[From Bill Powers (20089.09.30.0946 MDT)]
Martin Taylor 2009.09.30.10.35] --
MT: Bill [From Bill Powers (2009.09.29.1020 MDT)] managed to "refute" my argument by altering the fundamental equation of the control system, in that he introduced the noise that he usually strongly asserts to be irrelevant to control.
BP: I do? I believe that presence of system noise was my very first explanation, many years ago, of why there is a low correlation between control actions and controlled quantities, which gets lower as control improves, and also between disturbances and controlled quantities. If you can remind me of when I said noise is irrelevant to control, I'll be glad to admit that I no longer believe it. It is irrelevant, usually, to how well the perceptual signal tracks the reference signal because we're comparing the noise to the dynamic range of the reference signal. It's certainly not irrelevant to the error signal or the effect of the error on the output.
MT: We can certainly include noise in the analysis, but that is a red herring, as well. When the perceptual signal is so noisy that the perceptual signal is uninformative about the right sign of e (compared to the real-world status of "r - real(o+d)"), control isn't likely to be very good at all.
BP: But that would occur only when control is very good. I believe I used an example of a control system with a loop gain of 100, which would result in very good control (error signal about 1% of value of reference signal). If the system noise is also 1% of the value of the reference signal, this would make the noise in the error signal equal to the error.
MT: Usually, as Bill so often asserts, the noise in a controlled perception is not so great as to interfere with effective control, so we tend to ignore it.
BP: Of course: if the range of variation of the reference signal is 100 times as large as the system noise, we can pretty much ignore the noise in calculating the size of the output relative to the disturbance, or of the perceptual signal relative to the reference signal. But with a high gain, such a control system would have an error signal of about the magnitude of the perceptual noise; indeed, the unsystematic variations in the perceptual signal, small as they are, set a lower limit on how small the error signal can be.
MT: Sometimes, as in the Schouten study discussed in an earlier thread, it is possible to measure directly an upper bound on the noise characteristics of the perceptual signal, but usually we don't worry about. I'm a little surprised that it was brought up in this specific case when it is usually asserted to be irrelevant.
BP: I hope you can see now that you are oversimplifying what I said.
MT: One thing I had hoped that Rick or Bill would bring up, but they didn't, was that the signal "o" is a function of the history of the perceptual signal, if the reference signal is constant over time. Even if the output function G is a simple proportionality with no inherent time-binding, the signal paths in the loop have transport lags that ensure the output value added to the current disturbance is a function of a past value of the perceptual signal, not of its present value.
BP: The lags in tracking behavior are measured at about 8 60ths of a second -- about 130 milliseconds. The bandwidth of these control systems is roughly 2.5 Hz, a number that's been known for around 60 years. This means that the lags are about 35% of the period of a sine-wave disturbance at the upper frequency limit of control. This is also indicated by the phase relationships in the Bode plots of such systems.
The result is that over most of the bandwidth range, the feedback effects are opposed to effects of any disturbances that the control system can resist, and are not independent of the disturbances that caused them. If the feedback effects always came too late to affect the input variations, we would have not a control system but an oscillator.
This is obvious when you think about what we observe of the effects of disturbances on controlled quantities. In fact the changes in the controlled quantity due to disturbances are very much less than what they would be if there were no feedback. This says that the feedback effects occur in time to cancel most of the effect of the disturbance. If the effects of lags were as you describe them, so that the feedback effects came only from "past values" of the disturbance, the effects on the controlled quantity would not be reduced. Only if you use disturbances with bandwidth greater than the bandwidth of good control do you see a significant effect from the lags -- and then the effect is to allow greater effects of the disturbance and to reduce the ability to control. The variations in the perceptual signal at the higher frequencies involved would then be much larger and we would start to see information in the perceptual signal about the higher frequencies in the disturbance -- but not the lower frequencies.
MT: I wonder if Bill's original memory of the analysis by Allan Randall and me as being "open loop" was engendered by memory of our statement that one didn't actually need to use the signal "o" to recover the disturbance waveform from the perceptual signal.
BP: Very likely, since that would be my present view, too. If you would do me the favor of restoring my comments about system noise to their intended meaning, as explained above, you can see that there is no way to work backward from the perception to the disturbance. In effect, you're trying to solve an equation that has two variable quantities of about equal size, the perceptual signal and the reference signal, being subtracted one from the other, leaving a very small difference, amplified to drive the output function. If the perceptual noise is comparable in magnitude to that difference, the output will have a large random component. Since you have to subtract the output from the perceptual signal (in your simplified set of equations) to calculate the size of the disturbance, your estimate of the size of the disturbance will be very uncertain -- and get more uncertain as the loop gain increases.
And now you drop the bombshell:
MT: It might also be worth noting, though I apologise if it is too obvious to make explicit, that even though the entire disturbance waveform can be reconstructed from the entire perceptual waveform, there is almost no information about the current value of the disturbance in the current value of the perceptual waveform.
BP: Martin, that's beneath you, or should be. "Apologize if it is too obvious to be made explicit!" The whole discussion was about whether the current value of the disturbance could be deduced from the current value of the perception. I said it couldn't, and all my arguments were aimed at showing why it couldn't. Now, in effect, you're saying I was right all along about the point I was trying to make, but that you were really making a different point which was "too obvious to be made explicit."
MT: As Richard and I have mentioned many times over the years, if G is a pure integrator, o is completely uncorrelated with p; informationally, the information from p to o is distributed equally over all informationally independent samples in its history, a notionally infinite number unless we start at some time t0 with known values of p and o. Hence the information available about the current value of o in the current value of p approaches (using the usual mathematical sense of "approaches") zero.
BP: Now you're really off into some never-never-land. This kind of gobbledegook smacks of desperation, not erudition. It's a smokescreen, a snow job. "Notionally infinite" -- Jesus Christ.
MT: I mention this trivial fact because some might have seen an apparent contradiction between (1) there being essentially no information about the current value of d in the current value of p under conditions of excellent control, and (2) the fact that the d waveform can be totally reconstructed from the p waveform, and therefore that all the information about d is available in p and its history.
"Trivial fact." Well, that does it for me. If you know what you're talking about, you're the only one here who does. I doubt that you will ever be able actually to perform the calculations you make these claims about, and the only way you'll ever convince me that you can do this is to do it. Call back when you have something to show me.
I've been annoyed at you before, Martin, and got over it. This time it's going to be hard.
Bill P.