[Martin Taylor 940830 16:50]
Bill Powers (940810.0930 MDT)
I have a puzzlement about (a) what happens when two non-linear pull-only
conrol systems are opposed to form a virtual two-way control system, and
(b) why wrong answers are so seductive as to be agreed by competent and
well-informed people.
Before I went off on holiday, I re-posted some maths originally posted
in Dec 1993, which seemed to show that the two opposed one-way control
systems would form a two-way virtual control system with an output law
that was the derivative of the output law of the one-way real systems.
This was a formulation of a claim Bill P. had made, and supported that
claim. Bill praised it at the time, but now has said that it was in fact
wrong.
As I recall, you concluded that the loop gain of a
combined linear opposing pair would be zero, and that is certainly not
true. It's not even true in the conflicted case, as Kent McClelland
demonstrated at the CSG meeting last year: a pair of conflicted systems
can maintain control until one or the other output runs into a limit.In the Byte article, so far as I can see, it is the linear error, and
when you oppose two linear control systems, nothing happens to the CEV.Forget that conclusion that you drew last December. It is wrong. The
systems in the BYTE article were linear, too.
(Note that wordings "you concluded" and "conclusion that you drew last
December." It was actually a conclusion Bill drew in July 92, and
reiterated last November. My part was to supply some apparently faulty
maths that supported the same comclusion, based on the foundation of
what Bill and Greg had done.)
Something seems to be happening different from the setup we discussed
in December.No, it's just that the conclusions drawn last December, whether I agreed
to them or not, were wrong.
There's something puzzling about this, although I accept that the
conclusions we accepted in December were wrong. And from direct contact
with Kent McLelland, Bill's recent characterization of Kent's results is
correct, whereas what he said about them last November was wrong (see
below). Kent says that two conflicting linear control systems CAN maintain
control until one or the other output runs into a limit (at which
point it ceases to be linear, so that hardly counts). They maintain
control in the sense that the effects of a disturbance are resisted,
not in the sense that each achieves a perceptual signal equal to its
reference.
What's puzzling is that in July 92 and at the end of November 93, the
situation seemed to be quite otherwise. Now please be clear--I'm
not complaining that other people make mistakes, too. What I am puzzled
about is how so many people knowledgeable about the workings of control
systems can maintain such a wrong position for so long. This particular
position has something quite seductive about it, that allows algebraic
errors (to which I have always acknowledged being prone) to pass unchallenged.
I'm also still a bit puzzled about how two opposed square-law systems can
provide the equivalent of a gain-controlled linear system if two opposed
linear systems also allow linear control.
Here's the background to it all, and the sources of my puzzlement.
···
================
Bill Powers (920722.0800)
In our work on the arm model, Greg Williams found a reference that provided
force-length curves for various muscles. These curves can be fitted quite
closely with a second-power function over most of the force range (below
the saturation level of tension). Muscle tension is produced by stretching
the passive component of the spring, so muscle tension goes very nearly as
the square of the driving signal and the amount of contraction.
When you oppose two such muscles, the net force as a function of length is
represented as the difference between two offset square functions.
Let c = common-mode contraction, and d = differential contraction. Then
F1 = (c + d/2)^2 and
F2 = (c - d/2)^2
As a result, we have
F1 - F2 = (c^2 + 2cd/2 + d^2/4) - (c^2 - 2cd/2 + d^2/4) or
F1 - F2 = 2cd.
This says that the differential force produced by a differential
contraction is proportional to the common-mode contraction: that is, the
output sensitivity of this force-generator is so determined. If the rest of
the system is linear, the loop gain of this force-control system is
linearly proportional to muscle tone, and the differential force at
constant muscle tone is a LINEAR function of the differential contraction
in the two muscles (until one muscle or the other totally relaxes).
The above still looks correct to me. But is it?
The next segment of the present posting was based on the above. It is part
of an off-line exchange last Nov 27 and 28 between Bill Powers and me, in the
context of a four-way mini-CSG group that included Rick and Tom Bourbon.
Nobody at that time objected to any of the content. At the end of it, I
believed (a) that if two linear systems were pulling in opposite directions,
there was a dead zone with no control if the perceptual signal (in both)
was between the two reference signals, (b) that Kent McLelland had
simulated this situation and found it to be so, and (c) that (a) could be
generalized, in that the effect of two opposed non-linear control systems
was the same as that of a virtual two-way control system whose output
function was the derivative of the output functions of the individual
one-way systems, provided that the perceptual signal was between the
two reference levels.
I did the maths and seemed to find that (c) was, in fact so. But it now
appears to be false.
=====================
(Me, Nov 27)
I always keep in mind your demonstration that a pair of opposed square-law
systems will act like a virtual two-way linear system. Rick pointed out
the other day that a pair of opposed linear systems have zero control,
and are victims to any disturbance. This suggests a generalization to
me, and you might know whether it is true. The generalization is:
If two opposed one-way control systems have an output function f(e),
then the virtual two-way system that they create has an apparent output
function that is the derivative df(e)/de.
Do you know whether this is correct or off-the-wall? If you don't know,
I'll try to look into it, but if the answer is well known, I don't want
to bother.
==========================
(Bill, in response)
Martin:
I always keep in mind your demonstration that a pair of opposed
square-law systems will act like a virtual two-way linear
system.
With a balanced pair of identical one-way systems having linear
output functions, the resistance to disturbance will be zero in
the region of overlap, but it will be normal outside that region
(for larger disturbances), as Kent McClelland showed at the
meeting. The combination will be like a single output function
with a dead zone: errors will produce no net output until they
reach a threshold level, after which larger errors will produce
output as usual (from one of the systems).
Note that you have to have a leaky integrator or a proportional
gain element, not a true integrator, for the balanced-pair
concept to work. The steady-state gain can't be too high, or the
adjustment of the common bias will get too delicate.
It's interesting, isn't it, that the agonist-antagonist pairs of
muscles HAVE to have a nonlinear response if muscle tone is
nonzero and there is to be any fine control.
Martin:
If two opposed one-way control systems have an output function
f(e), then the virtual two-way system that they create has an
apparent output function that is the derivative df(e)/de.
This would appear to be true within the region of overlap for
one-way systems, but not outside it.
Martin:
Do you know whether this is correct or off-the-wall? If you
don't know, I'll try to look into it, but if the answer is well
known, I don't want to bother.
This isn't the sort of thing that would be well-known in advanced
control engineering, and as far as I know it hasn't been
mentioned in behavior modeling (except by us).
================
(Me, in response to the foregoing)
This would appear to be true within the region of overlap for
one-way systems, but not outside it.
"Appear to be true?" (Within the region of overlap I take for granted).
That seems to suggest that it looks that way to you, as it does to me,
but that you haven't analyzed the situation to see if it is true.
I'll see if I can make sense of it. One advantage in reinventing wheels
is that you find out how wheels work.
(Bill, responding to this)
Martin et al --
All right, it IS true within the region of overlap but not
outside it.
Bill
Upon which I performed the maths that showed Bill was right. I recently
re-posted the maths, but Bill now has said it is wrong (and I believe him
now, as I believed him then). In my defence, I prefaced that posting
(Martin Taylor 931219 17:40) with:
I should probably add my usual caveat when presenting algebraic results:
I make no guarantees as to the absence of sign inversions and similar typos.
There must have been a sign inversion early on in the derivation. Makes
a difference!
=========================
Now where are the facts? Do two opposed linear systems look like a linear
system with a dead zone if the perceptual signal is between the two reference
values? Do two opposed square-law systems? And why is the wrong answer so
seductive, easy to find, and hard to correct?
It's a puzzlement.
Martin