Stability; Request

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
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

[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

[Martin Taylor 931230 15:00]
(Tom Bourbon 931230.1238)

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 don't think it can be done in 2-D. At the very least, one needs radial
position, radial momentum and angular momentum, and I suspect one needs
angular position though the symmetry of the situation may let one get away
without this last, provided the bowl is circularly symmetric.

Consider: if the angular momentum is zero, there is a 2D plot of radial
position and velocity within which one can draw a set of orbits, all of
which represent tracks in which the ball oscillates through the centre
of the bottom of the bowl. If there is no frictional loss as the ball
rolls, these orbits are distinguished only by the energy in the ball, and
they will be half-ellipses in the phase space (there is no radial position
at negative values, and so as the ball passes through the centre it goes
instantaneously from a high negative to a high positive velocity. That
doesn't mean an infinite acceleration. It means that the phase space has
to be cut if it is to be laid out on a flat plane; at zero position,
negative velocity orbits connect to equal positive velocity orbits. Actually,
it is easier to see what is happening if you replicate the half-space
and mirror it upside down for negative radial position).

toward centre | away from centre
(mirrored) |
.**|>-_
/ | \
*..|<_-
away from centre| toward centre
(mirrored)

Now think of a situation in which the angular momentum is not zero.
The orbits now never reach zero position, and it is harder to visualize.
But imagine that the ball chases around and around the bowl at a constant
distance from the centre. Then the orbit in the above 2-D space is a single
point at radial velocity zero and some positive radial position. But in the
space of angular momentum and position, the orbit is a line through all
angular positions at a constant momentum. In the entire 4-D space, the
orbit is this same line through all angular positions at a constant radial
position, radial velocity (zero), and angular momentum.

Now think of the more general case, in which the radial and the angular
momenta are both non-zero. In the radial space above, the orbit is some
kind of near-circle around some point on the radial position axis. Such
an orbit would cut across the orbits for zero angular velocity. This in
itself shows that the 2-D space is inadequate, since orbits in a complete
phase space cannot cross.

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."

They aren't different, so far as I can see. They assert that there would
be no overshoot in the particular system you drew.

What would be different for a real control system as opposed to what we
might call a "passive" dynamic is that there would be no single orbit
through any specific point in phase space. In a completely described
passive dynamic, if you measure infinitely closely, there is only one
orbit through any point. Orbits do not cross. But in an incompletely
described dynamic, orbits can cross, as pointed out above. If a control
system happens to be generating the dynamic you are observing, and you
are not observing the internal states of the control system along with
the environmental variable, you do not have a complete description of
the dynamic. If there is control, what you will see is, for different
trials starting at the same point in the phase space, a tangle of orbits
that fan out from the starting point into a possibly narrow band, and
that arrive by some probably spiral route in a disk near the zero point.
Only if control is overdamped and therefore not as quick as it might be
will you avoid the spiral. The better the control, the less of that you
will see, and the higher the inward velocity will be until the orbit gets
near the zero-error position. But you should see the tangled fan-out from
the starting point, if your measuring apparatus is good enough, and there
really is control.

Martin

Tom Bourbon [931231.1144]

(I am exploring this topic as a prelliminary step toward testing my
understanding of the post in which Martin outlined his integration of
information theory, dynamical systems analysis, and PCT.)

[Martin Taylor 931230 15:00]
(Tom Bourbon 931230.1238)

Tom:

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?

Martin:

I don't think it can be done in 2-D. At the very least, one needs radial
position, radial momentum and angular momentum, and I suspect one needs
angular position though the symmetry of the situation may let one get away
without this last, provided the bowl is circularly symmetric.

Martin then gave an example of a 2D plot when angular momentum is zero and
there is no friction between ball and bowl. Thanks, Martin. As things turn
out, you confirmed my unposted interpretation of what would happen in that
case, which you diagrammed as:

toward centre | away from centre
(mirrored) |
.**|>-_
/ | \
*..|<_-
away from centre| toward centre
(mirrored)

(Martin then went on to describe cases in which angular momentum, or angular
and radial momenta, are not zero, which I will not address here.)

I had also arrived at that plot. My interest is in comparing a ball-bowl-
gravity example, when friction is present, with one involving cursor-target-
person. When there is *friction between ball and bowl*, my rendering of the
velocity-displacement relationship is shown below. (In this figure, pardon
my ASCII. The positions of the points are only approximations.)

Assume a ball released at a height (d) above the bottom of a
bowl, with the bowl resting on the surface of this planet. There
is friction between the ball and the bowl.

+d (distance above bottom of bowl)
>
0.
. |
. . 2
. . | .
. . . 4 .
. . | . .
-v ----+----+----|----+----+---- +v
1a 3a A 3b 1b

d = vertical distance of ball above bottom of bowl.
(Only positive values of d are showm; the ball
cannot move below the bottom of the bowl.)
-v = velocity toward bottom of bowl (down).
+v = velocity away from bottom of bowl (up).
A = The point at which d and v are both zero.
At A, the ball is motionless at the bottom of
the bowl. A could be called "a point attractor."

If the ball is released from a point +d above the bottom of the bowl at time
0, it will move toward the bottom of the bowl, arriving there with maximum
velocity at time 1a. At that instant (now shown as 1b), it will start
moving upward from the bottom of the bowl at the same velocity it had when
it arrived there. (As Martin said, this "jump" in phase space does not
imply infinite acceleration; the jump is only a consequence of plotting
this system in 2D.) At time 2, the ball will arrive at a new (lower)
maximum height above the bottom of the bowl, where it will have zero
velocity and begin to move down.

The process I described above will *necessarily* continue, with the ball
reaching a smaller value of +d on every cycle, until the ball comes to rest
at A. *If* the conditions remain as I described them, *then* the cycle
will *necessarily* occur as I described it.

If I am still "orthodox," after that interpretation, next year I will move
into a consideration of some of Martin's discussions of what control looks
like in phase space. I am especially interested in examples like the
following.

Now assume a cursor at a vertical distance d above a target on a
computer screen. Further assume a person and a control handle, the position
of which affects the vertical position of the cursor.

+d (distance above target)
>
. cursor at time = 0
>
>
>
-v --------------|-------------- +v
>
>
>
>
>
-d (distance below target)

What will *necessarily* happen next? (Is that a meaningful question? Under
*all* conditions? Under *any* conditions?)

In my first posts on this topic (Tom Bourbon [931227.1213], [932812.1545])
I showed the cursor moving on a simple arc from +d down to the target.
(When the cursor began at -d, I showed it following an arc up to the
target.) In his post on the subject, Martin Taylor (931230 15:00) did not
specifically address the example of a cursor and target, but he discussed
the idea that:

ยทยทยท

=============================================== "If there is control, what you will see is, for different trials starting at the same point in the phase space, a tangle of orbits that fan out from the starting point into a possibly narrow band, and that arrive by some probably spiral route in a disk near the zero point. Only if control is overdamped and therefore not as quick as it might be will you avoid the spiral. The better the control, the less of that you will see, and the higher the inward velocity will be until the orbit gets near the zero-error position. But you should see the tangled fan-out from the starting point, if your measuring apparatus is good enough, and there really is control."

Imbedded in that passage from Martin (and assumed but not stated in my
posts) is a key term. People who grasp the significance of the term will
probably have very different answers to my question, "What will
*necessarily* happen next?", than people who do not.

+++++++++++++++++++++++

Tom:

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."

Martin:

They aren't different, so far as I can see. They assert that there would
be no overshoot in the particular system you drew.

Yes. However, that was not the point I intended by my remark. I was simply
referring to what I take to be the marked preference in the behavioral
(neurophilosophical, neurocognitive, cognitive neuroscientific) literature
for "chaotic" phenomena, in which phase-space diagrams take on the "fancy"
appearances I mentioned.

Until next year,

Tom