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Modeling Human Multimodal Perception and
Control Using Genetic Maximum Likelihood
Estimation
P.M.T. Zaal,∗D.M. Pool,†
Q.P. Chu,‡M.M. van Paassen,§M. Mulder¶and J.A. Mulderk
Delft University of Technology, Delft, The Netherlands
This paper presents a new method for estimating the parameters of
multi-channel pilot models that is based on maximum likelihood estima-
tion. To cope with the inherent nonlinearity of this optimization problem,
the gradient-based Gauss-Newton algorithm commonly used to optimize
the likelihood function in terms of output error is complemented with a
genetic algorithm. This significantly increases the probability of finding
the global optimum of the optimization problem. The genetic maximum
likelihood method is successfully applied to data from a recent human-in-
the-loop experiment. Accurate estimates of the pilot model parameters and
the remnant characteristics were obtained. Multiple simulations with in-
creasing levels of pilot remnant were performed, using the set of parameters
found from the experimental data, to investigate how the accuracy of the
parameter estimate is affected by increasing remnant. It is shown that only
for very high levels of pilot remnant the bias in the parameter estimates
∗Ph.D. student, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058,
2600GB Delft, The Netherlands; p.m.t.zaal@tudelft.nl. Student member AIAA.
†Ph.D. student, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058,
2600GB Delft, The Netherlands; d.m.pool@tudelft.nl. Student member AIAA.
‡Associate Professor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058,
2600GB Delft, The Netherlands; q.p.chu@tudelft.nl. Member AIAA.
§Associate Professor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058,
2600GB Delft, The Netherlands; m.m.vanpaassen@tudelft.nl. Member AIAA.
¶Associate Professor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058,
2600GB Delft, The Netherlands; m.mulder@tudelft.nl. Member AIAA.
kProfessor, Control and Simulation Division, Faculty of Aerospace Engineering, P.O. Box 5058, 2600GB
Delft, The Netherlands; j.a.mulder@tudelft.nl. Member AIAA.
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is substantial. Some adjustments to the maximum likelihood method are
proposed to reduce this bias.
Nomenclature
AState matrix
AdDisturbance sinusoid amplitude deg
AtTarget sinusoid amplitude deg
BInput matrix
COutput matrix
DFeedthrough matrix
eTracking error signal deg
fProbability density function -
fdDisturbance forcing function deg
ftTarget forcing function deg
H(s) Transfer function
H(jω) Frequency response function
HnRemnant filter
Hnm Neuromuscular dynamics
Hpe Pilot visual response
Hpθ Pilot motion response
Hsc Semicircular canal dynamics
Hθ,δeControlled pitch dynamics
jImaginary unit -
LLikelihood function -
KmMotion perception gain -
KnRemnant filter intensity -
KvVisual perception gain -
Kδe,u Pitch stick gain -
MΘΘ Fisher information matrix
mNumber of samples -
NRemnant Fourier transform -
NeOrder of the error response state matrix -
nPilot remnant signal deg
ndDisturbance frequency integer factor -
ntTarget frequency integer factor -
SSpectrum
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sLaplace variable -
Tlead Visual lead time constant s
Tlag Visual lag time constant s
TmMeasurement time s
Tsc1,Tsc2Semicircular canal time constants s
tTime s
uPilot control signal deg
¯xState vector
Symbols
αLine search parameter -
δeElevator deflection deg
ǫPrediction error deg
ζnRemnant filter damping -
ζnm Neuromuscular damping -
Θ Parameter vector
θPitch angle deg
¨
θPitch acceleration deg s−2
µMean
σStandard deviation
τmMotion perception time delay s
τvVisual perception time delay s
φSinusoid phase shift rad
ωFrequency rad s−1
ωmMeasurement base frequency rad s−1
ωnRemnant filter break frequency rad s−1
ωnm Neuromuscular frequency rad s−1
Subscripts
epilot error response
ddisturbance
ttarget
θpilot pitch response
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I. Introduction
In manual control of aircraft, pilots combine information from cockpit instruments, their
view of the outside world and physical motion sensations to achieve a suitable control action.
Knowledge on how pilots use these different types of motion cues not only gives more insight
into human motion perception processes, but it is also crucial to the design of manual control
systems and the tuning of flight simulators.1,2
Many researchers applied multi-channel pilot models in their efforts to explain and quan-
tify the effects of different modalities on pilot control behavior.3–9 Human manual vehicle
control behavior is an inherently nonlinear and time-varying closed-loop process. For care-
fully designed control tasks, however, the control behavior of well-trained individuals can be
accurately described with quasi-linear pilot models.10 Such models consist of a linear part
that describes a pilot’s response to all perceived variables in terms of control-theoretical
elements, supplemented with a remnant signal that accounts for all nonlinear contributions
to the observed control behavior. For tasks where pilots perceive information from multiple
modalities – e.g., visual and motion – these linear pilot models generally have a multi-channel
structure, where the response to the different modalities is separated. The characteristics
of such linear pilot responses are defined by the model parameters, such as weighing gains
and time delays. Values for these parameters, which can be determined from experimental
data using mathematical identification techniques, help to explain the effects of different
perceptual modalities on pilots’ control behavior in control-theoretical terms.
Parameter estimation techniques currently employed to estimate the parameters of a
multi-channel pilot model use either Fourier transforms11 or linear time-invariant models8to
obtain non-parametric pilot describing functions in the frequency domain. In a second step,
the parameters of a multi-channel pilot model are then optimized to yield an optimal fit to
these frequency responses. These methods have two main disadvantages. First, the accuracy
of the parameter estimate is affected by biases that originate from both identification steps.
In addition, for these identification methods to give accurate results, highly specific demands
on the design of the control task – and specifically the adopted forcing functions – need to
be met.8
An alternative to these frequency-domain parameter estimation methods are time-domain
identification procedures, which allow for estimation of model parameters from time-domain
data directly. For such time-domain identification techniques, the requirements to ensure
the identifiability of a model are significantly less stringent.12 Few studies are described
in the literature that investigate the application of time-domain identification techniques to
pilot modeling.13–18 These studies only considered very simple pilot models and provide little
detail of the identification procedure.
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Maximum likelihood estimation is an example of a statistical time-domain identifica-
tion method that, for instance, has been successfully applied to the identification of aircraft
stability and control derivatives from flight test data19–21 and of air- and spacecraft struc-
tural modes.22, 23 This study focuses on the application of a maximum likelihood parameter
estimation algorithm to the problem of multi-channel pilot model identification. For the de-
velopment of this algorithm, an output-error structure is assumed for the quasi-linear pilot
model. Of particular interest are the steps that need to be taken to make the identification
technique suitable for coping with the inherent nonlinearity and many local minima of the
cost function. To cope with these nonlinearities and local minima, the maximum likelihood
method is enhanced with a genetic algorithm. Data from a recent human-in-the-loop ex-
periment,2which was performed in the SIMONA Research Simulator at Delft University
of Technology, is used to test the genetic maximum likelihood algorithm. To evaluate the
performance of this parameter estimation method with increasing levels of pilot remnant,
the average bias and standard deviation of the parameter estimates are assessed using pilot
model simulations.
The structure of the paper is as follows. First, the parameter estimation problem for
multi-channel pilot models will be discussed. Then, the genetic maximum likelihood estima-
tion procedure will be explained in detail and parameter estimation results from experimental
data will be given. The paper ends with a discussion and conclusions.
II. The Parameter Estimation Problem
A manual vehicle control task that was investigated in a recent human-in-the-loop exper-
iment will be considered as an example of a typical multi-channel pilot model identification
problem in this paper. The objective of the experiment was to investigate the effects of ro-
tational and vertical motion during aircraft pitch attitude control in a disturbance-rejection
task.2It was found that the presence of rotational pitch motion significantly affected pi-
lot control behavior, making aircraft pitch attitude control a clear example of a multi-loop
control task.
ftu
−
e
θ
n
Hpe
Hpθ
−
pilot
Kδe,u Hθ,δe
θ
fd
δe
+ + ++
+
Figure 1. Multi-loop representation of a closed-loop aircraft pitch control task.
5 of 29
II.A. The Multi-Loop Control Task
In Figure 1, a schematic representation of the pitch attitude control task that is studied in
this paper is depicted. As can be seen in this figure, the pilot acts as a feedback controller on
the pitch dynamics of an aircraft, Hθ,δe. For this experiment, the controlled pitch dynamics
were those of a Cessna Citation II aircraft as given by:
Hθ,δe(s) = −10.6189 s+ 0.9906
s(s2+ 2.756s+ 7.612).(1)
In addition to the tracking error e, which is the difference between the current and
the desired pitch attitude as perceived from the compensatory visual display, information
about the aircraft pitch attitude can be perceived via physical rotational motion. Hence,
the total pilot response consists of the contributions of two linear response functions, Hpe
and Hpθ. A remnant signal nis added to the linear model output representing the nonlinear
behavior. The gain Kδe,u in Figure 1 represents the scale factor between sidestick and
elevator deflection. Two forcing functions, a disturbance fdand target ft, are used to excite
the combined pilot-aircraft system. The target signal gives the desired pitch attitude; the
disturbance signal can be considered as turbulence acting on the aircraft, perturbing the
pitch angle.
II.B. The Multi-Channel Pilot Model
The linear response functions are parametrized by gains and time constants, the pilot equal-
ization; time delays and neuromuscular dynamics, the pilot limitations; and vestibular dy-
namics, the pilot sensor dynamics. Based on McRuer’s precision model10 and Van der Vaart’s
multi-channel pilot model,6appropriate models for the linear response functions Hpe and Hpθ
for control of the pitch dynamics defined by Eq. (1) are given by:2
Hpe (jω) = Kv
(1 + jωTlead)2
(1 + jωTlag )e−jωτvHnm (jω),(2)
and
Hpθ (jω) = (jω)2Hsc (jω)Kme−jωτmHnm (jω).(3)
In Eq. (2), which describes the response to visual motion cues, Kvis the visual perception
gain, Tlead the visual lead time constant, Tlag the visual lag time constant and τvthe visual
perception time delay. Rotational pitch motion is mainly perceived with the semi-circular
canals of the vestibular system, which are sensitive to pitch acceleration.7The model for
the pitch motion perception channel Hpθ (Eq. (3)) therefore includes the dynamics of the
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semi-circular canals Hsc, the motion perception gain Kmand a motion perception time delay
τm.
As can be verified from the presence of Hnm in both Eq. (2) and Eq. (3), the total
linear pilot response is attenuated by the dynamics of the neuromuscular system. The
neuromuscular system is modeled as a second-order mass-spring-damper system, of which
the damping ζnm and natural frequency ωnm are parameters to be estimated:
Hnm (jω) = ω2
nm
ω2
nm + 2ζnmωnm jω + (jω)2.(4)
Finally, the dynamics of the semi-circular canals, which are part of the pitch motion
perception channel of the pilot model, are given by:
Hsc (jω) = 1 + jωTsc1
1 + jωTsc2
,(5)
with Tsc1= 0.11 and Tsc2= 5.9 seconds. These values are taken from previous research7and
are assumed fixed when modeling pilot control behavior for the pitch control task defined in
Figure 1. The parameter vector Θ, with a total of eight parameters to be estimated, is given
by:
Θ = [KvTlead Tlag τvKmτmζnm ωnm]T.(6)
II.C. Forcing Functions
For reliable identification of both the pilot visual and motion responses with previously used
parameter estimation methods, two independent forcing function signals are required. For
the control task described in Figure 1, both the target and disturbance forcing function sig-
nals (ftand fd) were defined as sums of ten sine waves with different frequencies, amplitudes
and phase shifts:
ft(t) =
10
X
k=1
At(k) sin (ωt(k)t+φt(k)) ,(7)
fd(t) =
10
X
k=1
Ad(k) sin (ωd(k)t+φd(k)) .(8)
To allow for use of spectral methods in the analysis of the experimental data, the forcing
function sine wave frequencies were all defined as integer multiples of the experimental mea-
surement time base frequency, ωm= 2π/Tmwith Tm= 81.92 seconds. The corresponding
integer factors ntand ndare listed in Table 1, together with the target and disturbance
7 of 29
signal frequencies, amplitudes and phases.
Table 1. Multi-sine forcing function properties.
disturbance, fdtarget, ft
k, – nd, – ωd, rad s−1Ad, deg φd, rad nt, – ωt, rad s−1At, deg φt, r ad
1 5 0.383 0.344 -0.269 6 0.460 0.698 1.288
2 11 0.844 0.452 4.016 13 0.997 0.488 6.089
3 23 1.764 0.275 -0.806 27 2.071 0.220 5.507
4 37 2.838 0.180 4.938 41 3.145 0.119 1.734
5 51 3.912 0.190 5.442 53 4.065 0.080 2.019
6 71 5.446 0.235 2.274 73 5.599 0.049 0.441
7 101 7.747 0.315 1.636 103 7.900 0.031 5.175
8 137 10.508 0.432 2.973 139 10.661 0.023 3.415
9 171 13.116 0.568 3.429 194 14.880 0.018 1.066
10 226 17.334 0.848 3.486 229 17.564 0.016 3.479
The target signal was defined to have only a quarter of the power of the disturbance
signal to make the task primarily a disturbance-rejection task.2
II.D. Parameter Estimation Methods
Estimation of multi-channel pilot model parameters from measurement data is a highly
nonlinear optimization problem. First of all, a linear model is fit on data that is inherently
nonlinear. The fact that the models are nonlinear in their parameters further increases the
nonlinearity of the optimization problem. Due to these nonlinearities, the resulting cost
function contains many local minima in addition to the global minimum.
In addition, it is apparent from the model structure given in Eq. (2) and Eq. (3) that
the lead term in the visual perception channel and the lead resulting from the integrating
action of the semicircular-canal dynamics in the vestibular channel are more or less inter-
changeable. If multiple terms in a model can cause the same overall response, the model is
overdetermined, which adds to the number of local minima. It is found in previous research
that the occurrence of these local minima is highly dependent on the controlled dynamics,
the type of forcing function and even the subject who performed the experiment.24
Many different techniques can be applied to estimate the parameters of a multi-channel
pilot model from measurement data. Figure 2 illustrates that two groups can be distin-
guished, parameter estimation in the frequency domain and parameter estimation in the
time domain.
II.D.1. Frequency-Domain Techniques
Parameter estimation in the frequency domain requires an additional step, to transform
the measured time-domain data to the frequency domain, as illustrated in Figure 2. In
this additional step, Fourier coefficients (FC) or linear time-invariant (LTI) models (e.g.,
8 of 29
autoregressive exogeneous (ARX) models) are used to estimate non-parametric frequency
response functions.8, 11 These non-parametric frequency responses are then used in a second
step, in which a multi-channel pilot model is fit by adjusting its parameters. For estimating
the pilot model parameters a frequency-domain criterion is used, which results in a nonlinear
optimization problem.8, 9, 25
time
domain
frequency
domain
non-parametric
identification
parameter
estimation
parameter
estimation
frequency-domain
techniques
time-domain
techniques
pilot model parameters
measured time traces
Figure 2. Comparison of pilot model parameter estimation methods.
With some exceptions, in the field of human operator modeling, parameter estimation
has mainly been performed in the frequency domain.3, 4, 6–9, 11, 25, 26 An advantage of this
technique is that the first step serves as a data reduction step, which makes the parameter
optimization efficient in terms of computational power. Another advantage of this method
is that the non-parametric frequency response functions provide an indication of the pilot
model structure required for describing the pilot dynamics.
One of the disadvantages of using two steps to estimate the pilot model parameters is that
the bias and variance of the non-parametric frequency responses can cause increased bias
and variance in the obtained parameter estimates. A second disadvantage is that in order to
estimate multiple frequency response functions in the first step, an equal number of forcing
functions needs to be inserted into the closed-loop system to assure that the inputs to the
pilot model are uncorrelated (see Figure 1).8, 11 Inserting multiple forcing functions may yield
an unrealistic piloting task as compared to real flight and makes the data less comparable to
classical single forcing function target-following and disturbance-rejection tasks used in many
experiments described in literature.6, 7, 10, 26 When using Fourier coefficients or LTI models
for estimating the frequency response functions, the forcing functions need to be carefully
designed keeping in mind the requirements for identifiability of the frequency responses and
the pilot limitations.8
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II.D.2. Time-Domain Techniques
Time-domain parameter estimation techniques, such as the least-squares method and max-
imum likelihood estimation,27 directly fit a parameter model on the time-domain data (see
Figure 2). As time-domain signals generally have significantly more data points than the
frequency response functions used in the frequency-domain methods, fitting the parameter
model on the time-domain data requires more computational power. Another disadvantage
is that it is difficult to determine the correct model structure beforehand and an incorrect
model structure can not be easily detected by comparing the model output with the measured
pilot output.
One of the advantages of a time-domain method is that the parameter model is directly fit
using only one step, reducing the bias and variance of the estimated parameters compared to
the frequency-domain methods. Also, there are fewer constraints when designing the forcing
functions, as the frequency content of the signals has less influence on the identifiability of
the parameters. There is also no explicit requirement for uncorrelated input signals of the
pilot model as is the case using the frequency-domain methods.
In experiments, the differences in pilot control behavior for different experimental condi-
tions are often small in magnitude. The increased accuracy of the estimated parameters for
time-domain techniques could increase the significance of the results in these experiments.
Also, as time-domain methods allow for more design freedom of the forcing functions, tasks
which are more comparable to real piloting tasks can be used in experiments. This will
increase the insight into multimodal perception and control for such tasks.
III. Genetic Maximum Likelihood Estimation
Maximum likelihood estimation (MLE) is a time-domain parameter estimation technique
that has many applications.19–23, 28 One of the main reasons for this is that maximum likeli-
hood estimates have some attractive statistical properties. They are consistent and efficient,
which means that the parameter estimate converges to the true parameter set and that the
variance reduces to the Cram´er-Rao lower bound as the sample size increases.29 Furthermore,
the errors in the resulting parameter estimates have an unbiased Gaussian distribution.
In this section a procedure for estimating the parameters of multi-channel pilot models
that is based on the principle of maximum likelihood will be described. The structure of
such pilot models warrants the use of an output-error model structure, which results in a
simplification of the full maximum likelihood estimation problem. To increase the probability
of finding the global minimum of this nonlinear optimization problem, a genetic algorithm
is used to determine the initial parameter estimates for a classical gradient-based parameter
estimation procedure.
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III.A. The Maximum Likelihood Method
The pilot model considered in this paper incorporates separate pilot visual and motion
responses, as depicted in Figure 1. The pilot’s control output uis considered to be the sum
of a response to visual and motion cues (eand θ, respectively) and a remnant signal n.
Maximum likelihood estimation procedures generally require models that are written in
state-space form. For the multi-channel pilot model of Figure 1, a discrete time state-space
representation can be given by:
˙
¯x(k+ 1) =
Ae(Θ) 0
0Aθ(Θ)
¯x(k) +
Be(Θ) 0
0Bθ(Θ)
e(k)
θ(k)
(9)
u(k) = hCe(Θ) −Cθ(Θ) i¯xs(k) + hDe(Θ) −Dθ(Θ) i
e(k)
θ(k)
+n(k).
In Eq. (9), Θ represents the vector of unknown pilot model parameters as defined by
Eq. (6). The subscripts eand θdivide the total state-space system into the state-space
models of the error response function Hpe and the pitch response function Hpθ , respectively.
These state-space representations are easily obtained by converting the transfer function
models defined in Equations (2) and (3) using the controller canonical form, as illustrated
for Ae(Θ) by:
Ae(Θ) =
0 1 0 ··· 0
0 0 1 ....
.
.
.
.
..........0
0 0 ··· 0 1
Ae(Ne,1) Ae(Ne,2) ··· Ae(Ne, Ne−1) Ae(Ne, Ne)
,(10)
where Neis the order of Ae(Θ). For the conversion to a state-space representation, the pilot
model time delays are approximated by using Pad´e approximations of typically fifth order,
to ensure accurate description of the high-frequency phase roll-off. Especially due to these
Pad´e approximations, the matrices of the state-space model as defined by Eq. (9) contain
coefficients that are a highly nonlinear function of the pilot model parameters. For example,
Equations (11) and (12) give the first and last coefficients in the final row of Ae(Θ) for a
fifth order Pad´e approximation of the visual time delay τv.
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Ae(Ne,1) = −30240ω2
nm
Tlagτ5
v
(11)
Ae(Ne, Ne) = −2Tlag τ5
vζnmωnm −30Tlag τ4
v−τ5
v
Tlagτ5
v
(12)
The state-space pilot model given in Eq. (9) only has an additive noise term in the output
equation, i.e., no process noise is assumed. This output-error structure significantly reduces
the complexity of the MLE procedure. The pilot remnant n, which is accounted for by
the measurement noise in the chosen model structure, is modeled as an additive zero-mean
Gaussian white noise signal, whose properties are defined as:
E{n(k)}= 0; En(k)nT(k)=σ2
n.(13)
Maximum likelihood estimation techniques attempt to find an estimate ˆ
Θ of the pa-
rameter vector Θ that maximizes the likelihood function. The likelihood function L(Θ) is
defined as the joint conditional probability density function of the prediction error for m
measurements of u(k):
L(Θ) = f(ǫ(1), ǫ(2),...,ǫ(k),...,ǫ(m)|Θ).(14)
The prediction error, indicated as ǫ(k) in Eq. (14), is defined as the difference between
the measured pilot control signal u(k) and the modeled pilot control signal ˆu(k) at dis-
crete instants. Given the properties of the remnant as defined in Eq. (13), the conditional
probability density function for one measurement of ǫ(k) is given by:
f(ǫ(k)|Θ) = 1
p2πσ2
n
e−ǫ2(k)
2σ2
n.(15)
The set of parameters that maximizes the likelihood function is the maximum likelihood
estimate of the parameter vector Θ. For the MLE method it is common practice to minimize
the negative natural logarithm of the likelihood function instead of maximizing L(Θ), as
this results in a more straightforward optimization problem. When a global minimum of the
negative log-likelihood is attained, the resulting parameter vector is the maximum likelihood
estimate, indicated with ˆ
ΘML. By combining Eq. (14) and Eq. (15) and considering the fact
that the pilot model contains a single output, the following expression for this maximum
likelihood estimate can be obtained:
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ˆ
ΘML = arg min
Θ
−ln L(Θ) = arg min
Θ"m
2ln σ2
n+1
2σ2
n
m
X
k=1
ǫ2(k)#.(16)
Eq. (16) summarizes the parameter estimation problem that is investigated for the estima-
tion of multi-channel pilot model parameters in this paper. Similar to the frequency-domain
methods, this is a highly nonlinear optimization problem, as described in Section II.D. To
cope with this problem, a strategy is chosen here that uses a genetic algorithm to identify a
solution close to the global minimum.
III.B. Genetic Likelihood Optimization
Genetic algorithms are commonly used to find approximate solutions to nonlinear optimiza-
tion problems, and have been successful in many applications.18, 30 The problem solving
capabilities of genetic algorithms are based on the principle of “survival of the fittest” as
considered in evolutionary biology. In such an algorithm, a population of candidate solu-
tions to the optimization problem is subjected to random genetic functions such as selection,
mating, crossover and mutation. These functions cause the population to evolve toward in-
creasingly better, i.e., more fit, solutions. The inherent randomness of genetic algorithms and
the fact that a large number of different solutions are evaluated results in a high probability
of finding the global minimum of an optimization problem.
Zaychik and Cardullo18 used a genetic algorithm as the principal estimation procedure
for identifying the parameters of the Hess operator model5from simulation data. However,
in this research, maximum likelihood estimation was not used for the formulation of the
optimization problem. Abutaleb30 describes an application of a genetic algorithm to a max-
imum likelihood estimation problem. The negative logarithm of the likelihood function is
used as the fitness criterion and the parameter sets that produce the lowest log-likelihood
are defined to be the fittest members of the population.
In the maximum likelihood estimation method developed in this paper, a genetic algo-
rithm is used to find sets of pilot model parameters that minimize the likelihood function
defined by Eq. (14). At the start of a genetic optimization process a random population of
parameter sets is generated within a lower and upper bound for all parameters. The fitness
of all the members of the population is calculated. Next, iterations are performed with the
following steps:
1. First, a random selection of members within the population is made. The members
with the highest fitness will have the highest probability of being selected. The selected
members will be used for mating.
2. The parameters of each member are coded into genes given their lower and upper
13 of 29
bound. A gene is a binary representation of the parameter value. For example, if
the current value of a parameter is 1.6 and the lower bound and upper bound are
respectively 0 and 2, a binary representation using 8 bits will be 11001100. The lower
and upper bound will be 00000000 and 11111111, respectively. The higher the amount
of bits used for the binary presentation, the higher the resolution of possible parameter
values between the lower and upper bound.
3. For each parameter set two solutions are picked at random from the pool created in
step 1. These two solutions will mate and produce offspring.
4. During the mating process, crossover between the genes will occur with a certain
probability, usually set to 0.7. The crossover location is chosen randomly. For example,
if we have two genes, 11001100 and 10101010, and a crossover location of 3, the two
offspring will have genes given by 11001010 and 10101100.
5. Next, every bit in the genes of the offspring will mutate with a certain mutation
probability. If a bit mutates, it changes from 0 to 1 or from 1 to 0. The mutation
probability is typically very low, e.g., 0.01.
6. The offspring is decoded from binary codes into real values and their corresponding
fitness is calculated.
7. A new population is chosen based on the fitness of the offspring and the original
population.
The genetic optimization is terminated when a specified number of iterations of steps 1
to 7 has been performed.
Due to the limited gene size that is commonly selected for encoding of parameters in
genetic algorithms to reduce the computational burden, the resulting parameter estimates
tend to be relatively inaccurate. In addition, genetic algorithms are not deterministic and
will give different results every time they are run. Therefore, a genetic algorithm is believed
to be less suitable to be used as the sole estimation method for estimating pilot model
parameters from measurement data.
III.C. Unconstrained Gauss-Newton Optimization
The parameter sets that are obtained from the genetic likelihood optimization have a high
probability of being close to the global optimum of the optimization problem. To further
refine these parameter estimates, the solutions of the genetic algorithm are used as the initial
parameter estimates for an unconstrained Gauss-Newton optimization. This gradient-based
14 of 29
optimization method is the classical approach to solving maximum likelihood estimation
problems.23, 29 The iterative parameter update equation for the Gauss-Newton optimization
is given by:
ˆ
Θ (i+ 1) = ˆ
Θ (i)−α(i)M−1
ΘΘ(ˆ
Θ (i))∂L(ˆ
Θ (i))
∂Θ.(17)
The gradients of the likelihood function with respect to all parameters, ∂L/∂Θ, can be
evaluated using the Jacobians of all state-space matrices with respect to the parameter vector
Θ. The Fisher information matrix, indicated with the symbol MΘΘ in Eq. (17), is given by:
MΘΘ =1
σ2
n
m
X
k=1 ∂ǫ(k)
∂Θ2
.(18)
The Fisher information matrix is symmetrical and should be positive definite, i.e., full
rank, as it needs to be inverted for the parameter update (Eq. (17)). The inverse of the
Fisher information matrix yields the Cram´er-Rao lower bound (CRLB), i.e., the minimum
achievable variance of the parameter estimate. Both the Fisher information matrix and the
first-order gradient of the likelihood function vary with the magnitude of the estimation error
ǫ, yielding parameter update steps of larger magnitude if estimation errors are larger. In
some instances, this causes the Gauss-Newton optimization to become unstable if initial pa-
rameter errors are large: in that case the first parameter update may force the unconstrained
algorithm to a region of meaningless and inaccurate solutions from which it cannot recover.
Using the solution of a genetic algorithm as proposed here as the initial parameter es-
timate for the Gauss-Newton optimization reduces the occurrence of such undesired effects
significantly. A second measure that can be taken to avoid this behavior is by considering
the line-search parameter αin Eq. (17). This parameter, which typically varies between 0
and 1, is determined at each iteration before the actual parameter update to ensure optimal
(i.e., most rapid) minimization of the likelihood function. This is illustrated in Figure 3,
where a typical variation of the likelihood Lis depicted as a function of α. For the example
shown in Figure 3 it is clear that α= 0.8 yields a superior parameter set after the update
step than would be obtained for α= 1.
The resolution that is considered for αduring the optimization – indicated with circles
in Figure 3 – can be selected freely and is highly dependent on the application. For the pilot
model estimation problem considered here, it was found that for the first iterations a rather
coarse spacing of α, i.e. 0.1, already yielded enough added stability to the optimization
algorithm. When parameter estimates came closer to their optimal values, it was found that
small improvements could still be made for very small values of α. Therefore, a finer spacing
of α, i.e. 0.01, was considered below 0.2 (see Figure 3).
15 of 29
α, -
-ln(L), -
0 0.2 0.4 0.6 0.8 1
500
600
700
800
900
1000
Figure 3. Typical likelihood variation along search vector.
IV. Results
In this section, the genetic maximum likelihood method will be evaluated using experi-
mental data. In addition, the parameter estimation results from the experimental data will
be used in multiple pilot model simulations with increasing remnant levels to investigate the
robustness of the method to increasing pilot remnant (measurement noise).
IV.A. Application to Experiment Data
The experiment was performed in the SIMONA Research Simulator of the Delft University
of Technology, Faculty of Aerospace Engineering. The objective of the experiment was to
investigate the effects of pitch rotational motion and heave motion on multimodal pilot
control behavior in an aircraft pitch control task. The experiment had a full factorial design
with eight conditions with varying motion cues. In every condition, pitch rotational motion
and two distinct components of heave motion could be either on or off.
The genetic maximum likelihood method was used to estimate the parameters of the
pilot model using the experimental data. In this section, the performance results of the
genetic MLE method will be given for data of one subject for one of the experimental
conditions. In this condition, rotational pitch motion was available in addition to the cues
from a compensatory display. The subject for which the performance data is shown was
chosen randomly.
IV.A.1. Experimental Measurements
Seven subjects participated in the experiment. All had experience with similar manual
control tasks from previous human-in-the-loop experiments. Two subjects have additional
experience as aircraft pilots. The subjects ages ranged from 25 to 46 years old. Before
starting the experiment, the subjects received an extensive briefing on the scope and objective
16 of 29
e, deg
(a) Error signal
0 10 20 30
-2
-1
0
1
2
u, deg
(b) Control signal
0 10 20 30
-8
-4
0
4
8
t, s
θ/ft, deg
(c) Pitch signal
pitch attitude θ
target ft
0 10 20 30
-3.0
-1.5
0.0
1.5
3.0
t, s
¨
θ, deg s−2
(d) Pitch acceleration signal
0 10 20 30
-40
-20
0
20
40
Figure 4. Average time traces of various loop signals (sub ject 1).
See, deg2/(rad s−1)
(a) Error spectrum
10-1 100101
10-8
10-6
10-4
10-2
100
102
Suu, deg2/(rad s−1)
(b) Control spectrum
spectrum
disturbance freq.
target freq.
10-1 100101
10-6
10-3
100
103
ω, rad s−1
Sθθ , deg2/(rad s−1)
(c) Pitch spectrum
10-1 100101
10-8
10-6
10-4
10-2
100
102
ω, rad s−1
S¨
θ¨
θ, deg2s−2/(rad s−1)
(d) Pitch acceleration spectrum
10-1 100101
10-6
10-4
10-2
100
102
104
Figure 5. Average loop signal spectra (subject 1).
17 of 29
of the experiment. The main instruction they received before the experiment was to attempt
to minimize the pitch tracking error, i.e., the signal that was presented on the visual display,
as best as possible.
The experiment started with a considerable number of training runs. The training phase
ended when a constant level of performance was achieved. After the training phase, 5
repetitions of every condition were performed in a random order, of which the data was used
in the final results. Each individual experiment run lasted 110 seconds, of which the last
81.92 seconds were used as the measurement data. Participants generally need some time
to stabilize the disturbed aircraft model after the start of a run, therefore the data from the
first 28.08 seconds of each run were discarded. Data was logged at a frequency of 100 Hz.
Further details of the experiment setup can be found in Reference 2.
Time traces of all signals depicted in Figure 1 were recorded during the experiment. To
reduce the noise content in the signals and improve pilot model parameter estimates, the time
traces of the five repetitions performed by each subject were averaged for the identification
procedure.8Examples of averaged time traces of the error signal e, the control signal u, the
pitch attitude θand pitch acceleration ¨
θare depicted in Figure 4. The time trace of the
pitch target signal ftis shown alongside that of the pitch attitude for reference.
Note from Figure 4 that the subjects successfully made the aircraft pitch attitude follow
the target signal ft. Most of the additional oscillations in pitch attitude can be attributed
to the presence of the disturbance signal.
The power spectral densities of the time traces given in Figure 4 are given in Figure 5.
The input frequencies of the forcing functions, as given in Table 1, are indicated with circles
and triangles for fdand ft, respectively. The signal-to-noise ratio at the input frequencies is
high for all the signals in the control loop.
IV.A.2. Algorithm Performance
The error, pitch acceleration and control signals given in Figure 4 are used to estimate the
pilot model parameters with the genetic maximum likelihood method. Before the Gauss-
Newton optimization, 100 iterations with the genetic algorithm are performed. The parame-
ter lower bounds were set to zero and the upper bounds were taken positive and large enough
to account for all relevant solutions of the estimation problem (see Table 2). For the param-
eter conversion to binary strings, 20 bits are used. Values for the crossover and mutation
probabilities were set to 0.7 and 0.01, respectively. The solution of the Genetic algorithm
is used as the initial parameter set for the Gauss-Newton optimization. The Gauss-Newton
optimization is terminated if the derivative of the parameters with respect to the search
vector is smaller than 10−6.
Typically, 100 repetitions of the algorithm are performed on a single data set, of which
18 of 29
Table 2. Lower and upper parameter bounds used in the genetic algorithm.
KvTlead Tlag Kmτvτmωnm ζnm
- s s - s s rad s−1-
lower bound 0.0 0.0 0.0 0.0 0.0 0.0 5.0 0.0
upper bound 5.0 10.0 10.0 10.0 1.0 1.0 20.0 1.0
iteration #
-ln(L), -
(a) Total likelihood variation
Genetic Gauss-
Newton
Genetic optimization
Gauss-Newton optimization
0 25 50 75 100 125
0
2000
4000
6000
8000
(b) Gauss-Newton likelihood variation
iteration #
-ln(L), -
Genetic Gauss-
Newton
1×parameter set 2
9×parameter set 1
95 100 105 110 115 120
0
500
1000
1500
2000
2500
Figure 6. Changes in negative logarithm of the likelihood during optimization (10 repetitions,
subject 1).
the ten with the lowest likelihood are further evaluated using Gauss-Newton optimization.
The negative logarithm of the likelihood for these ten repetitions is given in Figure 6. It is
clear from Figure 6b that the solution of the genetic algorithm is different for all repetitions.
After refinement by the Gauss-Newton algorithm, two different sets of parameters are found.
The solution with the lowest negative logarithm of the likelihood is found nine times. In
one instance, the optimization ends up in a different (local) minimum, which clearly has a
higher likelihood.
This illustrates the nonlinearity of the optimization problem. For data from other sub-
jects, none to up to three of the obtained parameter sets were found to represent local minima.
Note that a local minimum was never attained more often than the global minimum.
To illustrate the significance of this result, pilot model parameters were also estimated by
using only the Gauss-Newton maximum likelihood algorithm. To illustrate the large influence
of the initial parameter estimate on the results of this gradient-based optimization, a large
number of initial conditions were evaluated. For all parameters, a number of different values
were selected between the upper and lower bounds defined for the genetic algorithm. The
full factorial of all these initial parameter values yielded 40000 different initial parameter
sets. Figure 7 depicts the likelihood of the final parameter estimates obtained with the
Gauss-Newton algorithm for all these different initial estimates, in ascending order.
The leftmost shaded area in Figure 7 indicates the set of initial conditions for which the
19 of 29
1436×parameter set 1
843×parameter set 2
initial parameter set #
-ln(L), -
0 1000 2000 3000 4000 5000 6000 7000
0
2000
4000
6000
8000
10000
Figure 7. Negative log-likelihood of Gauss-Newton parameter estimates for 7000 different
initial parameter sets, sorted in ascending order (subject 1).
Gauss-Newton algorithm attained parameter estimate 1, as depicted in Figure 6; the second
shaded area, which corresponds to a solution with a higher likelihood, are the occurrences
for which parameter set 2 was found. All parameter sets that are outside either of the shaded
areas in Figure 7 represent initial estimates for which the algorithm was unable to converge.
Note that the Gauss-Newton algorithm only converged to the global minimum (parameter set
1) for around 20% of all tested initial conditions. For all other initial conditions, the Gauss-
Newton parameter estimation procedure would either converge to an incorrect minimum,
or fail to converge to a stable solution. As illustrated by Figure 6, the addition of the
genetic algorithm ensures reliable estimation of the parameter set that represents the global
minimum.
IV.A.3. Pilot Describing Functions
Figure 8 gives the linear pilot response functions Hpe and Hpθ for the two different solutions
found by the algorithm depicted in Figure 6b. The Fourier coefficients, which can be analyt-
ically calculated from the measured time-domain data, coincide with parameter set 1. This
is a further indication that parameter set 1 represents the correct solution of the parameter
estimation problem. The parameter values and the square root of the diagonal of the CRLB
resulting from the Gauss-Newton optimization are given in Table 3. Note that the CRLB is
found to be higher for nearly all parameter estimates of parameter set 2. From the param-
eter values, it can be seen that the gain for the vestibular channel Kmis much higher for
parameter set 1. In parameter set 2 the lower gain for the vestibular channel is compensated
for by the higher values for the lead and lag time constants. The vestibular time delay is
higher than the visual time delay for parameter set 2, which disagrees with findings from
previous research.2,6, 7 Finally, the neuromuscular frequency found for parameter set 1 is
20 of 29
higher than that obtained for parameter set 2.
|Hpe(jω)|, -
(a) Visual, magnitude
FC at ωd
FC at ωt
parameter set 1
parameter set 2
10-1 100101
10-1
100
101
|Hpθ(jω)|, -
(b) Motion, magnitude
10-1 100101
10-3
10-2
10-1
100
101
ω, rad s−1
6Hpe(jω), deg
(c) Visual, phase
10-1 100101
-540
-360
-180
0
180
ω, rad s−1
6Hpθ(j ω), deg
(d) Motion, phase
10-1 100101
-540
-360
-180
0
180
Figure 8. Bode plot of the pilot visual and motion response for parameter sets 1 and 2 (subject
1).
Table 3. Values and standard deviation for parameter sets 1 and 2 (subject 1).
parameter parameter set 1 parameter set 2
ˆ
ΘqCRLB( ˆ
Θ) ˆ
ΘqCRLB( ˆ
Θ)
Kv, - 0.938 6.77·10−30.932 8.91·10−3
Tlead, s 0.278 6.47·10−30.337 9.21·10−3
Tlag, s 0.502 1.92·10−20.774 3.08·10−2
Km, - 0.692 1.29·10−20.147 1.11·10−2
τv, s 0.268 2.79·10−30.134 2.41·10−3
τm, s 0.175 1.94·10−30.679 1.29·10−2
ωnm, rad s−111.560 6.41·10−29.775 7.15·10−2
ζnm, - 0.175 4.60·10−30.168 9.56·10−3
The measured pilot control signal is compared with the model output for the two param-
eter solutions in Figure 9. It can be seen that the difference between both solutions in the
time domain is only very small. The variance accounted for (VAF), which is a measure of
the correlation of two time-domain signals,8is only marginally higher for parameter set 1.
Figure 10 gives the pilot frequency response functions corresponding to the optimal pa-
rameter estimate for all subjects for the experimental condition used in this paper. From
21 of 29
t, s
u, deg
measurement
parameter set 1: VAF = 85.30%
parameter set 2: VAF = 82.05%
0 2 4 6 8 10
-2
-1
0
1
2
Figure 9. Comparison of model control signals for parameter sets 1 and 2 (subject 1).
this figure it can be seen that all subjects give consistent results. The response functions
vary slightly between subjects as every individual has a different control strategy, which is
commonly seen in this type of experiments.
IV.A.4. Pilot Remnant
The pilot remnant is determined by the difference between the model output of the linear
pilot model and the measured pilot control signal, as depicted in Figure 9. A time trace of
the pilot remnant for the model with parameter set 1 is given in Figure 11a. The power
spectrum of this signal is given in Figure 11b. It can be seen that the spectrum has the
approximate characteristics of a third-order low-pass filter with a damping coefficient:
Hn(jω) = Knω3
n
((jω)2+ 2ζnωnjω +ω2
n) (jω +ωn),(19)
with Kn= 2.6 the remnant intensity and ζn= 0.26 and ωn= 12.7 rad/s the remnant
filter damping factor and break frequency, respectively. Similar characteristics are found in
previous research.8
The probability density function of the pilot remnant is given in Figure 12. This figure
clearly shows that the remnant signal has a distribution that is almost perfectly Gaussian
with a zero mean, as is assumed in Eq. (13). The low-pass remnant spectrum has no effect
on the normality of the signal. The fact that the remnant is distinctly non-white, as shown
in Figure 11b, may cause extra bias in the pilot model parameter estimates, which has been
further investigated using off-line simulations.
IV.B. Off-line Simulations
To investigate the robustness of the genetic MLE method to increasing levels of pilot remnant,
parameter set 1 is used to perform multiple simulations of the closed-loop control task with
22 of 29
|Hpe(jω)|, -
(a) Visual, magnitude
10-1 100101
10-2
10-1
100
101
102
|Hpθ(jω)|, -
(b) Motion, magnitude
10-1 100101
10-2
10-1
100
101
102
ω, rad s−1
6Hpe(jω), deg
(c) Visual, phase
subject 1
subject 2
subject 3
subject 4
subject 5
subject 6
subject 7
10-1 100101
-540
-360
-180
0
180
ω, rad s−1
6Hpθ(j ω), deg
(d) Motion, phase
10-1 100101
-540
-360
-180
0
180
Figure 10. Bode plot of the pilot visual and motion response for all experiment subjects.
t, s
n, deg
(a) Remnant time history
0 10 20 30 40 50 60 70 80
-4
-2
0
2
4
ω, rad s−1
|N|, dB
(b) Remnant spectrum
remnant spectrum
remnant at ωd
remnant at ωt
remnant model
10-1 100101
-80
-60
-40
-20
0
Figure 11. Pilot remnant characteristics (sub ject 1).
23 of 29
n, deg
f(n), -
measured remnant f(n)
normal f(n|µ= 0, σ= 0.936 deg)
-4 -3 -2 -1 0 1 2 3 4
0
500
1000
1500
Figure 12. Remnant probability density function (subject 1).
increasing levels of remnant. The remnant filter break frequency ωnand damping factor ζn
are kept constant and the remnant intensity Knis gradually increased to achieve the desired
ratio between control signal power and remnant power. The ratio is increased from 0 to 0.5,
while the value found for the averaged experimental data is approximately 0.15 for parameter
set 1. For each remnant level, eighty simulations are performed, each with a different noise
realization.
Figure 13 shows the average parameter bias found by applying the genetic MLE procedure
to the data of each simulation. Also, the 99% confidence interval for each parameter is
indicated in this figure. It can be seen that for increasing levels of pilot remnant, the average
bias of the parameter estimates increases. For all parameters, the average bias remains
within the 99% confidence interval, which indicates that the bias is not significant. At a
remnant-control signal power ratio of 0.15, found experimentally, the bias is relatively small
for all parameters.
V. Discussion
This paper presents a new strategy for the estimation of multi-channel pilot model pa-
rameters from time-domain data based on maximum likelihood estimation. The classical
MLE method, with only a gradient-based algorithm to estimate the parameters, is very
dependent on the initial parameter guess. This is highly undesirable given the fact that
the cost function is highly nonlinear and contains many local minima. For this reason, a
genetic algorithm was used in combination with the gradient-based Gauss-Newton optimiza-
tion procedure. The genetic algorithm does not require an initial parameter guess and is far
less sensitive to local minima, as a population of possible solutions is kept until the algo-
rithm terminates. The subsequent use of a Gauss-Newton algorithm increases the accuracy
of the parameter estimate, as gradient information is included. The method was successfully
24 of 29
bias Kv, -
σ2
n/σ2
u, -
(a) visual perception gain
0 0.1 0.2 0.3 0.4 0.5
-1.0
-0.5
0.0
0.5
1.0
bias Tlead, s
σ2
n/σ2
u, -
(b) visual lead constant
0 0.1 0.2 0.3 0.4 0.5
-0.2
-0.1
0.0
0.1
0.2
bias Tlag , s
σ2
n/σ2
u, -
(c) visual lag constant
0 0.1 0.2 0.3 0.4 0.5
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
bias τv, s
σ2
n/σ2
u, -
(d) visual perception time delay
0 0.1 0.2 0.3 0.4 0.5
-0.1
-0.1
0.0
0.1
0.1
bias Km, -
σ2
n/σ2
u, -
(e) motion perception gain
0 0.1 0.2 0.3 0.4 0.5
-2.0
-1.0
0.0
1.0
2.0
bias τm, s
σ2
n/σ2
u, -
(f ) motion perception time delay
0 0.1 0.2 0.3 0.4 0.5
-0.04
-0.02
0.00
0.02
0.04
bias ζnm, -
σ2
n/σ2
u, -
(g) neuromuscular damping
0 0.1 0.2 0.3 0.4 0.5
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
bias ωnm, rad s−1
σ2
n/σ2
u, -
(h) neuromuscular frequency
99 % conf.int.
mean bias
0 0.1 0.2 0.3 0.4 0.5
-4.0
-2.0
0.0
2.0
4.0
Figure 13. Average parameter bias for different remnant levels.
25 of 29
applied to data from an experiment investigating the influence of pitch and heave motion
cues on pilot control behavior.
Applying the genetic maximum likelihood algorithm to the experiment data multiple
times for all subjects resulted in the occurrence of solutions other than the global optimum.
These occurrences were infrequent, however, and the corresponding solutions always have a
higher likelihood than the global minimum. Hence, the optimization should be performed
several times for each condition from an experiment to ensure that the global optimum
solution is found. The often subtle differences between the parameter sets of a local or
a global optimum, and the equivalence of the resulting time-domain fits make it difficult
to distinguish between these solutions based on this information alone. Although, Fourier
coefficients can provide an indication of the frequency response of the global optimum, these
can only be computed when multi-sine signals are used as forcing functions.
Simulations have been performed to investigate the robustness of the MLE method to
increasing levels of remnant. It was found that the average bias of the parameters increases
for increasing levels of remnant power. The bias remained within the 99% confidence in-
terval of the parameter estimate, which indicates that the bias is not significant. For the
parameter estimation on averaged time-domain data of five runs (σ2
n/σ2
u≈0.15) the bias of
the parameter estimates is relatively small. When individual runs (σ2
n/σ2
u≈0.30) are used
in the parameter estimation procedure, the bias becomes more substantial, however.
As maximum likelihood estimates have the property to be unbiased for infinite mea-
surement times, the increase in bias for higher levels of remnant is an unexpected result.
Although this bias is partly caused by the finite measurement times, the fact that the rem-
nant is colored rather than purely Gaussian white noise may cause an additional bias. Also,
because of a correlation between the pilot model input and output in the closed loop – the
remnant, which is part of the output of the pilot model, travels around and is also present
in the input of the pilot model – an additional bias may be present. A solution for both
problems would involve inclusion of process noise into the state-space representation of the
pilot model, which requires a prediction-error algorithm to be added to the MLE method.
This allows for the properties of the remnant filter to be estimated, and could reduce the bias
in the parameter estimates at higher levels of remnant. The Kalman filter that is needed
to solve this problem, however, will increase the complexity of the parameter estimation
procedure and will significantly increase the computational burden.
In this paper, the genetic MLE method is evaluated with data from an experiment where
multi-sine signals were used as forcing functions. Parameter estimation in the time domain
also allows the use of novel types of input signals, however, such as ramp or step signals that
are often considered in other system identification problems. These signals can be used as
target signal and may yield a control task that is much more similar to real piloting tasks. An
26 of 29
investigation on the use of these signals is already performed,24 but more research is needed to
evaluate the performance of the MLE method with these signals. The maximum likelihood
parameter estimation algorithm can also be used to identify multi-channel pilot models
when only one forcing function is present, as long as all the inputs to the pilot model are
sufficiently excited. The method has already been successfully applied to measurement data
from human-in-the-loop experiments where pure target-following tasks were considered.31, 32
Finally, the use of a genetic algorithm allows for the introduction of nonlinear elements
into the pilot model, as no gradient information is required. An example can be the addition
of motion perception thresholds. This will also be a subject of future research.
VI. Conclusions
A pilot model parameter estimation technique, based on the maximum likelihood method,
was introduced and enhanced with a genetic algorithm to considerably improve performance.
The new method was successfully applied to data from an experiment investigating the role
of pitch and heave motion during pitch control of an aircraft. The parameters of the multi-
channel pilot model and the properties of the remnant could be accurately estimated from
the time-domain signals. Using multiple iterations of the algorithm on a single data set,
the global optimum solution was found for 90% of the cases. Using multiple simulations
with different levels of pilot remnant, it was shown that there is no significant bias in the
parameters for lower remnant levels. For high levels of pilot remnant bias is more substantial,
caused by the fact that remnant tends to be colored rather than purely white noise.
Acknowledgments
This research was supported by the Technology Foundation STW, the applied science
division of NWO, and the technology program of the Ministry of Economic Affairs. The
authors would like to thank Tom Berger, currently affiliated with the NASA Ames Research
Center, for his initial investigation into maximum likelihood estimation of pilot model pa-
rameters.
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