR, subjects were expected to behave more like
a compensatory controller, and modelled best by the single-
channel ARX(e) model, and the BCM. Beyond a certain, yet
unknown, value of the SDR the VAF in the BCM data ﬁts was
expected to decrease, indicating that subjects changed their
For higher SDR values it was hypothesized that subjects
would employ control actions on ft, and only the multichannel
ARX(ft,e) model would be applicable to ﬁt the data. Further-
more, only the proposed FFM, and neither the BCM nor the
FCM, is expected to be ﬁt accurately. The FFM frequency
response would nicely follow the multichannel ARX estimated
frequency response (H.III).
V. RE SU LTS
A. Measured time traces
For two characteristic conditions, representative time traces
of the measured control signal u, error eand output θare
plotted in Fig. 12 (subject 1). The time traces are shown only
between 17 and 60 seconds into the experiment, to better
demonstrate the behavior during ramp-tracking. All measures
were calculated over the full 81.92 seconds measurement time.
The pitch attitude plots show that subjects could accurately
track ft; the error was never larger than ±2.5deg for all
conditions. Around the onsets of the ramps, a peak in the error
signal was observed, generally 50 % larger than the largest
20 30 40 50 60
(a) Control signal, R0D100
20 30 40 50 60
(b) Pitch and error signals, R0D100
(error is scaled by a factor 5)
20 30 40 50 60
(c) Control signal, R4D100 (ftis
scaled by a factor 0.4)
20 30 40 50 60
(d) Pitch and error signals, R4D100
(error is scaled by a factor 5)
Fig. 12. Measured time traces of e,u, and θfor R0D100 and R4D100.
error during a disturbance-rejection task with the same Kd
gain. Note that during the ramps the error esimply oscillates
around zero, similar to pure disturbance-rejection tasks.
The time traces of the control signal ushow the distinct
‘plateau’ during the ramps that becomes particularly eminent
for larger SDR values. The maximum deﬂection limits of the
sidestick were never hit by any of the subjects.
B. Non-parametric analysis of conditions without ramps
Before observing the results as a function of the SDR, the
effects of the disturbance gain Kdon behavior are examined
separately by studying the results of the R0 conditions. A
better understanding of the effects of Kdshould ensure that
the observed changes in behavior as a function of the SDR
are attributed to the correct independent variable.
We hypothesized that, as Kdbecomes smaller, it is more
difﬁcult to perceive fdand respond accurately. Fig. 13(a)
shows the error variance, normalized with the variance of
fd, dissected into error at the frequencies of fd(tracking
e,fd) and error at all other frequencies (remnant,
e,n). Indeed, task difﬁculty increases for smaller Kd’s. Note
that in this ﬁgure, and the following, the error bars show
95% conﬁdence intervals. Fig. 13(b) shows the control signal
variance, also dissected into correlated control action and
remnant. Control actions are less effective for smaller Kd’s,
with relatively more non-correlated control inputs (remnant).
These ﬁndings support H.I: it is more difﬁcult to accurately
respond to smaller disturbances, in line with Ref. .
To assess the possible changes in behavior due to the
disturbance gain Kd, the crossover frequency ωc, and phase
margin ϕmwere obtained from the non-parametric describing
functions, Fig. 14. Clearly, the crossover frequency reduces
slightly for smaller disturbance signal gains, likely because
these are more difﬁcult to perceive and control accurately. The
average phase margins do not change signiﬁcantly with Kd.
0.4 0.7 1
(a) Error signal variance
0.4 0.7 1
(b) Control signal variance
Fig. 13. Normalized variances of error and control signals, all subjects.
0.4 0.7 1
(a) Crossover frequency
0.4 0.7 1
(b) Phase margin
Fig. 14. Crossover frequency and phase margin for R0 conditions.
C. Control activity and tracking performance metrics
1) Control activity: Fig. 15(a) shows the variance of the
control signal derivative, ˙u. Surprisingly, control activity in
the R1 and R2 conditions is lower than for the R0 condition.
During conditions with ramps, subjects apparently put less ef-
fort in attenuating disturbances than during conditions without
a ramp. The same effect was reported by Pool et al. 
0.4 0.7 1
(a) Control rate signal variance
0.4 0.7 1
(b) Error signal variance
Fig. 15. Variances of the control rate and error for all conditions.
2) Tracking performance: Fig. 15(b) shows the error vari-
ance. Performance reduces for conditions with ramps, as might
be expected when another signal is fed into the closed loop.
Error growth can be due to the delay phase just after the ramp
onset (phases I and IIa), which gives a larger effect for higher
ramp rates. Also, tracking the ramp itself (phase IIb) may be
more difﬁcult for larger ramp rates.
D. Compensatory modeling results
1) Basic compensatory model ﬁts to conditions without
ramps: For the conditions without a ramp (R0), only the
BCM was ﬁt to the data. The contribution of the additional
equalization terms in the FCM can not be identiﬁed from the
R0 data. VAF values, shown in Fig. 18(a) (R0 conditions),
correspond well to values obtained in earlier research; VAFs
are slightly smaller for lower Kd’s, similar to what was
reported in other studies, e.g., Ref. .
Since the VAF indicates that the model ﬁts are adequate, the
identiﬁed parameter values, shown in Fig. 16, can be observed
to study possible changes in control behavior as function
of Kd. The identiﬁed values of the controller gain Kpeare
slightly lower for low values of Kd, which corresponds to the
lower crossover frequencies, Fig. 14(a). The time delay τpe
shows no signiﬁcant changes with Kd, which suggests that the
subjects were able to accurately perceive fdfor all disturbance
gains. Variations in neuromuscular system parameters ωnms
and ζnms are slightly more pronounced, but not signiﬁcant.
0.4 0.7 1
0.4 0.7 1
(b) τpe, s
0.4 0.7 1
(c) ωnms, rad/s
0.4 0.7 1
Fig. 16. BCM identiﬁed parameter values.
These results, and those from Section V-B, demonstrate that
HC behavior is generally constant for all conditions of Kd.
2) Basic compensatory model ﬁts to conditions with ramps:
Fitting the BCM to the data was not possible for most
conditions with ramps. The simulated control signal differed
too much from the measured control signal, for the entire
range of parameter values. Clearly, there is no information
in ealone that allows for forming a uthat corresponds to
what was measured. This is illustrated by Fig. 17(a), showing
the best ﬁt of the BCM to the measured data for subject
1 of condition R4D40. The modeled control signal remains
close to zero during the ramp-tracking segments, whereas the
measured control signal has a clear plateau. Eq. 10 states that
the modeled control signal equals the error signal, scaled by
Kpeand delayed by τpe. With the BCM the measured control
signal simply cannot be related to the measured error signal
for any combination of Kpeand τpe, see Fig. 17(b).
20 30 40 50 60
(a) Best ﬁt on condition R4D40
20 30 40 50
(b) Control and error signals, R4D40
Fig. 17. Illustration of the difference between the BCM and measured data.
The BCM did not return any sensible results for either of
the R2 and R4 conditions, the fastest ramps. The quality of the
ﬁts for the R1 conditions gradually decreased with increasing
SDR. Fig. 18(a) illustrates the VAF of the BCM ﬁtted to
the R1 conditions. It shows that R1D100 still produces an
acceptable VAF, suggesting that the model might describe
the subject’s behavior. Closer inspection of the best ﬁt, see
Fig. 18(b), reveals that the simulated control signal has a
structural discrepancy during the ramp segments which is not
observed during the hold segments. The model control signal
is offset from the measured control signal by a small amount.
0.4 0.7 1
(a) VAF values
25 30 35
(b) Best ﬁt on condition R1D100
Fig. 18. VAF and typical model ﬁts of the BCM for R0 and R1 conditions.
3) Full compensatory model ﬁts to conditions with ramps:
The FCM was ﬁt to all the conditions without any problems
and reasonably high VAF values were measured, see Fig. 19.
The FCM manages to model the plateau in the control signal
during the ramp segments much better than the BCM. The
additional lag equalization has an integrating effect on low
frequency components in the error signal, which are apparently
present in the error signal during ramp-tracking segments.
However, there are several reasons to believe that the FCM
does not accurately model the measured control behavior.
Fig. 19. VAF values of all three models for conditions with a ramp signal
(ARX results are slightly offset for visibility of the errorbars). VAF values of
the R0 conditions are also plotted, for reference, although their SDR value
(zero) is not within the horizontal axis range.
First, by comparing a typical model ﬁt to the measured
control signal (see Fig. 20) it can be seen that the best ﬁt
is clearly different from the measured u, in particular around
the ramp-tracking segments. The relative difference is largest
for conditions with a medium SDR, as also expressed by the
dip in the VAF values in Fig. 19 for 2 <SDR <5. A typical
example is condition R2D70, see Fig. 20(a). For higher SDR,
the VAF increases again, suggesting that the FCM ﬁts the
measured behavior better. However, Fig. 20(b) shows that still
a distinct difference remains during the ramp segments.
20 30 40 50 60
(a) Best ﬁt on condition R2D70
20 30 40 50 60
(b) Best ﬁt on condition R4D40
Fig. 20. Typical model ﬁts of the FCM.
Second, there is a large variance in the VAF between
subjects, as expressed by the magnitude of the 95% error bars
in Fig. 19. This indicates a high sensitivity of the model to
particular nonlinear behavioral aspects of the subjects.
E. Feedforward modeling results
The results presented in this section apply for the FFM
model presented in Fig. 7 with YFFM
peas deﬁned in Eq. 16.
Parameter identiﬁcation of the FFM was successful in all
conditions. Fig. 21 shows typical model ﬁts of two conditions,
demonstrating that the FFM is able to model the measured
control signal very well for the complete range of SDR and
for both ramp and hold segments.
20 25 30 35
(a) Best ﬁt on condition R1D100
20 30 40 50 60
(b) Best ﬁt on condition R4D40
Fig. 21. Typical model ﬁts of the FFM.
The VAF of the FFM is compared to the FCM and BCM in
Fig. 19. The FFM yields the highest VAF for all conditions,
with higher VAFs for larger SDR values.
The VAF depends signiﬁcantly on the model (F2,10 =
136.992,p < 0.05). A pairwise Bonferroni-corrected post-
hoc test showed that it is indeed signiﬁcantly different be-
tween all three models (pBC M,F C M = 0.001, pB CM, F F M <
0.001, pF C M,F F M = 0.001). The between-subject variability,
expressed by the error bars, is also much smaller for the FFM.
Clearly, the FFM is more robust against differences in control
behavior between subjects, and against remnant in general.
Since the FFM accurately models the measured behavior for
all conditions, we now study the identiﬁed parameter values,
see Fig. 22. The ﬁgure includes the estimated parameter values
of the BCM for the R0 conditions, for comparison.
0.4 0.7 1
0.4 0.7 1
(b) τpt, s
0.4 0.7 1
0.4 0.7 1
(d) τpe, s
Fig. 22. FFM identiﬁed parameter values.
The feedforward gain Kptis estimated to be somewhat
lower than 1 for all conditions, see Fig. 22(a). Kpt= 1 was
expected because this would result in the best ramp-tracking
performance, see Eq. 17. The value of Kptincreases for both
larger ramp steepnesses (F2,10 = 6.694,p < 0.05) and lower
disturbance gains (F2,10 = 6.022,p < 0.05). It is particularly
interesting to note the small error bars for the conditions with
a high SDR. Apparently, the between-subject variability for
this parameter is low and behavior is fairly constant.
The uncertainty in the feedforward time delay τptestimate
is high for the low SDR conditions, see Fig. 22(b). Note that
τpthas a small effect on the simulated control signal and is
thus very sensitive to remnant. For higher SDR conditions the
‘plateau’ in the control signal becomes more pronounced and
thus the delay between the ramp onset and the plateau start
can be estimated better. An additional effect is that subjects
might have been anticipating for a ramp segment to end, e.g.,
by remembering at what value the target would stop moving
or by remembering (or even counting) the duration of the
target movement. This would correspond with a τptequal to or
even smaller than zero. It is estimated consistently for the R4
conditions around 0.2 s, a plausible value in manual control
The compensatory control gain Kpe, Fig. 22(c), depends
signiﬁcantly on Kd(F2,10 = 20.873,p < 0.05) and is
estimated lower for the ramp conditions than for the R0
conditions. This corresponds with the higher control activity in
the latter conditions. As the control signal time traces suggest,
the HC response to disturbances is less powerful during ramp-
tracking, expressed in a lower Kpe.
Estimates of the compensatory time delay τpeare sig-
niﬁcantly higher (F2,10 = 4.857,p < 0.05) for higher
ramp steepness, suggesting that a faster ramp makes it more
difﬁcult for the HC to detect disturbances and therefore has
a more delayed response. Note, however, that the numerical
differences are very small which makes it difﬁcult to comment
on the true importance of these ﬁndings.
The neuromuscular system parameters ωnms and ζnms
(not shown) generally show the same behavior for all ramp
steepnesses and do not follow any signiﬁcant trends.
F. ARX model results
The single-channel and multichannel ARX models of
Fig. 11 were ﬁt to the experimental data. Their VAFs were cal-
culated by simulating the model, see Fig. 19. The ARX(ft, e)
model described the data very well, with a VAF of ap-
prox. 90% for most conditions. The ARX(e) model, performed
less well, with VAF values between 60 and 80%.
Estimates of Yptand Ypeare shown in Fig. 23 for four
characteristic conditions, averaged over all subjects, for both
the single channel and multichannel model. Also shown are
the analytical transfer functions of Yptand Ypeof the BCM,
FCM, and FFM, with the model simulation parameter values
as in Table I, for a qualitative comparison.
(a) ARX(e), ˆ
(b) ARX(ft, e), ˆ
(c) ARX(ft, e), ˆ
Fig. 23. ARX model identiﬁcation results.
Fig. 23(a) shows that the estimate of the response on e,
Ype, by the single-channel model ARX(e) varies much across
conditions. For high SDR values, its gain increases at low
frequencies, its phase reduces slightly around 1 rad/s.
The estimate of the response on eusing the ARX(ft, e)
model is consistently a gain up until about 6 rad/s, see
Fig. 23(b), after which a lightly-damped resonance peak
appears; the phase response contains a transport delay. Com-
paring the estimated frequency response to the analytical
responses of the BCM and FCM, it is clear that the response
resembles the BCM. That is, no integrating action nor a non-
zero phase difference is found in the experimental data, which
would suggest an FCM control strategy.
Fig. 23(c) shows the estimated response on the target signal
Ypt, using the ARX(ft, e) model. For the R0 conditions it
is close to zero, as expected. For the ramp conditions, it is
characterized by differentiating action for frequencies up until
approximately 6 rad/s. Beyond this frequency again a lightly
damped resonance peak is observed. The estimated frequency
response is very similar to the FFM and very much resembles
1/Ycover a wide frequency range.
The phase of ˆ
Yptdiffers from the FFM at higher frequencies
(>6 rad/s). The same bias was found in ARX estimations
performed on simulated data of the FFM. It was found that the
deviation is due to the inability of the ARX method to correctly
estimate the time delay in the feedforward path. That is, the
effect of the feedforward time delay is apparently not clearly
present in the measured control signal, such that it cannot be
estimated correctly. The MLE estimation of the FFM was also
unable to estimate the feedforward time delay consistently, as
shown in Fig. 22(b) and discussed in Section V-E.
VI. DI S CU SSI ON
To correctly interpret the results of the conditions with
ramps, we ﬁrst veriﬁed that HC behavior is constant through-
out the conditions without ramps. For the range of 0.4≤Kd≤
1.0this was indeed the case, although remnant increases for
lower values of Kd. This affects performance, but does not
cause behavior to change signiﬁcantly. The assumption that the
HC can be modeled as a linear controller is therefore valid for
the Kd-range investigated. Crossover frequencies were slightly
lower than reported in Ref. 3, likely caused by the different
disturbance signal spectrum .
HC behavior in the ramp-tracking conditions differs from
behavior in the disturbance rejection tasks. Control activity
reduces, and subjects responded less to the disturbances during
the ‘ramp segments’ than during the ‘hold segments’. Either
intrinsic HC limitations or a deliberate change in strategy
could be the cause. Intrinsic limitations include a worse error
perception due to the motion in the visual image during ramp
segments. Also, ramp-tracking requires the HC to attenuate
the disturbances through stick movements around a different
‘neutral point’, where the stick feel is different.
The ARX analysis unequivocally showed that HC behavior
changed to feedforward control, operating on both ftand e,
for all conditions with ramps. Independent of the SDR value,
subjects actively used the predictable target presented on the
pursuit display. The multichannel ARX(ft,e) model showed
that the feedforward response closely resembled the inverse
of the single integrator system dynamics.
The parametric model estimation conﬁrmed that the hypoth-
esized FFM, relying on inverse system dynamics, describes
the measured control response for all ramp-conditions most
accurately. Notably, the feedforward gain Kptwas estimated
somewhat lower than 1, which would correspond to the ‘ideal’
feedforward controller. This matches the observation that the
average error during the ramps was not equal to zero, but
always slightly positive, which causes the compensatory loop
of the model to contribute to the ramp-tracking control inputs
as well. This contribution then results in a decrease in the
necessary contribution of the feedforward path, expressed in
a lower value of Kpt.
The compensatory model that describes the behavior in the
ramp conditions most accurately is the FCM (although worse
than the FFM). The improvement of the VAF values of the
FCM with respect to the BCM is due to the integrating action
on the low frequency components in the error signal during
the ramp segments. This integration action is, however, not
able to fully explain the changed behavior of the HC.
The increase in VAF values of both the FCM and the single-
channel ARX(e) model for SDR >5 can be explained by
observing the deﬁnition of the VAF more closely. The VAF is
deﬁned as the variance of the error normalized by the variance
of the control signal. The control signal variance increases
much due to the plateaus in the control signal and thus the
VAF values for fast ramp conditions will always be higher
than for slow ramp conditions. Thus, the result is basically an
artifact. A better metric of the quality of ﬁt should be explored.
There was a considerable contribution of the feedforward
path for all ramp-tracking conditions, in contradiction to our
hypothesis H.II. A ‘transition’ was expected from pure com-
pensatory behavior, where the feedforward gain Kptwould be
estimated around zero, to the activation of feedforward, where
Kptwould be signiﬁcantly different from zero. This point
was expected for SDR values larger than 1, but apparently
lies below 1. Future studies should investigate whether the
transition point occures at lower SDR values.
When interpreting the results it is important to realize that
all metrics were calculated over the complete 81.92 s of the
data. The behavior of the HC differs between the ramp and
hold segments, and the effects of these differences translate
‘averaged’ into the calculated metrics. For example, control
activity was found to be lower during the ramp conditions.
However, the metric gives no information whether this was
the case during the ramp or hold segments (or both).
Finally, it is acknowledged that the current study only varied
the velocity of the ramp signals, but did not independently vary
the amplitude or time duration of the ramps. It is expected that
there is a combined effect of velocity and amplitude on the
selection of a particular control strategy. A threshold effect
might be present, where the feedforward strategy only comes
into effect if the HC knows that the ramp will be ‘sufﬁciently
long’. On the other hand, a shorter ramp might be experienced
more as a step then a ramp by the HC, causing the utilization of
an entire different control strategy altogether, for example the
switch to a pure open-loop mode as hypothesized by Ref. 2.
VII. CO NC LUS ION S
This paper studied human manual control behavior in a
pursuit tracking task with predictable, ramp-shaped target
signals in the presence of an unpredictable disturbance signal.
Three models of control behavior were postulated, a basic
feedback model following McRuer’s adjustment rules (BCM),
an extended feedback model tailored to ramp targets (FCM),
and a model combining basic feedback with a feedforward
component (FFM). The relative magnitude of the ramp tar-
get and the unpredictable disturbance signal was varied and
characterized by the Steepness Disturbance Ratio (SDR). A
model simulation analysis showed that for SDR values up to
1, all three models yield the same performance. When SDR
increases, performance improves for the FFM which employs a
feedforward loop on target. From a human-in-the-loop tracking
experiment, conducted for a range of SDR’s, we conclude
that: 1) within the SDR range investigated, human feedforward
behavior was unambiguously identiﬁed for all conditions; 2)
the hypothesized transition from compensatory to feedforward
behaviour when SDR increased was not found; 3) the feedfor-
ward response on the target signal approximates the inverse
of the single integrator system dynamics; 4) the compensatory
response on the error signal closely resembles the response
found during compensatory tracking tasks with unpredictable
targets. Supported by an independent ARX model analysis we
conclude that the combined feedback and feedforward model
(FFM) describes the data best for all conditions investigated.
Future work will address human behavior for a wider range
of Steepness Disturbance Ratio values. The current work was
performed with a pursuit display, for a better description of
real-world control behavior, further investigation with preview
displays is needed.
REF ERE NC E S
 E. S. Krendel and D. T. McRuer, “A Servomechanics Approach to Skill
Development,” Journal of the Franklin Institute, vol. 269, no. 1, pp.
 D. T. McRuer, L. G. Hofman, H. R. Jex, G. P. Moore, A. V. Phatak, D. H.
Weir, and J. Wolkovitch, “New Approaches to Human-Pilot/Vehicle
Dynamic Analysis,” Systems Technology, Inc., Tech. Rep., 1968.
 D. T. McRuer, D. Graham, E. S. Krendel, and W. Reisener, “Human Pilot
Dynamics in Compensatory Systems. Theory, Models and Experiments
With Controlled Element and Forcing Function Variations,” Wright
Patterson AFB, Tech. Rep. AFFDL-TR-65-15, 1965.
 J. C. Van der Vaart, “Modelling of Perception and Action in Compen-
satory Manual Control Tasks,” PhD thesis, Aerospace Engineering, TU
 J. I. Elkind, “Characteristics of Simple Manual Control Systems,” PhD
thesis, Massachusetts Institute of Technology, 1956.
 H. P. Bergeron, “Investigation of Motion Requirements in Compensatory
Control Tasks,” IEEE Trans. on Man-Machine Systems, vol. MMS-11,
no. 2, pp. 123–125, 1970.
 R. L. Stapleford, D. T. McRuer, and R. E. Magdaleno, “Pilot Describing
Function Measurements in a Multiloop Task,” IEEE Trans. on Human
Factors in Electronics, vol. HFE-8, no. 2, pp. 113–125, 1967.
 R. S. Shirley and L. R. Young, “Motion Cues in Man-Vehicle Control,”
IEEE Trans. on Man Machine Systems, vol. 9, no. 4, pp. 121–128, 1968.
 R. A. Hess, “Pursuit Tracking and Higher Levels of Skill Development
in the Human Pilot,” IEEE Trans. on Systems, Man, & Cybernetics, vol.
SMC-11, no. 4, pp. 262–273, 1981.
 L. D. Reid, “An Investigation into Pursuit Tracking in the Presence
of a Disturbance Signal,” in Proc. of the 5th Annual NASA-University
Conference on Manual Control, 1969, pp. 129–169.
 A. Abdel-Malek and V. Marmarelis, “Modeling of Task-Dependent
Characteristics of Human Operator Dynamics Pursuit Manual Tracking,”
IEEE Trans. on Systems, Man & Cybernetics, vol. 18, no. 1, pp. 163
 R. Pew, J. Duffendack, and L. Fensch, “Sine-Wave Tracking Revisited,”
IEEE Transactions on Human Factors in Electronics, vol. HFE-8, no. 2,
pp. 130–134, Jun. 1967.
 R. E. Magdaleno, H. R. Jex, and W. A. Johnson, “Tracking Quasi-
Predictable Displays Subjective Predictability Gradations, Pilot Models
for Periodic and Narrowband Inputs,” in Fifth Annual NASA-University
Conference on Manual Control, 1969, pp. 391–428.
 T. Yamashita, “Effects of sine wave combinations on the development
of precognitive mode in pursuit tracking,” The Quarterly Journal of
Experimental Psychology Section A, vol. 42, no. 4, pp. 791–810, Nov.
 R. J. Wasicko, D. T. McRuer, and R. E. Magdaleno, “Human Pilot
Dynamic Response in Single-Loop Systems with Compensatory and
Pursuit Displays,” Wright-Patterson AFB, Tech. Rep. AFFDL-TR-66-
 D. T. McRuer and H. R. Jex, “A Review of Quasi-Linear Pilot Models,”
IEEE Trans. on Human Factors in Electronics, vol. HFE-8, no. 3, pp.
 D. M. Pool, M. M. Van Paassen, and M. Mulder, “Modeling Human
Dynamics in Combined Ramp-Following and Disturbance-Rejection
Tasks,” in Proc. of the AIAA Guidance, Navigation, and Control
Conference, Toronto, Canada, Aug. 2-5, no. AIAA-2010-7914, 2010.
 R. W. Allen and H. R. Jex, “An Experimental Investigation of Compen-
satory and Pursuit Tracking Displays with Rate and Acceleration Control
Dynamics and a Disturbance Input,” NASA, Tech. Rep. CR-1082, 1968.
 D. M. Pool, P. M. T. Zaal, H. J. Damveld, M. M. Van Paassen,
J. C. Van der Vaart, and M. Mulder, “Modeling Wide-Frequency-Range
Pilot Equalization for Control of Aircraft Pitch Dynamics,” Journal of
Guidance, Control, and Dynamics, vol. 34, no. 5, pp. 1529–1542, 2011.
 D. T. McRuer, R. E. Magdaleno, and G. P. Moore, “A Neuromuscular
Actuation System Model,” IEEE Trans. on Man-Machine Systems, vol. 9,
no. 3, pp. 61 –71, 1968.
 H. J. Damveld, D. A. Abbink, M. Mulder, M. M. Van Paassen, F. C. T.
Van der Helm, and R. J. A. W. Hosman, “Measuring the Contribution
of the Neuromuscular System during a Pitch Control Task,” in Proc. of
the AIAA Modeling and Simulation Technologies Conference, Chicago
(IL), no. AIAA-2009-5824, 2009.
 M. M. van Paassen, “Biophysics in Aircraft Control, A Model of the
Neuromuscular System of the Pilot’s Arm,” PhD thesis, Aerospace
Engineering, TU Delft, 1994.
 K. Ogata, Modern Control Engineering, 4th ed. Upper Saddle River,
NJ, USA: Prentice Hall PTR, 2001.
 O. Stroosma, M. M. Van Paassen, M. Mulder, and F. N. Postema,
“Measuring Time Delays in Simulator Displays,” in Proc. of the AIAA
Modeling and Simulation Technologies Conference, Hilton Head (SC),
no. AIAA-2007-6562, 20–23 Aug. 2007.
 L. Ljung, System Identiﬁcation Theory for the User, 2nd ed., T. Kailath,
Ed. Prentice Hall, Inc., 1999.
 F. M. Nieuwenhuizen, P. M. T. Zaal, M. Mulder, M. M. Van Paassen, and
J. A. Mulder, “Modeling Human Multichannel Perception and Control
Using Linear Time-Invariant Models,” Journal of Guidance, Control,
and Dynamics, vol. 31, no. 4, pp. 999–1013, 2008.
 P. M. T. Zaal, D. M. Pool, Q. P. Chu, M. M. Van Paassen, M. Mulder,
and J. A. Mulder, “Modeling Human Multimodal Perception and Control
Using Genetic Maximum Likelihood Estimation,” Journal of Guidance,
Control, and Dynamics, vol. 32, no. 4, pp. 1089–1099, 2009.
 S. W. Breur, D. M. Pool, M. M. Van Paassen, and M. Mulder, “Effects of
Displayed Error Scaling in Compensatory Roll-Axis Tracking Tasks,”
in Proc. of the AIAA Guidance, Navigation, and Control Conference,
Toronto, Canada, Aug. 2-5, no. AIAA-2010-8091, 2010.
 H. J. Damveld, G. C. Beerens, M. M. van Paassen, and M. Mulder,
“Design of Forcing Functions for the Identiﬁcation of Human Control
Behavior,” Journal of Guidance, Control, and Dynamics, vol. 33, no. 4,
pp. 1064–1081, 2010.