Civil Engineering Reference
In-Depth Information
system output
u
(t) — is added to
u
c
(t):
u
(t)
¼
u
c
(t)
þ
v
u
(t)
:
(12
:
28)
1) motor noise vector is defined as an independent, zero-mean, Gaussian white noise vector
with auto-covariance: E{
v
u
(t)
v
u
(t
The (
'
þ
t)}
¼
diag(V
u
)d(t).
12.5.2.3 Control Task Representation
It is assumed that the task is reflected in the operator's choice of a feedback control
u
(
) which, in
steady-state, minimizes the cost functional:
(
)
,
E
X
X
'
X
'
m
q
i
y
i
þ
r
i
u
c
i
þ
u
c
i
J(
u
)
¼
g
i
_
(12
:
29)
i
¼
1
i
¼
1
i
¼
1
conditioned on the perceived information
y
p
(
). The cost functional weightings (Q
0; R
0;
G
0) may be either objective (specified by the experimenter) or subjective (adopted by the operator).
The rate of the control input is also weighted in the cost functional. This term may also represent an
objective (to account for human physiological limitations on the rate at which a control action can be
effected) or a subjective (reflecting that trained operators are reluctant to make rapid control move-
ments) weighting on control rate. The control rate weighting introduces a first-order lag in the
optimal controller (Kleinman et al., 1971). Hence, the control rate weighting matrix G is used to
include a first-order representation of the neuromuscular system in the model.
.
12.5.3 Model Parameters, Outputs, Solution, and Identification
12.5.3.1 Model Parameters
In order to apply the OCM, the following quantities must be defined: (i) the system parameters: the
characteristics of the linear system (A, B, C, D, E, H), and the statistics of the system disturbance
(W); (ii) the task-related parameters: the weightings of the cost functional (Q, R); and (iii) the human
response parameters (t, t
N
, V
y
, and V
u
). The OCM parameters describe the control task and the operator
control behavior in terms of optimal control.
The formulation of the operator's characteristics in mathematical terms is difficult. First of all, the
task-related parameters — determining the balance of the model, or, equivalently, the control strategy
of the operator — are not easy to define beforehand. The OCM describes the operator as an optimal
controller with respect to task-related quantities, which do not necessarily relate to human-centered
optimization strategies. Second, the operator is assumed to have a good mental model of the system
dynamics and the disturbance characteristics. Although it is reasonable to assume that this model
indeed becomes well-developed through a learning process (Krendel & McRuer, 1960; Stassen et al.,
1990), the OCM results can and should be considered as the best possible operator performance for a
given quadratic cost functional.
12.5.3.2 Model Outputs
Unlike the structural pilot models of Section 4, which operate almost exclusively in the frequency
domain, the OCM is essentially a time-domain model. Nonetheless, it provides results in both
domains (Kleinman et al., 1970b, 1971). The time-domain outputs are the variances of all signals in
the closed loop, that is, E{u
i
}
j
i¼1,...,'
, E{x
i
}
j
i¼1,...,n
, and E{y
i
}
j
i¼1,...,m
. Due to the incorporation of the
pilot remnant through the observation and motor noises, these variances allow a direct comparison
with the values obtained in the actual experiment.
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