[From Bruce Abbott (2012.12.13.1555 EST)]

I’ve thought of a way to make the role of information in a negative feedback control system clear – I hope. I’m modeling my approach on Ashby’s use of discrete variables to clarify issues that are more difficult to comprehend in the continuous case.

Imagine we have a control system that is attempting to keep a certain voltage constant at 0 volts, and that the range over which the voltage can vary is from 0 to 10 volts. It’s a high gain controller, so the error is very small and for our purposes can be ignored. The control system counteracts any disturbance by producing an equal and opposite voltage (range 0 to -10 volts), which is fed back to stabilize the controlled variable.

We, the observers of the system, cannot measure the disturbance that is acting on the system, but we do have access to the voltage that is being fed back to counteract the effect of the disturbance, and are recording that voltage. At first the recorded voltage is zero, but after some time has passed it suddenly jumps to -3 volts, holds that value for one second, then jumps to -7 volts for one second, etc. This pattern continues until we have the following recorded values: -3, -7, -1, -5, -9, -2, -7, -4, -10, the -10 holding for two seconds rather than the usual one second. After that sequence, the output stays at zero.

Question: what information was “in” the disturbance waveform? The answer is my office phone number.

The zeros indicate that the disturbance voltage is zero (thus the output is zero); a zero may be taken to be the idle state when no number is being transmitted. When the monitored voltage jumps to -3 volts, this -3 volts is counteracting a positive disturbance of 3 volts. But in my code, one volt represents the number zero, two volts the number 1, and so on up to 10 volts, which indicates the number 9. Applying this code to the sequence of recorded output values yields 2604816399.

Given that we know that single digits are being transmitted, each successive digit could have any value ranging from 0 to 9. After that digit has been transmitted to the controlled variable (by the disturbance voltage), the control system acts to nullify the disturbance by creating an opposing voltage, which we can read and translate via our code back to the original number. Uncertainty as to the transmitted value has been reduced by our reading of the output from ten possible values to one. The information transmitted, in information theoretic terms, is equal to this reduction in uncertainty, although it would be expressed as a number of bits. (I don’t have a calculator handy right now but it would be a little more than three bits. Three bits = 2 raised to the 3^{rd} power, or 8 values, in this case ranging from 0 to 7. We have 10 possible values, more than 3 bits but less than 4 bits, which would allow 16 values). Let’s call it 3 bits.

Each successive second a new number is transmitted until the 10 digit series has been completed, so that’s a transmission rate of approximately 3 bits/second. In the end I have reduced the uncertainty about the possible 10-digit number to zero. This was accomplished without having any access to the disturbance values, nor any knowledge of the transformations going on inside the control system. (Recall that we are assuming our standard proportional controller here.)

We could add an additional wrinkle if we assume that a leaky integrator in the output function; in that case the output would require a little time to settle onto each new value, but otherwise the result would be the same after an exponential rise or fall toward the new value.

A spy attempting to read the coded phone number by intercepting the value of the controlled variable would see only a nearly constant voltage and probably conclude that no message was being transmitted.

Obviously, my office phone number is not being “used” by this control system for any purpose whatsoever, certainly not for the purpose of control. So much for that confusion. Obviously, information is being transmitted from disturbance to output. Obviously, the only channel through which this information can be transmitted is the control loop. Obviously, the information transmitted does not reflect the internal organization of the control system (no behavioral illusion at work here).

Bruce