ROBUST ANALYSIS AND CONTROL DESIGN OF THE HEAD-NECK SYSTEM
By
Andrey L. Maslennikov

A THESIS
Submitted to
Michigan State University
in partial fulfillment requirements
for the degree of
Mechanical Engineering – Master of Science
2013

ABSTRACT
ROBUST ANALYSIS AND CONTROL DESIGN OF THE HEAD-NECK SYSTEM
By
Andrey L. Maslennikov

Neck pain is the one of the most frequently reported musculoskeletal complaints. A
systematic evaluation of a neck condition may allow clinicians to diagnose possible problems that a subject might have in an objective manner. In this dissertation, we propose
a systematic methodology for measuring robustness of the head-neck system that can be
used as an objective measure of the head-neck system characteristics. Our method builds
on computing the structured singular value µ as a measure of robustness for a particular
subject. The µ value is computed under the consideration that the variability of head-neck
system model parameters for a tested subject comes from the estimation errors as well as
from the natural variation of biological parameters. Both sources of subject’s variability
determine uncertainty levels that are within some physiologically reasonable ranges. In
addition, we analyzed worst-case scenarios of the head-neck system model. The worst-case
analysis can provide us with a possible tool to predict what would be the combination of
parameters that may cause the worst-case performance. Finally, the design of the robust
controller for the head-neck system is also discussed in this dissertation.

ACKNOWLEDGMENTS

I would like to extend my sincere gratitude to my adviser Dr. Jongeun Choi, who led
me through my whole M.S. program. I would like to acknowledge faculty members with
whom I worked for the last two years: Dr. Clark Radcliffe from Mechanical Engineering
Department, Drs. Jacek Cholewicki, Peter Reeves, John Popovich from College of Osteopathic Medicine. Very special thanks to my lab-mate Cody Priess. I also want to say
thank you to my best friends Pavel Polunin and Oleksii Karpenko who supported me all
these years and also all my other friends at Michigan State University.
Finally, this work has been supported in part by grant number U19 AT006057 from the
National Center for Complementary and Alternative Medicine (NCCAM) at the National
Institutes of Health.

iii

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

2 Head-neck System Model Description . . . . . . . . . . . . . . . . . . . . . . . .

5

3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

4 Robust Stability and Performance Criteria . . . . . . . . . . . . . . . . . . . . . 13
5 Robust Analysis of the Head-neck System Components . . . . . . . . . . . . . . 18
6 Individual Parametric Uncertainty Effect on Robust Stability and Performance . 20
7 Worst-case Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 22
8 Robust Control Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Appendix A
Estimated Parameters and Subject Variabilities . . . . . . . . . 29
Appendix B
Worst-case Analysis Numerical Results and Plots . . . . . . . . 32
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iv

LIST OF TABLES

Table 1

µ values for the robust stability analysis with respect to uncertainties
in the head-neck system model blocks. . . . . . . . . . . . . . . . . . 18

Table 2

µ values for the robust performance analysis with respect to uncertainties in the head-neck system model blocks. . . . . . . . . . . . . 18

Table 3

µ values for the robust stability and robust performance analyses
with respect to all uncertainties. . . . . . . . . . . . . . . . . . . . . 20

Table 4

µ values for the robust stability analysis with respect to individual
uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Table 5

µ values for the robust performance analysis with respect to individual uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Table 6

Worst-case critical frequencies. . . . . . . . . . . . . . . . . . . . . . 23

Table 7

Worst-case S ∞ values. . . . . . . . . . . . . . . . . . . . . . . . . 23

Table 8

Parameters of the tuned robust controller of the head-neck system. . 25

Table 9

Subjects’ variabilities and parameters of the nominal system. Control block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Table 10

Subjects’ variabilities and parameters of the nominal system. Muscle
and plant dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Table 11

Subjects’ variabilities and parameters of the nominal system. Delays

Table 12

Worst-case parameter values of the head-neck system. . . . . . . . . 33

v

31

LIST OF FIGURES

Figure 1

Head-neck system model. . . . . . . . . . . . . . . . . . . . . . . . .

Figure 2

The experimental setup of a head-neck position-tracking task. Adapted
from [Priess et al., 2012]. (For interpretation of the references to
color in this and all other figures, the reader is referred to the electronic version of this thesis.) . . . . . . . . . . . . . . . . . . . . . . 8

Figure 3

Power spectral density of the input signal. . . . . . . . . . . . . . .

Figure 4

Head-neck system model for Monte-Carlo simulations . . . . . . . . 11

Figure 5

Input and output data collected from subject 1 and the response of
the estimated model. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Figure 6

Monte-Carlo simulations for the estimated model for trial 1, subject 1. 13

Figure 7

Input and output data collected from subject 1 and the response of
the estimated model. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Figure 8

Robust tests diagrams: a) robust stability test, b) robust performance test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Figure 9

Head-neck system model with performance scaling. . . . . . . . . . 16

Figure 10

µ values for the robust performance analysis with respect to uncertainties in the head-neck system model blocks. . . . . . . . . . . . . 19

Figure 11

Robust control design. Tuned control gains. . . . . . . . . . . . . . 26

Figure 12

Magnitude plot of the sensitivity functions. Subject 1. . . . . . . . . 34

Figure 13

Simulated responses of the worst-case head-neck system models.
Subject 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Figure 14

Magnitude plot of the sensitivity functions. Subject 2. . . . . . . . . 35

Figure 15

Simulated responses of the worst-case head-neck system models.
Subject 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 16

Magnitude plot of the sensitivity functions. Subject 3. . . . . . . . . 36

Figure 17

Simulated responses of the worst-case head-neck system models.
Subject 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vi

5

9

Figure 18

Magnitude plot of the sensitivity functions. Subject 4. . . . . . . . . 37

Figure 19

Simulated responses of the worst-case head-neck system models.
Subject 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Figure 20

Magnitude plot of the sensitivity functions. Subject 5. . . . . . . . . 38

Figure 21

Simulated responses of the worst-case head-neck system models.
Subject 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

vii

1

Introduction
One of the most frequently reported musculoskeletal complaints is neck pain [Cˆt´
oe

et al., 1998], [Holmstr¨m et al., 1992]. The current method of evaluation is primarily
o
based on (subjective) surveys from subjects; see [Langley and Sheppeard, 1985]. Hence,
researchers have been working on developing an objective evaluation of neck pain subjects
and classification of subjects between control and pain groups. Such measure can yield a
numerical value, which may estimate and/or predict conditions of the head-neck system.
Our motivation is to develop a rigorous procedure of computing such objective measure.
A mathematical description of the head-neck system, as a dynamical system, can
be described as an interconnection of transfer functions including an internal controller,
muscle dynamics, plant dynamics (skeleton with ligaments), and neurological delays. Previous studies proposed different models of the head-neck system dynamics, for example,
[Hannaford et al., 1986], [Huebner et al., 1992], [Peterson et al., 2001], and [Chen et al.,
2002]. In this dissertation we adopt a model with a fixed structure developed in [Priess
et al., 2012]. In their study, performance of the head-neck system was investigated during
a tracking task, where participants were asked to follow a moving target using a custommade head-mounted laser device. Parameters of the developed model were estimated from
time series data produced by a target trajectory as an input and a laser dot trajectory
as an output. The estimated model showed good fitting that makes it applicable to our
studies.
It should be noted that a fixed low-order linear model structure may neglect nonlinear
and/or high-order dynamics of the system that may result in modeling errors. In addition,
such models do not take into account a discrete decision making process that a subject
may have, which results in hybrid systems. Furthermore, parameters of the head-neck
system could vary within some physiological ranges due to, for instance, fatigue, task
assignment, control strategy and so on; see [Davids et al., 2006] and [Newell and Corcos,

1

1993]. Moreover, parameters of the head-neck system are not known and, consequently,
these parameters should be estimated using system identification techniques discussed in
[Ljung, 1999], [Forssell and Ljung, 1999], and [Gustavsson et al., 1977]. Assuming the
system is time-invariant, the estimated parameters using a finite number of observations
will have estimation errors. Therefore, in addition to the biological variability in system
parameters, the estimation errors need to be also taken into account when analyzing
stability and/or performance of the head-neck system.
From an engineering perspective, one of the most important questions is whether or
not the dynamical system is stable. On the other hand, from a medical point of view,
mathematical instability of the head-neck model could be related to an injury. By considering a set of all possible models due to the biological variability and parameter estimation
errors we need to investigate robust stability of an uncertain model rather than of stability
of a single true (nominal) model. Robust stability tests deal only with system’s internal
behavior assuming no existence of any external excitation. On the other hand, humans
aim to control their dynamics under the influence of external inputs/commands as can
be seen in position-tracking tasks. Robust performance analysis may be used to investigate how a dynamical system attenuates the signals of the performance channels from
exogenous influences in spite of uncertainties. In addition, robust performance analysis
may help us to classify subjects between control and pain groups as we will show further.
Robust stability and robust performance analyses of the head-neck system with respect to
its total variability could provide us with objective evaluations of the head-neck system.
There exist previous attempts to conduct robust analysis of human dynamics. The
robustness of a human postural control system focusing on stability margins under impulsive perturbations was analyzed in [Hur et al., 2010]. The robust space of a PD controller
structure of a human postural control system was investigated in [Masani et al., 2006].
A methodology of designing an optimal control for postural systems was proposed in [Xu
et al., 2010]. However, in those studies uncertainties either in a model or estimated sys2

tem parameters were not considered. Therefore, it is crucial to develop a new approach of
measuring robustness of the head-neck system with respect to a set of all possible models
for a particular subject. In the past, an estimated system was considered to be the true
nominal system, and the stability analysis was then followed on the estimated system.
There has been no effort to gauge uncertainties in such systems from experimental data
in order to analyze stability and performance.
The stability and performance analyses of systems subjected to structured uncertainties can be addressed with the structured singular value, µ [Zhou and Doyle, 1998]. The µ
value can be used as a measure of system’s robust stability and robust performance. Robust control theory and µ analysis are described in [Skogestad and Postlethwaite, 2005],
[Zhou and Doyle, 1998], [Balas et al., 2001], and [Balas et al., 2006]. Computing the
µ value with the exact combination of system parameters that will yield the worst-case
performance for a given set of models, is also important. From a medical point of view,
analyzing such worst-case performance and its associated parameters for a given subject
could provide the worst-case scenario, which the subject should avoid.
The contributions of this dissertation are as follows. We provide a description of the
proposed methodology for evaluating subjects’ head-neck system in an objective manner.
In comparison to approaches based on surveys [Langley and Sheppeard, 1985], our measure
will be objective. To begin with, we describe the experimental setup that is used to
collect data from subjects for a head-neck position-tracking task. We then show how
the parameters of the head-neck system model can be estimated using nonlinear least
squares optimization by minimizing the difference between the actual subject’s response
and the simulated model response. With a set of estimated parameters for each subject we
present a way to compute biological, estimation, and, eventually, subject’s total parameter
variabilities. Due to variability in parameters the dynamics of the head-neck system are
described by a set of dynamical models for which a robust analysis should be applied. We
define necessary criteria for the robust analysis of the head-neck system based on the H∞
3

norm of the sensitivity function. For the robust performance analysis we incorporate the
performance weighting function, which is used to define a boundary between satisfactory
and not satisfactory performance. Further, we give an interpretation of the µ value as
an objective measure of the head-neck system’s condition. Using the µ value we analyze
which block which individual parameter of the head-neck system are critical to system’s
robust performance. Then we use the worst-case robust analysis to show how a likelihood
of being injured for a tested subject can be suggested. Finally, we describe the robust
control of the head-neck system that possibly can be used as a guideline for physical
therapists in developing treatment for patients. In order to show the applicability of
the proposed methodology we perform all aforementioned analyses for experimentally
obtained data from 5 subjects. The computed µ values allow us to see similarities and
differences among subjects.
The remainder of this dissertation is organized as follows. Chapter 2 provides a description of the head-neck system with uncertainties. In Chapter 3 we describe an experimental setup that we use for collecting data from subjects and an estimation process
of the head-neck model parameters and describe how the biological, estimation and total
subject variabilities could be obtained. In Chapter 4 we briefly review criteria for robust
stability and robust performance of the system for our further analyses. In Chapter 5 we
show which components (control, muscle dynamics, plant dynamics or delays described
in Chapter 2) of the head-neck system are most critical to robust stability and robust
performance of the system. In Chapter 6 we investigate how each individual parametric
uncertainty affects robust stability and robust performance of the head-neck system. The
worst-case analysis of the head-neck system is presented in Chapter 7. The design of a
robust controller with a fixed structure for the head-neck system model is discussed in
Chapter 8. Finally, we provide concluding remarks in Chapter 9.

4

r(t) - D (s) - K(s)
i
−6

-

Q(s)

-

P (s)

y(t)
-

Df (s) 
Figure 1: Head-neck system model.

2

Head-neck System Model Description
In this chapter we provide a description of the head-neck system model. A variety

of the head-neck system models exist. For example, a sixth-order nonlinear model was
used in [Hannaford et al., 1986] and smooth pursuit of an eye-head tracking system with
their own model was analyzed in [Huebner et al., 1992]. We adopt a model from [Priess
et al., 2012] considering the good fit to preliminary experimental data. This model can
be used to represent head-neck dynamics in both flexion/extension and axial rotation. To
simplify the presentation, we investigate only flexion/extension case. However, the exact
same approach can be applied to the axial rotation case in a straightforward manner.
Our model, shown in Fig. 1, contains a neurological (visual) delay in the input channel
and a follow-up closed-loop subsystem. This subsystem contains a controller, muscle
dynamics, and plant (skeleton and ligaments) dynamical blocks in the forward loop and a
neurological delay in the feedback channel. Each component of the head-neck system can
be represented by a transfer function in the s-domain; see [Skogestad and Postlethwaite,
2005]. Here we denote the neurological delay in the input channel by Di (s), the controller
by K(s), muscle dynamics by Q(s), plant dynamics by P (s), and the neurological delay
in the feedback channel by Df (s).
The neurological delays, as a part of postural control system, were already considered
in previous works. For example, the step response of eye motion and the neurological
(visual) delay in the input channel were analyzed in [Carl and Gellman, 1987]. The delay
in the input channel as a component of the head-neck system model is also mentioned
in [Peterson et al., 2001] and [Chen et al., 2002]. A neurological delay in the forward
5

loop of the head-neck system model and the corresponding time constant were considered
in [Robinson et al., 1986]. The existence of the delay in the feedback channel is also
mentioned in [Masani et al., 2006]. Note that for computational simplicity, we used a
first-order Pad´ approximation for delays, but a higher-order approximation could also
e
be used.
The controller K(s) is assumed to have a fixed PID structure. A PID or PD controller
structure is one of the most popular choices for studies in postural control. For example,
a PD controller was used to simulate human posture control in [Masani et al., 2006]. Q(s)
is modeled by a first-order transfer function and its structure was proposed in [Peterson
et al., 2001] and [Chen et al., 2002].
Plant dynamics, P (s), is represented by a second-order transfer function and it was
used in [Peng et al., 1996], [Chen et al., 2002], and [Robinson et al., 1986]. The values
of stiffness and damping ratio were experimentally obtained in [Bourdet and Willinger,
2008] and [Pankoke et al., 1985]. Plant dynamics, by considering the structure of the
skeleton, was analyzed in [Reber and Goldsmith, 1979] and [Vette et al., 2011].
Finally, the described blocks of the head-neck system, in terms of transfer functions,
are defined as follows.
2 − τi s
,
2 + τi s
2 − τf s
−τ s
,
Df (s) = e f ≈
2 + τf s
K
K(s) = Kp + Kd s + i ,
s
1
Q(s) =
,
τm s + 1
1
P (s) = 2
,
Is + bs + k
Di (s) = e−τi s ≈

6

(1)

and the closed-loop transfer function M (s) of the head-neck system model as follows.

M (s) = Di (s)

3

K(s)Q(s)P (s)
.
1 + K(s)Q(s)P (s)Df (s)

(2)

Parameter Estimation
The parameters of the head-neck system model should be estimated. In order to

estimate these parameters a physical experimental setup for the position-tracking task
was constructed. In this experimental setup a tested subject sit on the chair in front
of the screen. A projector was placed above a tested subject and the laser was rigidly
mounted on subject’s head. During this position-tracking task, an individual subject was
asked to follow a projected cross on the screen in front of the subject with a dot from
a laser on the head. Positions of both dots were calculated from images captured by a
video camera and saved as to time series data. The experimental procedure and setup was
described in detail previously in [Priess et al., 2010]. The diagram of the experimental
setup is presented in Fig. 2.
In our particular case, five healthy subjects signed their informed consent form and participated in the experiments. Each individual subject completed five trials of a positiontracking task on two different days. This experimental procedure was approved by the
Michigan State University Biomedical and Health Institutional Review Board (IRB# 12456).
Parameters of the head-neck system model were estimated using these time series data
sets for each trial independently. Each estimation provides us with a vector of estimated
ˆ
parameters θ, which has the following structure
ˆ
ˆ ˆ ˆ ˆ ˆ b, ˆ ˆ ˆ
θ = [Kp , Kd , Ki , τm , I, ˆ k, τf , τi ] ∈ R9 ,

7

(3)

Figure 2: The experimental setup of a head-neck position-tracking task. Adapted from
[Priess et al., 2012]. (For interpretation of the references to color in this and all other
figures, the reader is referred to the electronic version of this thesis.)
where the inertia parameter I, is computed using subject’s weight with methods described
in [Winter, 2004] and [Yoganandan et al., 2009].
The parameter vector θ is estimated using the experimental input and output data.
Based on our preliminary studies, a pulse-width modulated (PWM) signal was chosen as
an input type for our experiments. The time between signal changes varied from 0.3 to
0.9 seconds randomly with a normal distribution in order to eliminate the predictability
aspect. In this way, we eliminate any feed-forward actions in the closed-loop system while
focusing on feedback mechanism. Another reason of using a time-varying step function
is to generate the signal with a desired frequency spectrum, as can be seen from Fig. 3.
The desired frequency spectrum of the input signal should be such that it covers the
bandwidth of the head-neck system, which is about 1 Hz, according our preliminary

8

Figure 3: Power spectral density of the input signal.
studies. The output signal is the actual subject’s response during the position-tracking
experiment. Both input and output signals were simultaneously captured by a video
camera at frequency 60 Hz in order to remove any delays in the projection.
The parameter vector θ is estimated based on nonlinear least-square optimization. It’s
description can be found in [Levenberg, 1944], [Marquardt, 1963], and [Dennis Jr., 1977].
In our case, we are looking for such vector θ within a set Θ that minimizes the difference
between the experimentally obtained subject’s response Y and the simulated subject’s
ˆ
response F (θ). In other words, vector θ contains the parameters such that the simulated
ˆ
response F (θ) fits best to the actual subject’s output Y . Mathematically this method can
be described as follows.
ˆ
θ := arg min F (θ) − Y 2 ,
2
θ∈Θ

(4)

where Θ is a compact set whose element-wise boundaries are defined by the parameterwise variability of vector θ over the whole population. In other words, the set Θ includes

9

all physiologically possible parameters of the head-neck system model.
To perform the robust analysis we need to characterize uncertainties in system parameters. In our case, each parametric uncertainty is within a range defined by the biological
and estimation variabilities. In order to determine the subject’s biological variability, we
perform a set of experimental trials. The parameters of the head-neck system model are
estimated using data from each individual trial. Furthermore, n denotes the number of
ˆ
trials, which in our experiments is 10 for each subject. Having a set of ten vectors θ,
ˆ
ˆ
ˆ
ˆ
we compute three vectors: θavg , θmin , and θmax . θavg is the vector with the average
ˆ
ˆ
values of the estimated parameters, and θmin and θmax defines the boundaries of the estimated parameters or the element-wise ranges in which the parameters of the head-neck
system could vary for a tested subject due to the biological variability. Here and further,
index i, such that i ∈ I := {1, · · · , n}, denotes the number of trial and index j, such
ˆ
ˆ
that j ∈ J := {1, · · · , 9}, denotes a particular element in θ. The parameter vector θavg
determines the nominal model. These three vectors are defined as follows.
ˆ
ˆ
θavg = E [θi ],
i∈I

ˆ
ˆi
θj min = min(θj ),
i∈I

(5)

ˆ
ˆi
θj max = max(θj )
i∈I

The total subject’s variability is defined as a union of the biological and estimation
variabilities. To this end, we also consider the estimation error for a given set of estimated parameters of a subject. To compute the total subject’s variability we perform the
ˆ
Monte-Carlo simulations, where for each estimated θi we simulate the model, as shown
in Fig. 4, N = 50 times with different realizations of additive white noise in muscles. The
parameters of the model are estimated for each Monte-Carlo simulation. Using the whole
ˆ
ˆ
set of these estimates the parameter-wise boundaries (vectors θmin, tot and θmax, tot ) of
the estimates are computed in a same way as it was done for the biological variability
10

r(t) - D (s) - K(s) - Q(s)
i
−6

w(t)
+? P (s)

y(t)
-

Df (s) 
Figure 4: Head-neck system model for Monte-Carlo simulations
using Eq. (5).
ˆ
ˆ
These two vectors θmin, tot and θmax, tot , which are parameter-wise boundaries, form
a compact set Θsubj uniquely for an individual subject. Finally, the total subject’s variability is defined as follows.
ˆ
θ ∈ Θsubj , with some probability α

(6)

where the associated probability α can be obtained from Θsubj and the estimation error
statistics.
Since our estimation is based on signal statistics and randomness in human behavior,
ˆ
the vector of the estimated parameters θ and the whole set Θsubj are estimated with some
probability α as in Eq. (6). Additionally, the natural description of the uncertainty level
for the estimation error variability can be made more explicitly for the robust stability
and performance analyses. This aspect was investigated in [Ljung, 1999], [Bombois, 2000],
[Bombois et al., 2001], and [Gevers et al., 2003]. According to those studies, the estimated
ˆ
parameter θ lies in the ellipsoidal uncertainty region Uol with some probability α(q, χ) =
P r(χ2 (q) < χ), where χ2 (q) the chi-square distribution with q parameters. The ellipsoidal
uncertainty region Uol defined as follows.
−1
ˆ
ˆ
Uol = {θ|(θ − θ) Pθ (θ − θ) < χ},

(7)

ˆ
where Pθ is the covariance matrix of θ, which is different from the formulation in Eq. (6).

11

Figure 5: Input and output data collected from subject 1 and the response of the estimated
model.
The estimated nominal parameters for each subject as well as its biological and total
variabilities are listed in three tables presented in Appendix A. An example of the input
signal and actual subject’s response with the best fitted simulated response is presented
in Fig. 5. As can be noted, the simulated response (green dashed line) of the model with
ˆ
the set of fitted parameters θ1 and the simulated response of the nominal model (cyan
dashed-dotted line) are close to the actual subject’s response at trial 1. This means that
the model structure with the estimated parameters allows us to capture dynamics of the
system. Nevertheless, small deviations between the simulated response and the actual
subject’s output, caused by the estimation and modeling errors, could be observed.
In general, we observed some general trends in the estimated parameters from subjects.
However, estimates for subject 5 are not consistent with others and the total variability
for that subject is higher than those of others. The results of Monte-Carlo simulations for
subject 5 are shown in Fig. 7. From this figure it could be noted that a set of simulated

12

Figure 6: Monte-Carlo simulations for the estimated model for trial 1, subject 1.
responses as well as a set of fitted models are not so close to the actual subject’s response
for subject 5 in comparison to the case of subject 1 (see Fig. 6). Such deviations can be
caused by error and uncertainty in modeling.

4

Robust Stability and Performance Criteria
An important question in analysis and control design of dynamical systems is about

stability of these systems. However, with the presence of uncertainties in a dynamical
system the robust stability analysis has to be involved. Robust stability is stability of an
uncertain dynamical system with respect to a set of all uncertainties specified for that
dynamical system. To answer this question, a dynamical system has to be transformed
to the form presented in Fig. 8 a) where G(s) is the generalized plant and ∆st is the
uncertainty matrix. In our case all uncertainties are parametric, in other words, structured, and; consequently, the matrix ∆st is block diagonal uncertainty matrix. Taking an
13

Figure 7: Input and output data collected from subject 1 and the response of the estimated
model.
advantage of structured uncertainties the criterion of determining whether a dynamical
system is robustly stable with respect to its uncertainties can be formulated in terms of a
structured singular value µ. With the µ value a dynamical system is said to be robustly
stable if and only if the following condition is satisfied (see Theorem 10.7 in [Zhou and
Doyle, 1998])
sup µ∆st (G(jω)) ≤ β,

(8)

ω∈R

with β > 0 such that ∆st ∞ < 1/β, where the µ value is defined as follows.

µ∆st (G) =

1
.
min{¯ (∆st ) : det(I − G∆st ) = 0}
σ

(9)

The value of 1/µ has a physical meaning of the smallest perturbation that makes the
generalized plant G(s) unstable. Therefore, µ has been used as a measure of system’s
robustness.

14

∆pf
-

∆f

-

∆st
Gp (s)

-

∆st



e(t) wp (s)



G(s) 
a)

G(s) 

r(t)

b)

Figure 8: Robust tests diagrams: a) robust stability test, b) robust performance test.
The robust stability of a dynamical system is an important question; however, the real
goal of the analysis and control design is to satisfy predefined performance characteristics.
It is possible that a system is internally stable, but it performs poorly. Thereby, the robust
performance analysis has to be involved. The robust performance analysis answers the
question whether or not a dynamical system meets (or has a good performance) desired
performance specifications. The diagram for the robust performance test is presented in
Fig. 8 b) and as it can be noticed the robust performance test is transformed to the robust
stability test with G(s) combined with a performance weighting function wp (s) into the
weighted generalized plant Gp (s), and ∆st combined with ∆f into ∆pf as follows.




0 
∆st
,
∆pf = 


0 ∆f

(10)

where ∆f is the fictitious uncertainty matrix, which characterizes the disturbance input,
and wp (s) is the performance weighting function, which establishes the desired performance objective. With this formulation, a good robust performance means that the
generalized plant Gp (s) with the disturbance input r(t) and the error output e(t) (see
Fig. 8 b) and Fig. 9) is robustly stable with respect to a set of all uncertainties specified

15

6
e(t)

wp (s)
6

r(t)- K(s)
−6

-

Q(s)

-

P (s)

y(t)
-

Df (s) 
Figure 9: Head-neck system model with performance scaling.
for that dynamical system in ∆pf .
In terms of the µ value, an uncertain dynamical system is said to have good robust
performance if the following condition is satisfied

sup µ∆

ω∈R

pf

(Gp (jω)) ≤ β,

(11)

where β > 0 such that ∆pf ∞ < 1/β. It is also clear that robust performance implies
robust stability but not vice versa. The performance of a MIMO dynamical systems
could be characterized in terms of H∞ norm of the sensitivity function that is the transfer
function from the input r(t) to the output e(t), as shown in Fig. 9. The robust performance
criterion using the sensitivity function can be specified as follows.

S ∞ ≤ β, ∆pf ∈ RH∞ , ∆pf ∞ < 1/β,

(12)

where S(s) is the “weighted” sensitivity function defined as follows.

¯
¯
S(s) = S(s)wp (s), where S(s) =

1
.
1 + K(s)Q(s)P (s)Df (s)

(13)

The performance weighting function wp (s) is described using parameters A, B, ωb , and q
as follows.
wp (s) =

(s/B 1/q + ωb )q
,
(s + ωb A1/q )q
16

(14)

where parameters of wp (s) are chosen such as A = 1.0, B = 4.0, wb = 0.01 Hz, and
q = 20.0. Note that the choice of the performance weighting function is not restricted
to the form presented in Eq. (14). In this form wp (s) acts as a weighting function,
¯
which suppresses the magnitude of the sensitivity function S(jω) in the frequency range
above ωb and amplifies it over the frequency range below ωb . The parameters A and B
define suppression/amplification levels and the parameter q is used to make the transition
between two frequency ranges more narrow. Mathematically, the effect of the wp (s) on
¯
the H∞ norm of the sensitivity function S can be shown as follows.
1
|wp (jω)|

¯
|S(jω)| ≤

¯
⇔ |S(jω)wp (jω)| ≤ 1

(15)

¯
⇔ Swp ∞ ≤ 1, for ∀ ω ∈ R
⇔ S ∞≤1
The performance weighting function provides a flexible way to define performance
specifications for the robust performance analysis of the head-neck system. An appropriate
choice of the form and parameters of wp (s) that reflects a performance boundary between
subjects in control and pain groups can be used to classify subjects. The difference
between value of S ∞ and 1 may be related to the likelihood for a particular subject
to be injured and a classification criterion with an appropriate choice of wp (s) can be
determined such that
S ∞ ≤ 1, for ∀ ω ∈ R, then the performance specification is met,
S ∞ > 1, for ∀ ω ∈ R, then the performance specification is not met.

17

(16)

Uncertain block

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

K(s)

1.554

1.149

1.058

1.176

1.754

Q(s)

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

P (s)

0.893

0.882

0.699

0.769

0.997

Df (s)

0.943

0.916

0.996

0.930

1.215

Table 1: µ values for the robust stability analysis with respect to uncertainties in the
head-neck system model blocks.
Uncertain block

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

K(s)

1.865

1.376

1.255

1.430

2.076

Q(s)

0.549

0.626

0.521

0.570

0.568

P (s)

0.957

0.924

0.746

0.792

1.060

Df (s)

1.204

1.064

1.058

1.046

1.491

Table 2: µ values for the robust performance analysis with respect to uncertainties in the
head-neck system model blocks.

5

Robust Analysis of the Head-neck System Components
In this chapter we provide the results of the µ analysis for the case when uncertainties

exist in each individual block of the head-neck system. We are interested in determining
which block with uncertainties (controller K(s), muscle dynamics Q(s), plant dynamics
P (s) or delay in the feedback channel Df (s)) is most critical to robust stability and
robust performance of the whole head-neck system. In order to do this, we consider that
only one particular block has uncertainties while all other system’s blocks are fixed at
their nominal parameters. The computational results of the robust stability and robust
performance tests for each individual uncertain block are presented in Table 1 for robust
stability and in Table 2 for robust performance tests.
The highest µ value corresponds to the block of the closed-loop system which is most
18

Figure 10: µ values for the robust performance analysis with respect to uncertainties in
the head-neck system model blocks.
critical to system’s stability and performance. We can notice that the computed µ values
for different uncertain blocks for the robust stability as well as for the robust performance
have approximately similar trends among all subjects.
For the robust stability case we can observe that the muscle dynamics block does not
play a significant role; on the contrary, uncertainties in control block K(s) are most crucial
to robust stability of the head-neck system for all subjects. At the same time, uncertainties
in plant dynamics and in the feedback delay affect robust stability approximately at the
same level. We can see the same trends from the robust performance analysis. The
illustration of µ values from the robust performance tests for all tested subjects presented
in Fig. 10.
It is also interesting to note that all µ values for all considered cases for subject 5
is higher than for all other subjects. This correlates with our observation about the
estimation variability of subject 5 that is higher than those of other subjects. It is also

19

Test

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Stability

3.344

2.450

1.917

2.265

3.405

Performance

3.683

2.666

2.102

2.503

3.766

Table 3: µ values for the robust stability and robust performance analyses with respect
to all uncertainties.
interesting to compare µ values related to the whole uncertain head-neck system. As can
be seen from Table 3 subject 5 has the highest µ values among others for both robust
stability and robust performance tests. At the same time, the µ values for subject 3 are
the smallest ones. In other words, subject 3 is the most robust subject, whereas subject 5
is the least robust among other subjects.

6

Individual Parametric Uncertainty Effect on Robust Stability and Performance
In this chapter we analyze how each individual uncertainty affects robust stability

and robust performance of the head-neck system. To perform such analysis we assume
that only one parameter of the system is uncertain. All other parameters are fixed at
their nominal values. Results of robust stability and robust performance tests for each
individual parameter are presented in Table 4 and Table 5, respectively.
The delay in the feedback channel is the almost primary parameter affecting robust
stability of the system. This is essential from the feedback control point of view. For two
subjects the analysis also shows that the integral gain may affect robust stability of the
head-neck system.
More interesting observations can be made from the results for the robust performance
tests. Delay in feedback still significantly influence system performance; however, depending on the subject it may not be the most crucial parameter. Control gains are now the

20

Parameter

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Kp

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

Kd

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

Ki

< 10−3

< 10−3

0.897

< 10−3

< 10−3

τm

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

I

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

b

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

k

< 10−3

< 10−3

< 10−3

< 10−3

< 10−3

τf

0.943

0.916

0.996

0.930

1.215

Table 4: µ values for the robust stability analysis with respect to individual uncertainty.
Parameter

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Kp

1.261

1.144

0.904

1.111

1.190

Kd

1.696

0.917

0.732

0.869

1.959

Ki

0.971

1.000

0.930

0.909

1.080

τm

0.549

0.626

0.521

0.570

0.568

I

0.463

0.379

0.359

0.374

0.502

b

0.911

0.740

0.739

0.694

1.042

k

0.881

0.870

0.713

0.783

0.966

τf

1.204

1.064

1.058

1.046

1.491

Table 5: µ values for the robust performance analysis with respect to individual uncertainty.
most influential. The parameters of plant dynamics could also affect robust performance
of the system significantly but according to µ values their impact on system’s performance
is slightly less than those of control block parameters.
It seems that uncertainties in muscle dynamics and in inertia of the plant are less
important for robust performance of the head-neck system. However, the corresponding
µ values are not too small to neglect parameters τm and I of these blocks.

21

7

Worst-case Performance Analysis
Worst-case analysis provides us with the system parameters that lead to the worst-case

performance. The procedure of computing such worst-case parameters was developed in
[Shin et al., 2001] with an application to the X-38 crew return vehicle. In the application
to the head-neck system computation of the worst-case scenario, we want to determine a
vector of parameters θwc such that the magnitude of the sensitivity function S(θwc , s) is
highest over the whole set of possible models defined by the subject’s variability Θsubj .
Mathematically this problem can be defined as follows.

θwc = arg max S(jω) ∞ .

(17)

θ∈Θsubj

The worst-case analysis is of interest since it shows the worst-performance of a tested
subject with that specific combination of parameters computed from the analysis. Assuming that physiological problems with the subject’s plant dynamics can be independent
from the rest of the system dynamics, we investigate three worst-case scenarios. The first
scenario (denoted as KQP Df WC) is related to the worst-case of the whole head-neck
system with uncertainties in all parameters. In the second scenario (KQDf WC) only
uncertainties in the control block, muscle dynamics and the delay in the feedback channel
are considered. Finally, the worst-case scenario of only the plant dynamics is considered
(P WC).
In terms of robust control, the worst-case is defined as the magnitude peak of the
sensitivity function (weighted sensitivity in general) of a system. The critical frequency
fcr and the magnitude corresponding to this peak. Consequently, our interest is not only
to compute the parameters of the system that lead to the worst-case scenario, but also
to identify the corresponding frequency at this magnitude peak. Values of S ∞ are
computed as lower and upper bounds. The upper bound is of interest. The parameters

22

WC Scenario

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

KQP Df

0.010

0.113

0.085

0.120

0.010

KQDf

0.056

0.209

0.195

0.230

0.064

P

1.111

1.510

1.618

1.232

0.015

Table 6: Worst-case critical frequencies.
WC Scenario

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

KQP Df

inf

Inf

Inf

Inf

Inf

KQDf

Inf

Inf

Inf

Inf

Inf

P

-2.44

-4.20

-6.86

-6.21

9.42

Table 7: Worst-case S ∞ values.
that lead to the worst-case scenario are related to the lower bound case, but the infinite
upper bound is related to the case when magnitude peak is sharp and the computation of
the exact value of S ∞ is problematic. The critical frequencies fcr are listed in Table 6
as well as the magnitudes of the peak S ∞ are in Table 7.
The computed worst-case parameters are listed in Table 12 in Appendix B. A couple
of observations can be made using this table for the worst-case scenarios KQP Df and
KQDf . The proportional gain Kp for all subject tends to be the smallest value within the
subject’s variability. The integral gain Ki tends to increase (except subjects 1 and 5 in
KQP Df worst-case scenario) but not necessarily up to the upper bound within subject’s
total variability. At the same time, the derivative gain Kp values are not consistent. The
values of the delay time constant τf and the parameter τm of the muscle dynamics block
are maximal within individual subject’s variability. There is no strong consistency in the
worst-case parameters of the plant dynamics block.
Other observations from the worst-case analyses can be made from Tables 6 and 7.
First, the worst-case scenarios KQP Df and KQDf primary occur at frequencies below
0.3 Hz. In addition, the critical frequency for the worst-case scenario KQP Df is smaller
23

than that of KQDf . For subjects 1 and 5 fcr for the worst-case scenarios KQP Df tends
to zero. At the same time, the worst-case scenario P occurs at frequencies above 1.0 Hz,
which significantly distinguish this worst-case scenario from two others.
The magnitude peaks for the worst-case scenarios KQP Df and KQDf are sharp and
the upper bound of the S ∞ is infinity. However, the magnitude peaks of the worst-case
scenario P are smooth and the corresponding values of the S ∞ are finite. In addition,
the S ∞ value for subject 5 is higher than 0 dB and for other subjects this values are
less than 0 dB. This means that the worst-case scenario P for subjects 1, 2, 3, and 4 is
still satisfy performance specifications.
Finally, the illustration of the frequency responses of the sensitivity functions, including worst-case sensitivity functions, is presented in Figs. 12, 14, 16, 18, and 20 in
Appendix B as well as the simulated time responses of the head-neck system model with
the worst-case parameters in Figs. 13, 15, 17, 19, and 21. Parameters of the head-neck
system model that lead to the worst-case are listed in Table 12 of Appendix.

8

Robust Control Design
In this chapter we present results of the optimally (or suboptimally) synthesized ro-

bust controller for the head-neck system. The synthesized robust controller could help
in planning rehabilitation. The robust controller is designed using the nonsmooth optimization, which has to be involved due to the structural constraints (PID structure in our
case) imposed on the controller. In our case, this procedure is a search of a combination of
the controller gains Kp , Kd , and Ki (within the subject’s variability) that minimizes the
H∞ norm of the sensitivity function S(jω). Mathematically, this problem is formulated
as follows.
k := arg min S(jω, θ, k) ∞ ,
k∈K

24

(18)

Par.

Subj. 1

Subj. 2

Subj. 3

Subj. 4

Subj. 5

Nom.

Tuned

Nom.

Tuned

Nom.

Tuned

Nom.

Tuned

Nom.

Tuned

Kp

0.287

0.116

0.923

0.196

0.890

0.196

0.835

0.179

0.180

0.089

Kd

0.196

0.048

0.263

0.056

0.359

0.066

0.241

0.054

0.235

0.044

Ki

0.030

0.000

0.115

0.000

0.083

0.061

0.096

0.042

0.048

0.000

Table 8: Parameters of the tuned robust controller of the head-neck system.
where k is a vector that contains the controller gains (Kp , Kd , and Ki ), and K is a closed
set prescribed by the parameter-wise variabilities of those control gains from Θsubj .
In particular the nonsmooth H∞ synthesis proposed by [Apkarian and Noll, 2006] and
[Bruisma and Steinbuch, 1990] is used, which is an effective way to solve a non-convex
H∞ synthesis problem where controller is subject to structural constraints.
The results are presented in Table 8.
The computed robust control gains are smaller rather than the estimated control gains.
In general, smaller control gains mean that robust control is less aggressive in comparison
to the estimated control. Another interesting observation is that the integral gain Ki in
the robust controller tends to be the smallest value within subjects’ variability. These
tendencies can be clearly seen in Fig. 11.

9

Conclusions
In this dissertation we proposed a new methodology of measuring robustness of the

head-neck system. This methodology is based on performing the robust stability and
robust performance analyses of the head-neck system with respect to uncertainties. Uncertainties come from the biological variability of the parameters of the system and from
the estimation errors. The robust stability and robust performance tests provide us with
the µ value that may be used as an objective measure in clinical evaluation.

25

Figure 11: Robust control design. Tuned control gains.
The general procedure for our approach can be summarized as follows:
1. Determine the biological variability of a subject via estimating a parametric model
using subject’s responses during position tracking experiments.
2. Determine the estimation variability of the dynamical model for a subject by analyzing estimation error statistics.
3. Determine the total variability as the union of the biological and estimation variabilities.
4. Perform robust analyses of the system including robust stability and robust performance tests.
5. Perform the worst-case performance analysis of the system.
6. Design an optimal (or suboptimal) robust controller for planning rehabilitation.
26

We also applied our methodology using the experimental data collected with five adult
subjects. Using the experimental data we showed that the estimated parameters and
variabilities for the four subjects have a trend and the simulated responses are very close
to the actual subjects’ responses. Our experimental setup allows us to capture dynamics
of the head-neck system. Subject 5 was exceptional showing relatively large modeling
errors.
The robust analysis of each individual uncertain block of the head-neck system showed
a strong trend in the µ values for different blocks among tested subjects.
In the robust analysis of each individual uncertainty we noted that robust stability of
the system was affected by the delay in the feedback channel and for some subjects by
the integral gain of the PID controller. In the case of robust performance, all individual
uncertainties affected system’s performance. The control gains and delays in feedback are
more crucial; however, uncertainties in plant dynamics as well as in the muscle dynamics
block cannot be neglected.
With the worst-case analysis we can clearly see that the worst-case scenario for the
whole system occurs at frequencies below 0.150 Hz and the worst-case scenario with
uncertainties in control, muscle dynamics and delays in feedback only occurs in frequencies
within range from 0.150 to 0.300. At the same time, the worst-case scenario, in the case
when uncertainties exist in the plant only, occurs at frequencies above 1 Hz and its peak
has finite value.
The designed robust control design showed the tendencies that the control gains of
the robust controller are smaller than those of the estimated nominal system. In other
words, the robust controller tends to be less aggressive. The synthesized robust controller
could help in planning rehabilitation.

27

APPENDICES

28

A

Estimated Parameters and Subject Variabilities
In this Appendix we provide tables with the estimated parameter values and deter-

mined subject’s biological and total variabilities. Parameters of the control block are
listed in Table 9, parameters of the plant and muscle dynamics blocks are in Table 10 and
the delay time constants are listed in Table 11.
Kp

Bio. variability

[0.087, 0.562]

[0.130, 0.401]

[0.000, 0.142]

[0.000, 1.406]

[0.057, 1.365]

[0.000, 0.591]

0.923

0.263

0.115

Bio. variability

[0.404, 1.304]

[0.232, 0.317]

[0.000, 0.502]

[0.274, 3.543]

[0.111, 0.904]

[0.000, 3.828]

Nominal value

0.890

0.359

0.083

Bio. variability

[0.320, 1.584]

[0.210, 0.784]

[0.000, 0.236]

Total variability

[0.180, 2.981]

[0.096, 1.143]

[0.000, 1.617]

Nominal value

0.835

0.241

0.096

Bio. variability

[0.431, 1.117]

[0.202, 0.355]

[0.000, 0.399]

Total variability

[0.241, 3.136]

[0.073, 0.903]

[0.000, 1.371]

Nominal value

0.180

0.235

0.048

Bio. variability

[0.033, 0.402]

[0.102, 0.474]

[0.000, 0.146]

Total variability

Subj. 5

0.030

Total variability

Subj. 4

0.196

Nominal value

Subj. 3

0.287

Total variability
Subj. 2

Ki

Nominal value
Subj. 1

Kd

[0.000, 1.355]

[0.085, 7.225]

[0.000, 1.587]

Table 9: Subjects’ variabilities and parameters of the nominal system. Control block

29

τm

0.158

0.030

Bio. variability

[0.097, 0.500]

[0.024, 0.026]

[0.107, 0.261]

[0.006, 0.122]

[0.050, 0.500]

[0.024, 0.026]

[0.100, 0.916]

[0.000, 0.424]

0.148

0.033

0.228

0.073

Bio. variability

[0.123, 0.186]

[0.032, 0.035]

[0.100, 0.282]

[0.006, 0.209]

[0.050, 0.500]

[0.032, 0.035]

[0.100, 0.643]

[0.000, 0.998]

Nominal value

0.168

0.032

0.191

0.079

Bio. variability

[0.105, 0.420]

[0.031, 0.034]

[0.100, 0.302]

[0.014, 0.224]

Total variability

[0.050, 0.500]

[0.031, 0.034]

[0.100, 0.712]

[0.000, 0.466]

Nominal value

0.160

0.027

0.176

0.074

Bio. variability

[0.128, 0.217]

[0.026, 0.029]

[0.100, 0.223]

[0.022, 0.182]

Total variability

[0.050, 0.500]

[0.026, 0.029]

[0.100, 0.536]

[0.000, 0.591]

Nominal value

0.230

0.029

0.125

0.030

Bio. variability

[0.050, 0.500]

[0.028, 0.031]

[0.100, 0.239]

[0.000, 0.090]

Total variability

Subj. 5

0.025

Total variability

Subj. 4

0.229

Nominal value

Subj. 3

k

Total variability
Subj. 2

b

Nominal value
Subj. 1

I

[0.050, 0.500]

[0.028, 0.031]

[0.100, 5.281]

[0.000, 1.493]

Table 10: Subjects’ variabilities and parameters of the nominal system. Muscle and plant
dynamics

30

τi
Nominal value

[0.000, 0.447]

[0.130, 0.258]

[0.000, 0.500]

0.188

0.039

Bio. variability

[0.165, 0.207]

[0.000, 0.064]

[0.152, 0.247]

[0.000, 0.237]

0.150

0.011

Bio. variability

[0.133, 0.171]

[0.000, 0.036]

Total variability

[0.116, 0.181]

[0.000, 0.194]

Nominal value

0.177

0.024

Bio. variability

[0.158, 0.189]

[0.000, 0.045]

Total variability

[0.140, 0.211]

[0.000, 0.186]

Nominal value

0.192

0.106

Bio. variability

[0.155, 0.244]

[0.000, 0.285]

Total variability

Subj. 5

[0.155, 0.209]

Nominal value

Subj. 4

Bio. variability

Total variability
Subj. 3

0.135

Nominal value
Subj. 2

0.181

Total variability

Subj. 1

τf

[0.108, 0.362]

[0.000, 0.500]

Table 11: Subjects’ variabilities and parameters of the nominal system. Delays

31

B

Worst-case Analysis Numerical Results and Plots
In this Appendix we present the numerical results for the worst-case analysis of the

head-neck system. The computed worst-case parameter values are listed in Table 12. The
magnitude plots of the worst-case sensitivity functions are shown in Figs. 12, 14, 16, 18,
and 20. The corresponding simulated responses of the worst-case head-neck system models
are shown in Figs. 13, 15, 17, 19, and 21.

32

Nominal KQDf P WC KQDf WC

P WC

Kp
Subj. 1 Kd
Ki
τm
b
k
τf

0.287
0.196
0.030
0.229
0.158
0.030
0.135

0.004
0.623
0.006
0.500
0.916
0.001
0.500

0.000
0.946
0.141
0.500
nominal
nominal
0.500

nominal
nominal
nominal
nominal
0.100
0.424
nominal

Kp
Subj. 2 Kd
Ki
τm
b
k
τf

0.923
0.263
0.115
0.148
0.228
0.073
0.039

0.274
0.111
0.337
0.500
0.643
0.000
0.237

0.274
0.121
0.515
0.500
nominal
nominal
0.237

nominal
nominal
nominal
nominal
0.100
0.998
nominal

Kp
Subj. 3 Kd
Ki
τm
b
k
τf

0.890
0.359
0.083
0.168
0.191
0.079
0.011

0.180
1.012
0.476
0.500
0.712
0.000
0.194

0.180
0.805
1.469
0.500
nominal
nominal
0.194

nominal
nominal
nominal
nominal
0.100
0.466
nominal

Kp
Subj. 4 Kd
Ki
τm
b
k
τf

0.835
0.241
0.096
0.160
0.176
0.074
0.024

0.241
0.073
0.311
0.500
0.536
0.000
0.186

0.241
0.505
1.352
0.500
nominal
nominal
0.186

nominal
nominal
nominal
nominal
0.100
0.000
nominal

Kp
Subj. 5 Kd
Ki
τm
b
k
τf

0.180
0.235
0.048
0.230
0.125
0.030
0.106

0.000
0.085
0.021
0.500
5.281
0.027
0.500

0.000
0.641
0.126
0.500
nominal
nominal
0.500

nominal
nominal
nominal
nominal
5.281
0.000
nominal

Table 12: Worst-case parameter values of the head-neck system.
33

Figure 12: Magnitude plot of the sensitivity functions. Subject 1.

Figure 13: Simulated responses of the worst-case head-neck system models. Subject 1.

34

Figure 14: Magnitude plot of the sensitivity functions. Subject 2.

Figure 15: Simulated responses of the worst-case head-neck system models. Subject 2.

35

Figure 16: Magnitude plot of the sensitivity functions. Subject 3.

Figure 17: Simulated responses of the worst-case head-neck system models. Subject 3.

36

Figure 18: Magnitude plot of the sensitivity functions. Subject 4.

Figure 19: Simulated responses of the worst-case head-neck system models. Subject 4.

37

Figure 20: Magnitude plot of the sensitivity functions. Subject 5.

Figure 21: Simulated responses of the worst-case head-neck system models. Subject 5.

38

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