[From Bill Powers (2006.12.16.0830 MST)]
Martin Taylor 2006.12.16.10.15 –
Without commenting directly on
the discussion (which I think a bit misinformed) of
“traditional” statistical methods, I suggest that if you are
interested in the question of correlation between observable disturbance
and observable control system output, you have a look at
<
http://www.mmtaylor.net/PCT/Info.theory.in.control/Control+correl.html
I remember seeing the original, but don’t remember why I didn’t comment
on it. I think you got misled by working at too high a level of
abstraction and not really working out the details.
In the first place, your analysis shows no sign of feedback effects,
perhaps because you solved for an independent variable in developing your
first starting point.
p = o + d = Ge + d = G(r-p) + d = Gr
- Gp + d (where G represents the output function)
which yields d = p + Gp - Gr.
That’s starting point 1.
The disturbance is not, of course, determined by the perceptual
signal or the reference signal. It is an independent variable. Here is
the correct derivation:
p = o + d
…
p = Gr - Gp + d, which then leads to
p(1 + G) = Gr + d, or
G 1
p = ----- r + ----- d,
1 + G 1 + G
So the perceptual signal is a dependent variable which depends on just
two independent variables, r and d.
Note that G/(1+G) approaches 1 as G becomes much greater than 1. The
90-degree phase shift which you say reduces correlations to zero is
greatly modified by this expression (see below for the case in which G is
an integrator).
Even with the perfect integrator, the output varies so it remains about
equal and opposite to the disturbance, with a phase shift that varies
from zero at very low frequencies to 90 degrees at very high frequencies
where the amplitude response approaches zero. The negative feedback makes
the frequency response of the whole system different from the frequency
response of the integrating output function. For one thing, if the time
constant of a leaky-integrator output function is T seconds, the time
constant of a response of the whole system to a disturbance is T/(1+G),
where G is the loop gain.
In Jagacinski and Flach, Control Theory for Humans, there are discussions
of the relationship between Laplace transforms and the frequency domain.
A system with a first-order lag (in the limit of large lags, an
integrator) is discussed in chapter 4 of that book. Chapter 5 is useful
as well. To change a Laplace equation into a time-domain equation, simply
substitute jw (j = square root of minus 1, w = omega = 2pifrequency).
So if the output gain is G/s, the time-domain expression would be
G/(jw)
Suppose the gain is all contained in the output function as you
assume and all the other functions are unity multipliers. We can then see
what the expression G/(1+G) amounts to in the time domain.
The expression G/(1+G) then becomes
G/jw G
-------- = ---------
1 + G/jw G + jw
Multiply numerator and denominator by G - jw to get the imaginary part
into the numerator only:
G(G -
jw)
= ------------
(G+jw)(G-jw)
G^2 - Gjw
= ------------
. G^2 + w^2
Separating the real and imaginary parts, we have
G^2
Gw
= ------------- - j
···
G^2 +
w^2 G^2 + w^2
From this we can see that as the integrating factor G increases. and as
the frequency decreases (remember that w is 2pifrequency), the real
part of the factor G/(1+G) approaches 1. As G increases and w
increases, the imaginary (90-degree phase shifted) part approaches
zero.
The tangent of the phase angle is the imaginary part divided by the real
part, or w/G.
For a given frequency of disturbance, as the gain factor increases the
perceptual signal approaches equality to the reference signal, and the
effect of the disturbance on it decreases toward zero.
If you like, you can work out a comparable expression for the output o,
and then calculate the ratio of o to d. It will approach -1, with the
imaginary part going to zero at high gains or low frequencies.
What you say about correlations applies only to the output integrator,
not the whole system. The correlation of the error signal with the output
of the integrator will always be zero. However, a correlation lagged 90
degrees will be perfect, which is not true for any other form of output
function. And the main point here is the the system as a whole does NOT
behave as a pure integrator, but in fact approaches the behavior of a
pure proportional system as the gain increases. Then everything we have
been saying about correlations holds true.
Hope my derivations don’t contain mistakes – it’s been a long time since
I last tried to do that stuff with complex numbers, though I had a little
practice in corresponding with Flach.
Best,
Bill P.