Civil Engineering Reference
In-Depth Information
pilot
H
p
system
H
c
open loop
Y
OL
10
1
10
1
10
1
k
10
0
10
0
10
0
w
c
10
−1
10
−1
10
−1
10
−1
10
0
10
1
10
−1
10
0
10
1
10
−1
10
0
10
1
10
1
10
1
10
1
k
/
s
10
0
10
0
10
0
w
c
10
−1
10
−1
10
−1
10
−1
10
0
10
1
10
−1
10
0
10
1
10
−1
10
0
10
1
10
1
10
1
10
1
k
/
s
2
10
0
10
0
10
0
w
c
10
−1
10
−1
10
−1
10
−1
10
0
10
1
10
−1
10
0
10
1
10
−1
10
0
10
1
frequency, rad/sec
frequency, rad/sec
frequency, rad/sec
s
2
.
The crosses indicate the frequencies of the sinusoids in the forcing function R( jv). The dashed lines in the right
column indicate integrator-like dynamics.
FIGURE 12.11
Identification of pilot control behavior when controlling the three basic systems k, k
/
s, and k
/
column shows the magnitude of the estimated open-loop, that is, the product of the estimated pilot FRF
and the system FRF:
Y
OL
(jv)
¼ H
p
(jv)H
c
(jv)
:
The dynamics of the center column show the proportional (top), integrator (center), and double inte-
grator (bottom) properties of the system to be controlled. The left column shows that the pilot FRF is
different for all three systems, a clear indication that the pilot is adapting to the dynamics of the
system to be controlled. The right column, however, shows that the shape of the open-loop is the
same. Independent of the system dynamics, the open-loop FRF resembles “integrator- like” dynamics
near the frequency where the open-loop equals one, that is, near the cross-over frequency. As will be
discussed later, the value of the cross-over frequency depends, among others, on the dynamics of the
system and the bandwidth of the forcing function.
Apparently, human operators adapt to the system to be controlled in such a way that the open-loop,
that is, the human dynamics times the system dynamics, becomes an integrator. McRuer et al. generalized
this systematic adaptation of human control behavior with the postulation of their COM theorem
(McRuer et al., 1965).
The COM theorem states that human controllers adjust their control behavior to the dynamics of the
controlled element in such a way that the dynamic characteristics of the open-loop transfer function in
the cross-over region can be described by:
v
c
Y
OL
(jv)
jv
e
jvt
e
,
¼
H
p
(jv)H
c
(jv)
(12
:
16)
where v
c
is the cross-over frequency and t
e
is a time delay lumping the information-processing delays of
the human operator.
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