[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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