Ashby's law of Requisite Variety

From[Bill Williams 9 May 2004 9:50 PM CST]

[From Bill Powers (2004.05.09.1729 MST)]

in discussion with

Bruce Abbott (2004.05.09.1720 EST)--

Powers says,

The law of requisite variety, like many other abstract >generalizations

about control, is one of those ideas >that follows from, but does not
explain control, or help >us understand it better.

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?

I know that there are many people who have a different >idea about this. I

intend to be just as stubborn as they >are.

Rather than stubbornness which never seems to be in short supply, it would
seem to me that more interesting question is who is better informed. I am
prepared to think that you are more likely to be informed about whether
Ashby's arguments are sound, than would be Ashby or his supporters.
However, I wonder about what was going on. If Ashby had it wrong on some
important points, where were the critics?

But if you can drop the subject, I can, too. I have had >enough

disagreements that I think I can make it through >the rest of the year
without needing any more.

I can understand why you might not have thought it was a prudent to begin
_B:CP_ with a critique of Ashby and some unfortunate aspects of cybernetics.
However, to someone who isn't familiar with vacuum tubes it isn't easy to
see where Ashby went wrong. The way I remember the arguments in the text,
it appeared as if his argument was derived from concrete experiments that
had actually been conducted. I think, just for the record, it would be
useful to have a fuller explanation of where the problem was in Ashby's
work. I don't need convincing that there was a problem, but I would like
to know in more detail where the problem developed. But, I don't "want to
know" to the extent that I would familiarize myself with vacuum tubes so
that I could figure this out for myself.

Bill Williams

[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”):

84a4c1.jpg

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.

From[Bill Williams 9 May 2004 11:30 AM CST]

[From Bill Powers (2004.05.10.0723 MST)]

Bill Williams 9 May 2004 9:50 PM CST–

Thanks for the explanation, saves me the trouble of going back and figuring out where Ashy went wrong. The crucial parts of your explanation seem to me to be that,

  1. Ashby, as far as I know, never built circuits based on his

later ideas about control.

And, there is a small but fatal flaw in Ashby’s reasoning.

  1. 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.

But, Ashby may have thought it could work, because …

  1. 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

So, there really was a feedback loop in Ashby’s system after all.

But, it was Ashby as you say, “tweaking the controller.”

Bill Williams

84a4c1.jpg

···

----- Original Message -----

From:
Bill Powers

To: CSGNET@listserv.uiuc.edu

Sent: Monday, May 10, 2004 10:42 AM

Subject: Re: Ashby’s law of Requisite Variety

[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”):

84a4c1.jpg

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.