[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 = 2*pi*frequency).

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 2*pi*frequency), 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.