[Martin Taylor 931219 17:40]
It has many times been noted, usually in passing, that the perceptual signals
in a neural control network cannot take on negative values. If a signal at
any stage in the internal part of the control loop would take on a negative
value, its actual value is zero. In effect, real neural control systems pull,
rather than pull and push. The effect of a push is achieved by a separate
control system that pulls in the opposite direction.
A long time ago, Bill Powers and Greg Williams pointed out that if two opposed
control systems had square-law output functions, the combined pull-pull system
would act like a single linear control system in the region of overlap when
they both provided output. The equivalent reference level for this equivalent
linear system is the average of the two reference signals of the two one-way
ECSs, and its gain is a function of the difference between the two reference
signals. One must note that the "equivalent linear ECS" is a construct of an
outside observer. Neither of the individual pull-only ECSs is affected by the
existence of the other, except insofar as the opposition ensures that neither
can individually bring its perception to its reference value. Sustained error
is assured in such a system, so long as both real ECSs are generating output.
This sustained error may induce continuing reeorganization, but that is not a
thread I wish to pursue here.
The fact that two opposed pull-only systems can act like one simple two-way
system has been taken as indicating that it is unimportant to worry about the
one-way nature of the underlying "real" ECSs. It is thought that models can
equally well be built with two-way systems that are nice and linear, because
we know we can build equivalent linear systems out of the opposed one-way
square-law systems.
I think that pull-only systems permit a good deal more flexibility than that,
and that it is worthwhile to consider their behaviour in its own right.
One reason that it may be useful to consider the one-way systems as such is
that they do not have to be arranged on opposed pairs. Indeed, they probably
would not normally be so arranged. Consider as a mechanical analogue the
classic rubber-band demo, but with three persons holding three rubber bands
connected at a central knot, as in the figure. These three can control the
two-dimensional location of the knot, whereas it would take four one-way
systems to control the two-dimensional location if they were arranged as
opposed pairs.
\ |
\ |
O----- rather than -----O-----
/ |
/ |
In three dimensions, four one-way systems suffice, or three opposed pairs (six
one-way systems). In general, it takes 2N one-way systems to achieve control
in N dimensions if the one-way systems are arranged in opposed pairs, and only
N+1 if they are used individually at the corners of an N-dimensional
triangular pyramid.
On the general principle that nature tends to prefer efficiency over waste, we
may guess that our own one-way ECSs are arranged in opposed pairs only where
their outputs directly generate mechanical movements that are restricted to
angles in a single plane. This happens only at the physical output surface of
the body, not at any higher level of perceptual control, where the
dimensionality of the control possibilities is limited only by the total
number of available degrees of freedom.
It seems worthwhile to extend Bill and Greg's result for opposed square-law
systems. These extended results may be well known, but as far as I am aware,
they have not been described on CSG-L.
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. But I
believe the main results to be correct.
Consider two one-way ECSs whose sensory inputs derive from the same physical
variable, such as the position, s, of the knot in a rubber band. The
perceptual signal in ECS 1 is p1(s), and in ECS 2 it is p2(s). s is defined
in the reference frame of an external observer, which we will for convenience
align with that of ECS 1. Accordingly, positive changes in s will result in
positive changes in the perceptual signal of ECS 1 and negative changes in the
perceptual signal of ECS 2. Ignoring resting-level constants that keep the
perceptual signals positive, and that are matched in the reference signals, we
can take the two perceptual input functions as +1 and -1 respectively, so that
p1 = s and p2 = -s.
The reference signals are different in the two ECSs--call them r1 and r2,
ignoring again the resting-level constants that are the same as for the
corresponding perceptual signals. In the reference frame of the ECS, these
values are positive, but in the observer's reference frame (which we assume to
be aligned with that of ECS 1), r2 is negative apart from the resting-level
constant. The resulting error signals are e1 and e2. No signal can have a
negative value, so in ECS 1, e1 = r1-s or zero, whichever is greater, and in
ECS 2, e2 = s-r2 or zero. The outputs O1 and O2 are functions G1 and G2 of
their respective error signals (presumably G1 and G2 are extended time
functions, but this does not affect the following derivation, so we will
assume that they are functions only of the current value of their respective
error signals).
Finally, the value of the physical variable s (not observable by either ECS
except as transformed into p1 or p2) is given by D+G1(e1)-G2(e2), where D is
the disturbance. There is a minus sign associated with G2 because in the
observer's frame of reference, the output of ECS 2 opposes that of ECS 1.
The loop calculation can be done as readily with this configuration as with a
normal simple ECS, as follows:
s = D + G1(e) - G2(e) = D + G1(r1-s) - G2(s-r2)
If the form of G is known, this can be carried further. Here are three well-
known cases:
(a) G2=0, G1(x) = G*x (an ordinary linear control system)
(b) Gn(x) = G*x (two opposed linear systems)
(c) Gn(x) = G * x^2, x>0, Gn(x) = 0, x<0 (two opposed square-law systems)
Case (a), simple ECS:
s = D + G*(r1-s)
s(1+G) = D+G*r1
s = D(1/1+G) + r1*(G/1+G)
A useful way of looking at this is to look at how much change there is in s
for an infinitesimal change in D--to compute ds/dD--the index of control as
seen by an outside observer who may be applying the Test by introducing D. In
the case of the simple ECS it is X = 1/(1+G). The equivalent output gain of
any ECS is G = (1-X)/X, an expression used in what follows.
Case (b), two conflicted linear ECSs of equal gain:
s = D + G*(r1-s) - G*(s-r2)
= D + G*(r1+r2)
which gives an index of control ds/dD = 1, or an equivalent gain of zero. The
value of s fully reflects any change in the disturbance. Linear control
systems in conflict cannot resist external disturbances, as Rick Marken
recently pointed out.
Case (c), pull-pull square-law ECS pair:
s = D + G*(r1-s)^2 - G*(s-r2)^2
There are four regimes corresponding to the signs of r1-s and of s-r2, which
can vary independently. Let us consider only the case in which both
differences are positive, and both one-way ECSs are producing positive output.
s = D + G * (r1^2 - 2sr1 + p^2 -p^2 +2sr2 - r2^2)
= D + G * (r1^2 - r2^2 - 2s(r1-r2))
= D + 2G*(r1-r2) * ((r1+r2)/2 - s)
Comparing this with the case of the simple ECS, we find that the form is the
same, with the gain of the equivalent simple ECS being 2G*(r1-r2), and the
reference level of the equivalent ECS being (r1+r2)/2. This is Bill and
Greg's result. Let's extend it a little further.
Case (d), Arbitrary output functions:
s = D + G1(r1-s) + G2(s-r2)
There is not much that can be done with this general form directly, but we can
compute the equivalent index of control as seen by the observer, ds/dD.
ds/dD = 1 + dG1/dD + dG2/dD
= 1 + dG1/ds*ds/dD + dG2/ds*ds/dD
ds/dD(1 - dG1/ds - dG2/ds) = 1
ds/dD = 1/(1 - dG1/ds - dG2/ds)
Noting that G1 is a function of r1-s, whereas G2 is a function of s-r2, we can
further expand the derivatives, giving
ds/dD = 1/(1 + dG1(x1)/dx1 - dG2(x2)/dx2)
where the x in each function is the value of its local error signal. Taking
the combined pull-pull system as an equivalent simple ECS, the simple ECS has
a gain Geq = (dG1(e1)/d(e1) - dG2(e2)/de2). For the square-law system above,
this gives Geq = 2G(r1-s) + 2G(s-r2) = 2G*(r1 - r2), as before.
The reason this formulation may be useful is that the functions G may in
reality be applied in any direction, as in the case of the three rubber bands.
One can compute the equivalent control gain for any direction at all in the
space, not just in the direction of one of the individual pulling ECSs, by
taking the function G to represent the component of the output in the
direction of interest. For example, given the three ECSs pulling in the
directions 1, 2, and 3 in the figure, an observer can determine the index of
control in the direction A-B, in a direction in which none of the ECSs are
pulling. An experimenter wishing to apply the Test to see whether there is a
controlled variable along A-B will find that there is, even though no ECS
actually controls in this direction.
1
\ .B
\ . .
. O----- 3
. . /
A /
2
The effect of a disturbance along A-B, Dab, is opposed by changes in the
projections of the outputs of ECSs 1, 2, and 3 onto the direction A-B. This
projection is the source of the minus sign associated with G2 in the
expression for the opposed square-law pair. Likewise, the effects of changes
in the physical value of the variable corresponding to the controlled
perception in each ECS is the projection of the change in the A-B direction
projected onto the direction defined by the PIF of the individual ECS. As
before, all the effects of the combined set of ECSs must be taken in the
reference frame of an external observer, because each individual ECS has only
its own perceptual signal to work with.
Skipping the preliminary steps, because they are strictly analogous with the
foregoing, we have, in the direction A-B:
Xab = ds/dD = 1 + dG1/ds*ds/dD + dG2/ds*ds/dD + dG3/ds*ds/dD
or
Xab = 1/(1 - dG1/ds - dG2/ds - dG3/ds)
Alternatively, the output gain of the equivalent simple two-way ECS operating
in the direction A-B is
Geq = -(dG1/ds + dG2/ds + dG3/ds)
where the derivatives are taken in the three directions of pull, at the
momentary projected values of s in the directions of pull.
One can, of course, make a similar analysis of the equivalent gain in the
direction perpendicular to A-B in the plane of the three pulls, arriving at
the same kind of formula, but with the projections taken orthogonal to A-B
instead of onto A-B. Label the direction A-B as x1, and the orthogonal
direction as x2, and label dGn/ds in the direction x1 as Gn'1, and we have
Geq1 = - sum_over_n(Gn'1))
Geq2 = - sum_over_n(Gn'2))
Provided that Gn(e) is locally smooth in the region of e, then
Gn'1 = cos(theta.n.1)*Gn'
where Gn' is dGn/ds in the direction of Gn and theta.n.1 is the angle between
the direction of pull Gn and direction x1. And this can be extended to any
number of dimensions. The equivalent gain is given by the sum of the
direction cosines of the derivatives of the individual gains, taken at the
projected values of the errors in the reference frame of the observer. In
general, the equivalent gain will depend on the value of the physical
variable.
Notice that if r<p in a pull-only ECS, the error signal is zero and thus the
output is unaffected by changes in the disturbance. The equivalent set of
two-way ECSs made from a group of pull-only ECSs can be made to have a central
dead zone by setting the reference levels appropriately. But in that case,
non-linearities in the individual ECSs do appear in the equivalent output
function directly. A pair of opposed square-law pull-only ECSs can be altered
on-line from being an equivalent linear two-way ECS to being a guard-control
system that does not care about small deviations from a central reference
level but that opposes with ever increasing strength larger disturbances. It
may not keep you driving near the centre of your lane, but it will keep you
from driving out of the lane over the cliff edge.
Non-orthogonal systems of one-way ECSs are not only physiologically more
reasonable than are orthogonal sets of push-pull ECSs, but also they offer
many interesting possibilities for flexible and robust designs. For example,
if there are more than N+1 pull-only ECSs controlling in a space of N
dimensions, the withdrawal of one of them will not affect the ability of the
set to control. This allows not only for robustness against loss of parts of
the system, but also for flexible reorganization. The reference level for one
ECS could be set so that its output is zero while its linkages are altered,
thus avoiding the transients that accompany link alterations while an ECS is
actively controlling. While this is happening, the remaining pull-only ECSs
could retain control of their perceptions in all relevant dimensions.
Redundant systems of pull-only ECSs deserve more attention than they have
hitherto received.
Martin
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