[spam] Re: PCT-Specific Methodology

[From Bill Powers (2006.12.20,1625 MST)]

Martin Taylor 2006. –

Bill, maybe this whole episode
might have been avoided had I written:

p + Gp - Gr = d

instead of

d = p + Gp - Gr.

See my latest (today) post to Bruce. Another reason I’m being prolix here
is that I’m snowed in without a car, am in a cozy warm apartment looking
out at a bleak landscape with snow going by horizontally, and would just
as soon be typing as doing anything else.
What I’m most used to, because of years of modeling, is solving equations
for single variables on the left as functions of the other variables on
the right. Too bad it took me so long to think of the word
“function.” Of course you could reverse the convention and put
the single variable on the right, as you do above, but that wouldn’t
change anything.
Perhaps the core of this matter as far as I am concerned is the fact that
in the expression p + Gp - Gr, p and r cannot vary independently if the
original model is to remain unchanged. I’m sure you can see why this is
so. On the other hand, when we solve for the dependent variable p, we
p = [G/(1+G)]r + d/(1+G)
with the one dependent variable on the left, and the two independent
variables on the right. This means we can evaluate the expression on the
right for any pair of values of r and d, and obtain a value of p that is
valid without changing the original model. We can also solve the system
of equations for e, qo, and (trivially) qi, with the same result: each
one is a function of d and r only, with none of the other dependent
variables appearing.
When you isolate an independent variable on one side of the
equation, you can no longer think of the expression on the other side as
a function, because the arguments of the function are no longer
independent. And obviously, the value of an independent variable does not
depends on any other variable in the system. In the present case, you
have one dependent variable on the other side, along with the other
independent variable, r.
Are you sure that in your animated diagram, each frame does not represent
the state of a *different model?*Best,

Bill P.

Re: PCT-Specific Methodology
[Martin Taylor 2006.]

[From Bill Powers (2006.12.20,1625
Are you sure that in your animated
diagram, each frame does not represent the state of a different


I think that should be clear from the accompanying text, right
beside the animated diagram: “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


[From Bill Powers (2006.12.21.0300 MST)]

Martin Taylor 2006. –

I find, in going back to your cited article, that I can’t make heads or
tails of it.

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 biggest problem is that when you say “the
correlation” and “the output” you do not indicate
which correlation and which output you mean. It seems that most of
the reasoning stays in your head and never gets out onto the paper. This
makes it very hard to understand what you’re talking about, particularly
for someone like me who is very shaky about Laplace transforms.

I do understand that Laplace transforms allow doing algebraic
manipulations instead of solving differential equations. But in the
examples I have seen, the transformed equations do not resemble the
original equations – for example, the Laplacian of an integration is
1/s, not s, so you end up manipulating a variable in the denominator, not
the numerator. That makes quite a difference in the resulting algebra. In
fact, to transform back to the normal form of the equations requires
partial-fraction expansions and gets very complicated. You seem to be
vastly oversimplifying that process. Of course I could simply be unable
to grasp the ideas you’re talking about. It would help if you didn’t omit
all the intermediate steps in your derivations.

For example, I can’t see, after much trying, how you get the statement
that the disturbance signal is correlated almost -1.0 with the output
signal. I presume that by “output signal” you mean the output
of the integrator. The equation you use is

d = p + Gp - Gr

In that equation, d varies in the same direction as Gp, which is the
output of the control system, and in the animated vector diagram in
your article, p and Gp both have components in the same direction as d,
not the direction opposed to it. So it would appear that Gp actually aids
the disturbance instead of opposing it.

I think the basic reason I have so much trouble understanding you is that
you state your conclusions without using normal mathematical derivations
to show how you got to the conclusions from the starting point. Without
those intermediate steps, I have no idea how you arrived at the
end-point, and can’t satisfy myself that I grasp your conclusion, or even
that it is correct.


Bill P.