Information and control

[From Bill Powers (930312.1400)]

Allan Randall & Rick Marken (930312) --

The confusion here comes from not specifying what you mean by
"information about" the disturbance. Obviously, if the controlled
variable were not disturbed, the control system would do nothing.
So in the variations of the controlled variable, there is
"information" to the effect that there is a disturbance of some
sort acting.

However, the control system reacts directly to changes in the
controlled variable, regardless of what is causing them (even if
the system itself is causing them). Also, it acts directly, at
the same time, ON the controlled variable. The result is that the
variations in the controlled quantity do not represent the state
of the disturbance alone, but only the combination of the
disturbance plus the system's own output actions. As these
effects are almost exactly opposed when control is good, the
variations in the controlled variable actually represent only the
DIFFERENCE between the system's output and the disturbing
variable.

From this we can perhaps clarify the argument.

The actual state of the original disturbing variable is NOT
represented in the state of the controlled quantity. If at a
given moment the disturbing variable has an amplitude of 100
units, the controlled quantity might actually be deviating from
its undisturbed state by only 1 unit, or even by -1 unit. The
explanation is, of course, that the system's own output is at the
same time producing -99 units, or -101 units, of effect on the
controlled quantity.

The output of the system can carry small random variations due to
noise sources inside the system. Hence, when a nominal -99 units
of output is required to cancel a disturbance of 100 units (in a
control system with a loop gain of 100), the output can actually
vary spontaneously by an amount greater than the amount of
uncancelled disturbance. As a result, the controlled variable
will reflect these endogenous noise variations, and those
variations are normally comparable to the difference between the
system's mean output and the disturbing variable's amplitude.

The practical result is that residual variations in the
controlled variable will show only a very low correlation with
variations in the disturbing variable, a correlation approaching
zero. They will also show a low correlation with the system's own
output. These low correlations are easily demonstrated in
tracking experiments. With disturbances of low bandwidth, there
is no apparent relationship between the observed variations in
the controlled variable and variations in the disturbance.
Statistical analysis shows correlations that range from +0.1 to -
0.1 (or so). Lagged correlations are not much larger.

If control were only a matter of information transmission from
the disturbing variable into the control system, there would have
to be enough information surviving the trip from the disturbing
variable, through the controlled variable, through the sensors,
to the output variable to account for the close match of the
output effects to the disturbing effects. Yet what we observe is
that the output can match the disturbance within one or two
percent, while the controlled variable through which all this
information has to pass shows random variations as large as the
remaining variations due to the disturbing variable -- a noise
content of at least 50%, within exactly the same bandwidth.

ยทยทยท

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I think that what the information-theoretic approach may be
missing is the fact that the only information path that really
matters is the one around the closed loop: from controlled
variable to sensor to output to controlled variable. The control
system is organized so as not to depend on knowledge of the
causes of disturbances. In fact, it rejects information from
outside the control loop by systematically opposing the effects
of any external influence. This rejection is not based on knowing
the causes of those effects, but only on knowing the states of
the variables inside the control loop. No matter where in the
control loop an outside agency tries to disturb a variable, the
remainder of the loop will respond to prevent such disturbances
from having any important effect at the point of injection.
Normally, of course, only the controlled variable in the external
part of the loop is accessible to outside influences.

So I believe that the mistake Ashby made, and that information
theorists in general have made, is to suppose that the action of
a control loop is somehow driven by external influences. If that
were true, you would have to explain the action of the system in
terms of information passed to it from outside it. But in fact,
control loops are driven by information completely inside the
loop; they act to reject the effects of information from outside.
This is essentially the same mistake made by stimulus-response
theorists, when they saw disturbances being rejected by the
actions of control systems, and concluded that the disturbances
were stimuli and the actions were responses caused by the
stimuli.

-----------------------------------------------------------
Best,

Bill P.

[Martin Taylor 930313 17:40]
(Bill Powers 930312.1400)

I'm not sure whether I made the following point in a private posting to
Bill or in public to CSG-L. I think the former, but if it was the latter,
I apologise for the duplication.

I think that what the information-theoretic approach may be
missing is the fact that the only information path that really
matters is the one around the closed loop: from controlled
variable to sensor to output to controlled variable. The control
system is organized so as not to depend on knowledge of the
causes of disturbances. In fact, it rejects information from
outside the control loop by systematically opposing the effects
of any external influence.

The essential point about any stable system is that the uncertainty
observed at any pointin it will be stable over time. The closed
loop ensures this by opposing uncertainties that might be introduced
by the disturbance.

Overall, what this means is that numerically there will be no information
supplied by the disturbance to the preceptual signal. This may seem odd,
or even magical, but it has to be so. If it were not, the whole system
would have an ever increasing uncertainty wherever it might be observed.
Entropy increases in a closed system. Closed systems are not stable.
Stability comes from the flow of energy through a system. If the system
is a control system, it is the use of this energy that allows the actions
to oppose the disturbance. If the actions suffice, then the system is
stable both mechanically and informationally. The uncertainties may
on average reflect the uncertainties in the disturbance, in that control
is imperfect, but that is a question of the dynamics of the system
as a whole. Even Rick agreed that the dynamics of the system allow
the dynamics of the disturbance to be detected, and the same is implicit
in Bill's comment.

Sorry, the spurious character rate is getting to be higher than I can
cope with. I'll stop here and try to edit out the ones I can see. But
I think I've made most of my point.

Martin