[Martin Taylor 980303 10:20]

(The date stamp is when I created the first draft of this message. I have

witheld

posting it because I am not 100% sure of its correctness. I post it now so that

people more mathematically adept than I can check its correctness, and so that

those with simulation data can see whether in fact the prediction is accurate,

even if it is mathematically flawed).

I have about 100 messages unread since Saturday, but I thought I'd better

send this

out anyway.

## ···

-----------------------

This message was stimulated by the following exchange between Bill Powers

and Bruce Abbott.

+[From Bill Powers (980226.0148 MST)]

+

+Bruce Abbott (980225.2005 EST)--

+

+>>You haven't yet replied to my description of the correlations that are

+>>actually observed, which contradict what your common sense is telling you.

+>

+>I thought that this was it:

+>

+>>>In an

+>>>ideal control system, this immediate response means that e would never get

+>>>large enough even to detect, even though d and o varied over a wide range.

+>>>At these levels of variation, _observed_ changes in e would constitute

+>>>nothing but measurement error, and variation in e would appear uncorrelated

+>>>(except by chance) with d and with o.

+

+This isn't the actual explanation of the correlation phenomenon to which I

+refer. It's not a question of measurement error in observing an ideal

+control system. We see this effect in real data, measures of definitely

+non-ideal human control. We can measure the cv and the output more

+accurately than they can be controlled (cv) or varied (output) by the

+organism. The real explanation lies in system noise -- the fact that

+outputs and other variables show small spontaneous variations uncorrelated

+with changes in the disturbance.

I wanted to see if it was possible to say anything about the correlation

between the

perceptual signal and the disturbance signal of a control loop in the

absence of system

noise. It turned out to be possible, unless I've made some serious mistake.

So here's

the message as originally drafted.

-----------------------

The recent reincarnation of the discussion on the relation between the

disturbance signal and the perceptual signal has made possible an

interesting analysis. We can compute, at

least for the simplest control loop with a perfect integrator output

function, the maximum

possible correlation between the two signals as a function of the degree of

control.

If my analysis is correct, this maximum is 1/CR, where CR is the control

ratio--the

ratio between the magnitude of the amplitude of the variation in the perceptual

signal in the absence of control to its amplitude when control is in operation.

The correlation would be lower, if there is noise in the system.

I'll make a large caveat here. To me, this analysis seems too simple, and

I'd be

grateful if anyone can find an error that _increases_ the maximum

correlation possible

between p and d in this idealized maximally simple control loop, or show by

simulation

that the actual correlation is higher than the analysis shows. I submit the

analysis

because I can't by myself, see anything wrong with it, other than

approximations that

work in the direction of increasing the maximum.

----------------------------

We start with the classic diagram of a control system:

perceptual signal (p) ^ | reference signal (r)

> V

>____________-___error signal(e)__

> comparator |

perceptual signal (p) | |

> output function (G)

input function |

> >output signal (o)

CCEV + -----environment function-------|

> >

disturbance signal (d)| |output signal (o)

^ V

And we will make the usual assumptions that are made when using the

expression "p = o + d". That is to say, all the functions in the loop

except the output function are unit transforms.

Since it is short, I will repeat the loop derivation I wrote for Rick.

p = o + d = Ge + d = G(r-p) + d = Gr - Gp + d (1)

which yields d = p + Gp - Gr

That's starting point 1.

In the equation, d = p + Gp - Gr, all the "variables" are actually Laplace

transforms, which can be treated as if they were algebraic variables if the

system is linear (an integrator is a linear system). A Laplace transform is

a way of describing a time function, to put it crudely. The output

function, G, is a pure integrator in the example I

analyze below.

------------------------

Starting point 2 comes from an interchange between Bill Powers and me,

summarized by Bill (980224.1755 MST):

If the output function is a pure integrator, all frequency components are

phase-shifted by >90 degrees, which reduces the correlation to zero. For a

system with a large leakage term, >the correlation would be high unless

there was noise added in the output function or >comparator.

For now, I'll consider only the case with a perfect integrator output

function, but the

argument works (giving a different final result) with any function. One

just has to

know the correlation between the function's input and output. For G a pure

integrator,

x is uncorrelated with Gx if Bill's and my speculation is correct. For

other functions,

the correlation is likely to depend on the frequency spectrum of the input

of the function.

-----------------------

The analysis:

Firstly, it may help you to understand the analysis if you note that

symbols like "p"

and "d" represent waveforms extended over a long (notionally infinite)

time. I use

the same symbol also to represent the Laplace transform of the signal. When

the signals

are treated as vectors and we talk of correlation, the time variation (the

waveform) is

what you should think of. In developing the equations (e.g. "d = p + Gp -

Gr") the

Laplace transforms are used. The two are essentially interchangeable in the

current

context, which is why I haven't worried about using a script font for the

transform

(as if I could, in an e-mail message:-).

If G = 0 (no control), then p = d. Clearly then the correlation between p

and d is 1.0.

To start the next phase of the derivation, consider the case of r = 0

forever. In this

case, Gr (the integral of r) is also zero forever, and we have

d = p + Gp

Now we use the assertion that a variable is uncorrelated with its integral.

The

variable "d" is composed of two orthogonal components, p and Gp (remember,

"G" is

the output function, which is a perfect integrator).

Gp * *d

> /

> /

> /

> /

> /

> /

> /

>/_______* p

The squares of the lengths of the vectors represents the mean-square

amplitude variation

in the signal values. When you think of the variations on this diagram that

occur as

the output gain changes, or as the reference signal is allowed to vary,

remember that

it is the "d" vector that stays constant, while the others may change their

magnitude

and direction--not the other way around.

The correlation between any two vectors is the cosine of the angle between

them. That

is why Gp and p are drawn at right angles. Their correlation is zero. If Gp

is large

compared to p, d has a correlation of nearly 1.0 with Gp and nearly zero

with p. Since

we are dealing only with the case in which the reference signal is fixed at

zero, the

output signal is Ge, which is -Gp. So the disturbance signal is correlated

almost

-1.0 with the output signal--as we know to be the case for good control.

The control ratio (CR) is the ratio between the fluctuations that would

occur in p in

the absence of control and the fluctuations in p when the perception is

controlled. In

other words, CR = d/p when the transform between qi and p is the unit

transform.

The cosine of an angle in a right-angled triangle is the length of the

opposite side

over the hypotenuse. That is to say Corr(d:p) = p/d.

From this, the correlation between d and p is 1/CR.

-------------------

Variation in the reference signal

Why did I say this was a maximum correlation rather than the precise

correlation?

In part this is because there is always noise, as Bill pointed out. But

more because

of the role of the reference signal. Above, we considered the case of r

permanently

zero. Now we let r vary, independently of d, of course.

The control system is linear. What this means is that the superposition

theorem holds--

the contributions of different components can be added in the time domain,

in the

frequency domain, and in the Laplace domain. In particular, the relation d

= p + Gp

can be used as a starting point, onto which can be added any effects due to the

variation of r. Most importantly, p and Gp will change when r varies.

Gp (the integral of p in the simple system we are analyzing) can, because

of superposition,

be divided into two parts, which I will label G_d.p and G_r.p. G_d.p is

just what

we had before, when r was fixed permanently at zero. G_r.p is the variation

in Gp that

is extra, due to the variation in r.

Now we can look at the full expressions that was shown at the head of this

message:

d = p + Gp - Gr

and rewrite it

d = p + G_d.p + (G_r.p - Gr)

The first part of this is exactly what we had before, when r was

permanently fixed

at zero. The part in brackets is the contribution of variation in r.

What does the part in brackets contribute to the correlation between d and

p? Since the

reference signal varies independently of the variation in the disturbance

signal, and

any contribution of (G_r.p - Gr) is due to the reference signal, that

contribution is

orthogonal to d (and to G_d.p). It cannot increase the correlation between

d and p,

except by accident over the (very) short term. Furthermore, the better the

control,

the more nearly does p match r, and therefore the more nearly does G_r.p

match Gr.

The two tend to cancel one another, so if they have an effect, it tends to

become

small when control is good.

What this means is that even when the reference signal is allowed to vary

freely, the

maximum correlation that should be observed between the perceptual signal

and the

disturbance signal is 1/CR.

-------------------------

Extension to more realistic control loops.

The above applies mathematically only to a control system that is linear,

has no loop

transport delay, has a perfect integrator as its output/feedback function,

and has no

other time-binding functions in the loop (i.e. all the other functions are

simple

summations or multiplications by constant factors). Most control systems

are not

like that. What then?

Some cases can be examined heuristically. For example, if the output

function G is a

leaky integrator rather than a perfect integrator, the angle between p and

Gp depends

on the frequency spectrum of p. If low frequencies dominate, then Gp

correlates well

with p, but if high frequencies dominate, G acts like a good integrator. So the

correlation between d and p can be greater than 1/CR if the output function

is a

leaky integrator.

If there is some loop transport delay, it has to be incorporated into the

initial

expression for d. It could all be included in the output function, G. With loop

delay, the correlation between the input to G and its output will vary between

positive and negative correlation as a function of the frequency of p. This

will

show up in a correlation between Gp and p that varies with the spectrum of

p. In the

diagram, Gp leans left and right as the frequency of d varies. If the

magnitude of

G is large enough, this can lead at some frequencies to p being larger than

d--there

is no control, and the cosine takes on an imaginary (or at least complex)

value; the

loop is oscillating, not controlling.

Other cases can be examined similarly, provided the system is composed of

linear

components that allow the use of Laplace transforms. And for some

non-linear systems

one can make reasonable heuristic approximations by appealing to

small-amplitude

linearity.

--------------------------

All of the above depends on the correctness of two things: (1) the

statement that the

output of an integrator is, in general, uncorrelated with its input, and

(2) whether

I have done the analysis correctly. As I said up front, the result seems

too simple

not to be well known, and from prior discussion it clearly is not well

known. I'm

sure Bill has masses of simulation data that could be used to test the result

claimed, even though simulation data necessarily include transport loop delay

(which I think tends on balance to lower the correlation, but in a way that

depends on the spectrum of p). Perhaps Bill's data could show that the

correlation

control loops between the disturbance signal and the perceptual signal in such

"perfect integrator" is characteristically greater than 1/CR. If it is, it

would

show that there is a flaw in the analysis somewhere, even though I can't see it

right now, and I've asked a couple of control engineers, who also seem to

think it

is OK.

Martin