Technical sense of thermostat stability

CSGnet Archive CSG1993
1

Continuing this issue publicly, which I think still relates to matters
of general interest...

Again, if it gets tedious, we can take it off-list. I will use LaTeX math
notation. The book I use is "Mathematics for Dynamic Modeling", Edward
Beltrami, Academic Press, 1987.

First, "stability" or "instability" is a property not of systems, or
orbits, but of equilibrium points. Thus the ball in the bottom of the
bowl at rest is at a stable equilibrium, the pencil balanced on a point
at an unstable equilibrium. In the thermostat example,

I agree that it is a property of points, or more precisely of regions.

>> 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.

where T is temperature, the point T = 20 C, dT/dt = 0 C/sec is
actually an UNSTABLE equilibrium. In this case, however, there is a
limit cycle, such that points near the unstable eq. spiral outward to
the cycle you described, and points within a neighborhood OUTSIDE the
cycle spiral INWARD to it.

Here Cliff hits exactly my point. At 20C the derivative is NOT zero!
When the furnace is on it is positive, when it is off it is negative.

What I think is important to recognize here is the difference between
continuous feedback (such as exerted by a spring, or by the
gravitational force on a ball in a bowl) and discrete feedback generated
by a thermostat. Discrete feedback is delayed feedback (it takes a
finite amount of time for the temperature to reach the switching point
of the thermostat and for the furnace to fire up or go off), which can
generate `unstable' behaviour if the actual derivatives dT/dt become too
great in magnitude.

Bill Silvert [931230 08:30]

2

This may fit the `description of stability' but not the definition.

OK, the gauntlet has been thrown down, and I must defend my honor. ;->
I will do so with a tactical retreat and a thrust in a different
direction, still asserting that the ilmit cycle can be technically
considered as stable.

Honestly, the following is in the interest of accuracy and full disclosure.
Again, if it gets tedious, we can take it off-list. I will use LaTeX math
notation. The book I use is "Mathematics for Dynamic Modeling", Edward
Beltrami, Academic Press, 1987.

First, "stability" or "instability" is a property not of systems, or
orbits, but of equilibrium points. Thus the ball in the bottom of the
bowl at rest is at a stable equilibrium, the pencil balanced on a point
at an unstable equilibrium. In the thermostat example,

>> 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.

where T is temperature, the point T = 20 C, dT/dt = 0 C/sec is
actually an UNSTABLE equilibrium. In this case, however, there is a
limit cycle, such that points near the unstable eq. spiral outward to
the cycle you described, and points within a neighborhood OUTSIDE the
cycle spiral INWARD to it.

Now dealing with equilibrium points, stability is when "every open
neighborhood \Omega of \bar{x} [the equilibrium] in U [the phase
space] the orbit remains in \Omega for all t \ge 0 whenever it starts
close enough to \bar{x}." [p. 10]

So by only a SLIGHT abuse of language, rather than a departure on a
metaphorical flight of fancy (hey, at least I'm TRYING to use the
mathematics!), the limit cycle, when considered as a point, is in fact a
stable equilibrium with respect to this outer environment which spirals
inward to it.

This suggestion is a simple systems-theoretic move, where the limit cycle
and its interior in phase space are taken to be an invariant point. The
cycle, its interior, and its exterior partition the space, defining
equivalence classes, and allowing us to redefine a new system at that
state-aggregated level (I hope that makes sense).

This is acceptible because (in a two-dimensional system T vs. dT/dt)
no orbits can cross the limit cycle. And as the gain increases, the
cycle tightens, the interior shrinks. As Bill reminds us, in good
control systems, this interior and the variation of the cycle becomes
harder to observe, and effectively becomes irrelevant.

So under these conditions, then at this higher level, the cycle is
TECHNICALLY STABLE.

O----------------------------------------------------------------------------->

Cliff Joslyn, Cybernetician at Large, 327 Spring St #2 Portland ME 04102 USA
Systems Science, SUNY Binghamton NASA Goddard Space Flight Center
cjoslyn@bingsuns.cc.binghamton.edu joslyn@kong.gsfc.nasa.gov

V All the world is biscuit shaped. . .