Control ratio and correlation

[Bill_Powers1]

[From Bill Powers (2009.06.21.0513 MDT)]

to Martin Taylor 2009.06.17.17.26 --

BP: I've been studying your derivation of the correlation between disturbance and perception as the control ratio, and have some comments and questions. You have retained the ability to make your way around in this mathematical area quite a lot better than I have, so it's taken me a while to grasp how you did it and what it means. And you know things about correlation that I never thought about and didn't know. Here's a reference I came across on the Web while searching for something relating to your way of getting the correlation from an integral. The relationship to the convolution theorem is suggestive, since we also deal with delays in control systems, but I can't go any further than that.

http://www-structmed.cimr.cam.ac.uk/Course/Convolution/convolution.html

The PDF file you sent is a sketch of a tutorial on the subject of correlation and control. I hope you will take the time to expand it into a hand-holding development for teaching. A reader like me needs to be told not just what the steps are, but how and why they are used.

I also suggest -- strictly for my own sake -- that you rearrange the equations a little to reflect more faithfully what the independent and dependent variables are. For example, you begin by defining the disturbance as a sinusoid:

d = sin(wt + phi)

where w (or omega, I presume) means 2*pi*frequency, t is elapsed time, and phi is a phase shift.

This appears to make both time and the phase into independent variables, since d depends on nothing else. However, there is no way to determine the phase shift because there is nothing independent to which the sine wave can be compared. When you generate a sine-wave disturbance outside the system in question, it is simply a sine wave. It is the form relative to which you would measure phase shifts in other variables that depend on d. So I recommend putting the phase shift into the perceptual signal as something to be determined from system properties, rather than as part of the main independent variable. In translating system equations into programs, this is a convention that makes the organization much clearer. I spent quite a while trying to figure out what determined phi in the first line, and not finding it.

So the first line would become

d = sin(wt)

followed by

p = m sin(wt + phi) (or -phi if you want the signs to come out the same)

The meaning of the "control ratio" line is not clear -- I suspect it's a matter of notation, in that |d| probably doesn't mean "absolute value of d". Or does it? Is |d|/|p| the ratio of the amplitudes of the two sine waves, or truly the ratio of absolute magnitudes? And if absolute magnitudes, how determined -- averaged over one cycle, or what?

In the "standard method of analyzing the control loop" we have one step like this:

d + k*integral((r-p)dt) = d - k*integral(p dt)

I assume that means you're setting r to zero. If not, shouldn't there be a term k*r*t on the right side?

Then you eliminate d from the equations to compute phi and I guess I follow the rest, except for the details of the scaling.

Now I think I see how you would get from this to a generalized (random) disturbance: represent the disturbance as a sum of phase-shifted sine waves (Fourier series), and compute the control ratio at each frequency. I actually use this way of generating disturbances now with random phases for successive harmonics and an exponentially declining amplitude as frequency increases. Varying the negative exponent seems to give a fairly linear relationship with RMS control error, which we can take as a measure of difficulty. I haven't even tried to do the math on that. And I don't know what it would mean to calculate a single correlation coefficient that represents the sum over the whole Fourier series, as we've been accustomed to doing. As you indicate, the correlation will be different at different frequencies.

As you can see, I am still paying attention within my personal bandwidth. Perhaps you can see from my discussion just what that bandwidth is -- what I need by way of explanations, which is clearly more than what you expect. When we get back to discussing information theory, which despite my skepticism I'm still willing to try, perhaps you can remember to talk down enough to make yourself clear to the dunces.

Best,

Bill P.

[MartinT]

[Martin Taylor 2009.06.21.11.25]

[From Bill Powers (2009.06.21.0513 MDT)]

to Martin Taylor 2009.06.17.17.26 –

BP: I’ve been studying your derivation of the correlation between
disturbance and perception as the control ratio, and have some comments
and questions.

Thanks. I appreciate your comments and suggestions. Some I can answer
here, but some need to be incorporated in a rewrite of the derivation.

Here’s a reference I came across on the Web while
searching for something relating to your way of getting the correlation
from an integral. The relationship to the convolution theorem is
suggestive, since we also deal with delays in control systems, but I
can’t go any further than that.

Thanks for the link. I have bookmarked it and will look at it later.

The PDF file you sent is a sketch of a tutorial on the subject of
correlation and control. I hope you will take the time to expand it
into a hand-holding development for teaching. A reader like me needs to
be told not just what the steps are, but how and why they are used.

I’ll try to do as you suggest.

I also suggest – strictly for my own sake – that you rearrange the
equations a little to reflect more faithfully what the independent and
dependent variables are. For example, you begin by defining the
disturbance as a sinusoid:

Only for the one specific example. In the more general case, it’s an
arbitrary waveform.

d = sin(wt + phi)

where w (or omega, I presume) means 2pifrequency, t is elapsed time,
and phi is a phase shift.

This appears to make both time and the phase into independent
variables, since d depends on nothing else. However, there is no way to
determine the phase shift because there is nothing independent to which
the sine wave can be compared.

I did that only for notational convenience. As I wrote it, Phi is
relative to the perceptual signal “p”, which in a linear system is
necessarily a sine wave of the same frequency as the disturbance. Since
“p” appears rather a lot in the analysis, I thought it visually less
confusing to use p rather than d as the reference zero phase. In
practice, what you say next is quite correct.

When you generate a sine-wave disturbance outside the
system in question, it is simply a sine wave. It is the form relative
to which you would measure phase shifts in other variables that depend
on d. So I recommend putting the phase shift into the perceptual signal
as something to be determined from system properties, rather than as
part of the main independent variable. In translating system equations
into programs, this is a convention that makes the organization much
clearer. I spent quite a while trying to figure out what determined phi
in the first line, and not finding it.

Sorry about that. I should have explained right at the beginning why I
put phi into the disturbance signal rather than the perception, despite
the fact that the disturbance is the arbitrarily generated signal.
Logically there’s no difference, and putting it into the disturbance
makes some of the equations in the derivation shorter and easier on the
eye. On my Web page that describes the general case with an animated
GIF, the disturbance is held fixed while the perceptual and output
phases change with the gain, which is the logically correct way, but
which makes for a visually confusing GIF.

The meaning of the “control ratio” line is not clear – I suspect it’s
a matter of notation, in that |d| probably doesn’t mean “absolute value
of d”. Or does it? Is |d|/|p| the ratio of the amplitudes of the two
sine waves, or truly the ratio of absolute magnitudes? And if absolute
magnitudes, how determined – averaged over one cycle, or what?

d> does not mean “absolute value of d(t) at time t”. It means the
length of the N-dimensional vector that represents the total time
course of d(t), which for a sampled signal is sqrt(sum_i(d(t_i)^2))
(for a continuous signal, substitute integral dt for sum). This length
would actually be proportional to the duration of the signal, but since
the time span is the same for all signals in the analysis, we can scale
by 1/t to keep things finite over infinite time. Doing that equates |d|
to the RMS value of the signal. Equivalently, for a sine wave, you
could use its amplitude as I did implicitly in the PDF, since that
changes only the scale factor for all the signals. Amplitude would not
work in the more general case, though. RMS always works.

In the “standard method of analyzing the control loop” we have one step
like this:

d + kintegral((r-p)dt) = d - kintegral(p dt)

I assume that means you’re setting r to zero. If not, shouldn’t there
be a term krt on the right side?

Yes. Didn’t I mention that this derivation was always for a steady zero
reference? No, looking back, I see that I did not. I said it only in
the general derivation. Sorry. (But the reference will be variable for
the informational analysis, as I currently conceive it).

Then you eliminate d from the equations to compute phi and I guess I
follow the rest, except for the details of the scaling.

Now I think I see how you would get from this to a generalized (random)
disturbance: represent the disturbance as a sum of phase-shifted sine
waves (Fourier series), and compute the control ratio at each
frequency. I actually use this way of generating disturbances now with
random phases for successive harmonics and an exponentially declining
amplitude as frequency increases. Varying the negative exponent seems
to give a fairly linear relationship with RMS control error, which we
can take as a measure of difficulty. I haven’t even tried to do the
math on that. And I don’t know what it would mean to calculate a single
correlation coefficient that represents the sum over the whole Fourier
series, as we’ve been accustomed to doing. As you indicate, the
correlation will be different at different frequencies.

Yes, it will depend on the spectrum of the disturbance.

As you can see, I am still paying attention within my personal
bandwidth.

Thanks for that. Sometimes I get quite discouraged with my failures to
communicate.

Perhaps you can see from my discussion just what that
bandwidth is – what I need by way of explanations, which is clearly
more than what you expect. When we get back to discussing information
theory, which despite my skepticism I’m still willing to try, perhaps
you can remember to talk down enough to make yourself clear to the
dunces.

I see no dunces here. As a perceptual aside, you may be amused to know
that I initially read your last few words as “clear to the dunes”,
which made me think of my childhood playing among the great sand dunes
of the Scottish village in which I lived. I interpreted your words as a
metaphor for being clear to everybody in range. You actually meant the
same thing, but used a much more literal form!

Martin

···

http://www-structmed.cimr.cam.ac.uk/Course/Convolution/convolution.html