[From Bill Powers (931222.0730 MST)]
Bill Silvert (931222.0824 AST) --
Nice to hear from you!
This dialogue reflects the dynamical systems theory mindset,
but feedback systems can be regulated in an unstable way. For
example, a thermostat switches a furnace on and off. In either
state the system is far from stability (the stable temperature
is perhaps 40 C when the furnace is on and -10 C when it is
off), but the temperature oscillates around 20 C. So one can
have control without stability.
Unfortunately, we are using "stability" in two different senses
on this net. One is "stability against disturbance," and the
other is "dynamic stability." The first usage means only that if
a disturbance tends to drive a controlled variable away from its
reference level, the system pushes back and prevents the change.
The second usage means that the system itself, with or without
disturbances, maintains a steady state of the controlled variable
instead of spontaneously creating oscillations of large
amplitude.
Now you are introducing a third meaning of stability, which is
"absence of even small oscillations of the controlled variable
about its reference level." This kind of stability is what all
designers hope to achieve, but the cost of achieving it is
sometimes too high. A home thermostat is an example of a design
in which a compromise was reached between cheap-and-simple and
the best possible control.
A thermostat is known affectionately in control engineering as a
"bang-bang" system: bang, it's on, and bang, it's off. The heat
output from the furnace is either turned on full blast or is
completely absent. What enables this device to control
temperature in a house with reasonable accuracy (and stability)
is the large thermal inertia of the air, walls, furniture, and so
forth in the house. Between the furnace and the sensor in the
thermostat unit on the wall, there is, effectively, a low-pass
filter that prevents the large fluctuations in furnace output
from reaching the sensor. The temperature of the sensor, which is
what is actually controlled, rises and falls just enough to make
the movable contacts move between just touching and a separation
of a few hundredths of an inch. The air temperature fluctuates
only a couple of degrees at the sensor, and only slightly more in
the middle of the room. This is instability in your third sense,
but it is not dynamic instability. Also, the _average_ output of
the furnace rises and falls (the proportion of on-time to off-
time changes) as heat losses change, preventing any large
departures of the average temperature from the set-point, so this
system is also stable in the sense of opposing disturbances.
If a thermostat is misused, it can become dynamically unstable.
If the furnace is very close to the room containing the
thermostat, and the room is small and poorly insulated, the low-
pass filtering becomes insufficient. Now when the furnace turns
on, it heats the air so rapidly that the contacts separate by a
large amount, prolonging the off-time, and when the furnace is
off the air cools so rapidly that the contacts press hard
together, prolonging the on-time. This leads to alternating
surges of too hot and too cold that get progressively larger,
until some nonlinearity limits the excursions. The room
temperature may rise and fall by tens of degrees. This is how a
dynamically unstable system behaves: it drives itself into
oscillations of considerable size even without external
disturbances.
There is a third (fourth?) kind of stability, "conditional
stability." This can arise in a control system with a nonlinear
output device. For small disturbances, the output changes by a
small amount and accurately counters the disturbance of the
controlled input. But for large disturbances, the output device
is driven into a region where it has a much higher sensitivity to
errors, and the loop gain increases so much that the stabilizing
compensation becomes inadequate. Now when the disturbance is
removed, the whole loop oscillates endlessly. It has become
dynamically unstable, and is useless as a control system.
I seem to recall seeing cases of conditional stability in some
ecological simulations, like predator-prey relationships
involving certain settings of parameters like reproduction rate.
Below a certain population, the predator and prey populations
come to a steady state, but for larger populations the boom-and-
bust cycle takes over. I suppose the opposite is also
hypothetically possible for other forms of nonlinearity.
All this, by the way, comes out of control-system analysis that
existed decades before what is now called "dynamic systems
analysis" was invented. I have been somewhat amused by the
innocent discussions of "limit cycles" in modern dynamic systems
analysis, at least in connection with control theory. Any
engineer with practical experience in control system design and
maintenance, upon seeing any significant limit cycles in a
controlled variable, would quickly have the cover off the box to
see what had gone wrong.
In your field, ecosystems, control theory would come into the
picture only for analyzing the behavior of individual organisms.
A ecosystem consists of interactions among independent individual
organisms and the nojnliving environment, and those interactions
can take on any form whatsoever. If there is stability, it is not
the stability of a control system, but the stability of an
equilibrium system, a balance of independent opposing forces.
Predator-prey relationships are of this nature. Such systems can
show oscillations, stability, or any other kind of behavior.
Systems analysis applies, but not the specific kind of system
analysis you would use with a control system. Control theory
would enter into that analysis only in the sense of helping to
describe behavioral characteristics of individual components of
the system. To predict what the whole system will do, you need to
apply system analysis in the general sense: solving or simulating
sets of differential equations to see the implications of the
interactions you can define, given the properties of the
component individuals.
Modern dynamic systems analysis might be useful in certain
situations. If you observe systemic behaviors that seem random or
only very approximately repeatable, it's possible that a chaotic
system exists -- not quite random, yet not quite orderly, either.
In that case, some of the things that have been discovered about
chaotic systems, for example the conditions that can switch a
system from regular to chaotic behavior and back, might prove
illuminating. But save for that application, ordinary old-
fashioned systems analysis, using differential equations and
common sense, will tell you just as much about what is going on.
You don't HAVE to use phase-space plots; you're allowed to plot
variables against time.
When there is irregular behavior, however, one shouldn't be too
quick to haul out the apparatus of chaotic system analysis,
fractals, and all that. You may be using the wrong theory. It
could be that when you find the right theory, the apparent
irregularities will turn into beautiful regularities. This
happens when we apply control theory to certain behaviors -- we
find not variability, but simple and clear relationships that
explain what seemed irregular before. Applying chaos theory to
the irregularities of behavior would have been a waste of time.
···
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When you say "one can have control without stability" you're
technically wrong. The more the instability of the whole control
system, the less control is possible. The object of control is
not to stabilize an _output_, but a controlled variable. In a
furnace the heat output varies wildly from maximum to zero, but
the controlled variable, the room temperature near the sensor,
rises and falls smoothly over a very small, and acceptable,
range.
In talking about ecosystems, you're not talking about control
systems, so control isn't an issue (except with respect to
individual organisms).
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Best,
Bill P.