From Tom Bourbon [931230.1238]
[From Bill Powers (931229.1900 MST)]
Cliff Joslyn (931229) --
Bill to Cliff:
I appreciate your attempts to find a mathematical and technical
definition of stability. To help you toward your goal, I have
some technical definitions of a stable control system that you
might find useful. I am concerned here only with "dynamic
stability," not stability in the sense that a system with high
loop gain can "stabilize" a controlled variable against
disturbances.1. Unstable (the opposite of control)
[details omitted here]
2. Piss-poor excuse for stable control
When you turn on the control system, the controlled variable is
brought somewhere near the vicinity of the intended reference
level, but it is continually quivering and oscillating and
jumping back and forth, and never does settle down within specs.
Tom:
The sort of thing sometimes published in _Science_, where, as we all know,
perceptual control theory is "old hat."
3. Sort of stable control, or OK, yeah, I guess you could say
it's controlling.Any change in the reference signal, or any disturbance, causes an
error that oscillates in diminishing phase-space circles until
eventually, after a few seconds, hours, or days, it disappears up
its own point attractor.
What a sight *that* would be. Right down into its own basin of attraction,
then right up its own point attractor!
4. Pretty good stable control; or, it's a good cost-performance
tradeoff so why don't we just go ahead and tell the customer it's
ready.
[details omitted here]
5. Stable control; or that, man, is what I call a control system.
When a step-change in the reference signal occurs, the controlled
variable changes at the same time to a new steady state, without
any overshoot or undershoot, so quickly that you'd have to look
closely to see the transition. When a step-disturbance suddenly
occurs, the controlled variable jumps immediately (by a small
amount) to a new value, without overshoot or undershoot, and
stays there. A dynamically stable control system is extremely
undramatic in its behavior.
. . .
Most of the phase-space diagrams I have seen, if they represented
the behavior of a real control system, would fall under 2. or 3.
above.
Tom:
Bill, I just returned from an all-morning session in the library at Rice
University, where there is a large collection of books and journals on
topics like dynamical systems, systems analysis, control theory, optimal
control theory, phase space analysis, and so on. I was there to do a
"reality check," looking through the recent material to determine if my
lay-person's memories of that literature are still accurate. There were
very few phase-space trajectories for real control systems; the literature
leans heavily toward other kinds of systems, or toward theoretical and
idealized functions. The few plots I found for real control systems belong
in your categories 2 and 3, just as you recalled from your own experience.
(By the way, while I was in the library, I looked at the first chapters of
about two dozen introductory texts on control theory -- the real kind. Not
one of them mentions control by living things. These days, first chapters
seem to be populated by capacitors, inductors and mass-spring systems, not
by living things.)
I have never seen enough data on a system represented by
the usual loops and spirals to estimate the static stability that
is represented. The phase-space diagram of what I call a good
control system would not be very interesting (a single arc from
point A to point B, as in Tom Bourbon's recent plots).
You noticed the plain-vanilla trajectories in my recent examples. I had
hoped someone would comment on how different they are from the nifty spirals,
orbits, doughnuts and butterfly wings that seem to impress people in
behavioral science. The behavior of good control systems, living things for
example, just isn't very "fancy." In fact, it can seem so unremarkable and
undramatic that it goes unnoticed -- for centuries.
A request:
Speaking of simple plots in phase space, could someone post an acceptable,
orthodox 2-D plot for the ball-in-a-bowl example, showing the trajectory for
velocity and vertical displacement? I confess my neophyte status at working
with plots in phase space. None of the books I looked through today were
any help to me; they did not show anything as simple as that example. I'm
looking for a simple ASCII rendering in coordinates like these:
+d (displacement of ball from bottom of bowl)
>
>
>
>
>
-v ----------|---------- +v (velocity)
Any help will be appreciated.
Until later,
Tom