[From Bill Powers (2004.05.10.0723 MST)]

Bill Williams 9 May 2004 9:50 PM CST–

For quite some time, I was of the

opinion that I didn’t understand Ashby’s

work because I didn’t understand vacuum tubes.

And, when Op-amps were so easy to work with I decided it wasn’t worth

the

trouble to learn to work with vacuum tubes.

So, how was it possible that Ashby could write as if what he was

saying

about circuits if the actual circuits he seemed to be describing

couldn’t

really work?

Ashby, as far as I know, never built circuits based on his later ideas

about control. His “Uniselector,” which was largely made of

relays, employed four negative feedack control systems with internal

connections that could be switched around by stepper relays. This

demonstrated his principle of “ultrastability,” which was my

starting point in developing the idea of a reorganizing system.

The alternative to negative feedback that Ashby proposed looks like this

(drawn from page 210 in the 1963 (Wiley) paperback reprint of “An

introduction to cybernetics”):

E is the “essential variable,” the state of which is to be

regulated. D is a disturbing variable which can act on E through the

transmission channel T. And R is a regulator designed to prevent

disturbances of E.

R receives information about the state of D, and emits an output that

enters the transmission channel where it adds to the effects of D.

Clearly if the effect of R on T is equal and opposite to the effect of D

on T, the net effect on E will be zero – EXACTLY zero. Elsewhere (I

haven’t found the place but it’s there, I think in *Design for a Brain*)

Ashby points out that in a negative feedback controller, the compensation

for disturbances can never be exact because there must be some error to

drive the action, whereas in this design, it is possible for exact

compensation to occur.

Suppose the effect of D on T is 2 units for every unit of D, and the

effect of D on R is 1 unit for every unit of D. Let the effect of T on E

be 2 units for every unit of T. From this we can deduce that 3 units of

disturbance D will have 12 units of effect on E. In order to

prevent that amount of disturbance from having any effect on E, how much

output should R produce for every unit of input from D? The answer is -2.

Work it out and you’ll see that the net effect on E will then be exactly

0 for any value of D. When you give your examples only in terms of

symbols for variables, or in small whole numbers, you can achieve great

exactness.

But now let us build a real controller to illustrate this principle. The

first thing we have to do is measure the effect of D on T and T on E.

Using our expensive 1% accurate voltmeter, we find that 1.00 unit of D

(as measured) has 2.07 units of effect on T, and each unit of T has 1.98

units of effect on E. Similarly, the sensors in the regulator are such

that each measured unit of D has 0.95 units of effect on R, so we have

to adjust the output of R so that for each measured unit of D, the

measured output of R will be minus 2.18 units (all this at 20 degrees C

and 6900 feet of altitude, with the line voltage at 116.5 volts, air

moving at 1 meter per second outside the house, and the voltmeter

calibrated by the National Bureau of Standards three years ago), Setting

up this circuit according to these calculations, we can conclude that we

have successfully protected E from being disturbed by any amount of D.

Can’t we?

Well, no. It would be a very good idea to actually measure the state of E

to see whether we need to tweak this controller a little bit. But look at

the diagram. There is no arrow coming out of E to give anything else

information about its state. This controller is blind to its own effects.

If something changes a little in T or the connection from D to T, the

controller will not know about this, and will still put out -2.18 units

of output for each unit of effect on itself from D, so the state of E

will no longer be protected completely against disturbances. Also, if the

effect of D on R changes, the regulator will not know about this, either

– all it knows is the effect it experiences at the arrow end of the line

from D to R. If that effect should double, God forbid, R will put out

twice as much effect as before, thinking that the disturbance has

doubled, and this will cause E to change just as much as if R were not

there, although in the opposite direction. All this simply because R has

no information about the state of E.

The point I am making in such a heavy-handed way is that Ashby’s design

for a controller is hypersensitive to any change in the environment about

which it is not told, as well as changes in its own properties and the

inner properties of other entities like T. The simplest way to put this

is the way we normally do: this sort of system can work only in an

environment free of unpredicted disturbances, and when built of

components that never deteriorate or change their characteristics when

environmental factors change. If a perfectly stable environment and

changeless materials of construction could be found, we would have to

agree that this sort of system could work slightly better than a negative

feedback control system could, in principle, though we would never know

the difference if there were a contest to pick a winner.

Why would we never know the difference? Because we would design our

regulator to ignore D, but to sense the state of E using the best

measuring instrument available. Using an integrating output function, our

negative feedback control system could keep the error at the lower limit

of detection, which means that we could not tell the difference between

the amount of control actually achieved and perfect control. So even if

the Ashby controller worked a smidgen better, nobody could measure any

improvement. Remember, we have the best sensing instrument

available.

Given the susceptibility of the Ashby controller to all sorts of

disturbances and changes in conditions, it is highly unlikely that this

controller, even if adjusted to perfection, could continue to control

perfectly for more than a minute or two. Then its owner, lurking in the

background with an eagle eye on the state of E, would step in and tweak

the controller until he saw that E was back in the correct state (using,

of course, the best instrument available). And the moral of that point is

that it is totally impossible to build and operate an Ashby regulator

without using negative feedback control.

Ashby’s design, and all those that resemble it, is actually the method of

regulation used by most engineers in the 18th and 19th Centuries. A

“compensated pendulum” in a clock, for example, is protected

against temperature changes by having a bob made of two materials with

different coeffients of thermal expansion. One lowered the center of mass

of the bob as the temperature increased, the other raised it, and with

careful adjustment the two effects could be made to cancel, leaving

the effective length of the pendulum the same over a reasonable

temperature range. The disturbance had two effects, one directly on the

period of the pendulum, and the other one indirectly through the

compensating part of the pendulum (R) – a column of mercury filled to

exactly the right height, in one design. The basic principle is a

“balance of forces” idea, which becomes more susceptible to

disturbances the larger the opposing forces are.

Best,

Bill P.