In the 1940s aircraft
designers appreciated the need to characterize the transfer function of the
human pilot in terms of a differential equation. Indeed, this is necessary for
any vehicle or controlled physical process for which the human is the
controller, see Figure 6.1.2. In this case both the human operator H and the physical
process P lie in the closed loop (where H and P are
Laplace transforms of the component transfer functions), and the HP combination
determines whether the closed-loop is inherently stable (i.e., the closed loop
characteristic equation 1+HP = 0 has only negative real roots).
In addition to the
stability criterion are the criteria of rapid response of process state x
to a desired or reference state r with minimum overshoot, zero “steady-state
error” between r and output x, and reduction to near zero of the
effects of any disturbance input d. (The latter effects are determined
by the closed-loop transfer functions x=HP/(1+ HP)r+
1/(1+ HP)d
, where if the
magnitude of
H is
large enough
HP /(1+
HP) approaches unity and 1/(1+ HP) approaches 0. Unhappily, there
are ingredients of
H which
produce delays in combination with magnitude and thereby can cause instability.
Therefore, H must
be chosen carefully by the human for any given P.)
Research to
characterize the pilot in these terms resulted in the discovery that the human
adapts to a wide variety of physical processes so as to make HP=K(1/s)(e–sT). In
other words, the human adjusts H to make
HP constant.
The term K is an overall amplitude or gain, (1/ s) is the Laplace
transform of an integrator, and ( e-sT) is
a delay T long (the latter time delay being an unavoidable property of
the nervous system). Parameters
K and
T vary modestly in a predictable way as a function of the physical process
and the input to the control system. This model is now widely accepted and
used, not only in engineering aircraft control systems, but also in designing
automobiles, ships, nuclear and chemical plants, and a host of other
dynamic systemsŲ²
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