[From Bill Powers (2004.06.25.1137)]
Attached is a figure for a paper to be given at this year's CSG conference.
It will be part of the book I mentioned, which I hope will be co-authored
by Richard Kennaway and Rick Marken -- and anyone else in the CSG who can
contribute to this subject.
This figure shows the behavior of a negative feedback control system with a
perceptual delay and a leaky integrator for an output function. The
duration of each plot is 10 seconds. The columns from left to right show
plots for a perceptual delay of 0, 0.2, and 0.4 seconds. The rows from top
to bottom show plots for gains of 50, 25, and 12.5, with the output time
constant (1/slowing_factor) set to 2.5 sec, 1.25 sec, and 0.625 sec,
preserving a constant ratio of gain to time constant.
The result of changing the time constant and gain together in this way is
to make the behavior of the perceptual signal the same in each column. In
the left column, the time constant is longer than it needs to be, so the
rise of the perceptual signal to the reference signal is slow. In the
middle column, this time constant just compensates for the delay in the
feedback path of 0.2 seconds, so the rise of the perceptual signal is about
optimal. In the third column, the delay of 0.4 seconds is too long to be
compensated by the time constant-gain combination, and damped oscillations
are apparent, the signs of borderline instability.
The rise of the perceptual signal in the first column could be made very
fast by reducing the time constant in all three rows; the instability in
the third column could be removed by making the time constants longer or
the gain lower in all three rows. For any delay and gain, there will be a
time constant that will provide the quickest possible rise of the
perceptual signal to the reference signal without overshoots. Likewise, for
any combination of delay and time constant, there will be a gain that will
give the optimum rise time of the perceptual signal, again without overshoots.
Clearly, by adjusting the gain, time constant, and delay parameters, it is
possible to cause the perceptual signal to "hunt" back and forth around the
reference level. But it is equally clear that simply by picking the best
combination of gain and time constant for any given amount of delay, it is
possible to eliminate the hunting and obtain smooth and accurate performance.
Going down the rows, the gain decreases. As the gain decreases, the
steady-state value of the perceptual signal (indicated by numbers for the
times shown by tick-marks) falls farther below the value of the reference
signal, and the effect of the negative-going disturbance in the steady
state becomes larger.
While gain changes do affect the ability to resist disturbances, the effect
of the changes is far from proportional. A decrease in the output gain to
one-fourth its maximum value (from 50 to 12.5, a 75% drop) causes the
steady-state perceptual signal to change from 98% of the reference signal's
value to 92%: a 6% drop. This illustrates the relative independence of the
controlled quantity from changes in the output properties of the control
system itself, as well as showing the defense of the input quantity against
disturbances over a wide range of output gains and time constants..
Other composite figures of this kind will be used to illustrate the
relationships between actions and disturbances with proportional
environmental functions and with nonlinear functions. The same will be done
to show how nonlinear perceptual input functions (for example,
Weber-Fechner laws or power laws) result in apparent nonlinearities between
the controlled quantity and the perceptual signal, while still keeping the
perceptual signal near the reference signal, and maintaining a nearly
linear relationship between action and disturbance.
Then we can go to multidimensional control at a single level, which I call
"intraordinal" control, showing how perceptual signals that are functions
of multiple environmental variables can still be controlled by outputs that
affect one or more of the environmental variables, with disturbances
affecting some or all of the same environmental variables. The meaning of
orthogonality of perceptual input functions will be illustrated in two
dimensions, three dimensions, and many dimensions. This will establish the
fact that controlled perceptions can exist when there is no corresponding
single variable in the environment, and how the nature of a controlled
perception can be changed just by changing the perceptual input function.
And finally we can start looking into hierarchical control, or
"superordinal" control, in which higher systems control lower ones. The
meaning of "logical types" (or better, "perceptual types") will be illustrated.
There are many illustrations of these properties of control systems,
devised and programmed by Marken, Kennaway, Bourbon, and me. Other examples
can be found in the literature and can be referred to the principles laid
out in previous parts of this book.
Alice Powers McElhone will be the publisher of this book; I owe it to her.
But Dag Forssell will be the co-editor with Alice, and Greg Williams will
be invited to be a co-editor as well. Everything, however, is subject to
negotiation. There will be close coordination among all the parties
involved, and between the print version of the book and electronic versions
on Dag's web page. I estimate that it will take two years for the book to
be ready for publication.
Now all I have to do is last that long.
Best,
Bill P.
PropFig2.bmp (96.1 KB)
