# transport lag

[Martin Taylor 970428 13.45]

Hans Blom, 970428g]

Thus, if the delay is large,
control must necessarily be slow. It is in these cases where open
loop and feedforward help most. That this has not been appreciated in
PCT is, maybe, due to the fact that delays in the loop have not yet
received attention -- at least not that I'm aware of.

It hasn't been ignored, but I can't remember when it was last the focus
of a thread. Perhaps it is time. The usual turn of discussion when
loop delay or transport lag is brought up is to say that we are dealing
only with disturbances that are slow compared to the transport lag, so it
doesn't matter and it can be ignored. But it isn't ignored when the
control loops are subjected to Laplacian analysis or in talking of the
"Marken effect" (which I'll let Rick explain--it's dependent on there
being a transport lag).

I mentioned one point about transport lag the other day, before we got into
this strange by-way about directional correlations. What I pointed out
was that if there was a transport lag of tau seconds, then the best
possible control was limited by the autocorrelation function of the
disturbance signal at tau seconds. If A(d, tau) = c, then the variance
of the disturbance signal not accounted for in the output signal is at least
1-c^2 (the coefficient of alienation). The RMS variation in the input
variable qi with no output is var(d). With the countervailing output of
the control system, it is at least (1-c^2)*var(d), so the best possible
stability factor for any control system is 1/(1-c^2) (unless I made one
of my infamous algebraic errors :-()

The autocorrelation being the inverse Fourier Transform of the Power
Spectrum, for a band limited white noise disturbance signal of bandwidth
W, the autocorrelation function is (I think) c(t) = sin(pi*W*t)/pi*W*t .
Plug in the disturbance bandwidth and the transport lag, and you get the
best possible correlation between output and disturbance signals, and
from that the maximum possible control stability factor.

Some examples for a disturbance bandwidth of 5Hz:

Lag seconds 1-c(t)^2 best stability factor
0.01 .0082 122
0.02 .032 30.8
0.03 .072 13.9
0.05 .189 5.3
0.07 .343 2.9
0.1 .595 1.7

For longer lags, there's hardly any control at all. These can all be scaled
linearly to different disturbance signal bandwidths and lags. For a disturbance
signal bandwidth of 1 Hz, multiply the listed lags by 5.

These aren't the stability factors that will be achieved by any particular
control system. If the calculations are correct (always a dubious proposition
with my calculations) these are the best stability factors that can be
attained by ANY control system (MCT or PCT) controlling against a white
noise disturbance of bandwidth 5 Hz and transport lag as listed.

Martin

[From Bill Powers (970428.1738 MST)]

Martin Taylor 970428 13.45--

Some examples for a disturbance bandwidth of 5Hz:

Lag seconds 1-c(t)^2 best stability factor
0.01 .0082 122
0.02 .032 30.8
0.03 .072 13.9
0.05 .189 5.3
0.07 .343 2.9
0.1 .595 1.7

For longer lags, there's hardly any control at all.

This conclusion assumes that the control system would be ineffective over
the entire 5-Hz bandwidth, which isn't true, is it? Below a certain
frequency, control would be good, and the effective gain would be high. It's
only as the disturbance frequency components approach the point where the
phase shift begins to exceed 90 degrees that we begin to see larger errors.
So the error spectrum would not be 5 Hz, but some band between X and 5 Hz.
The error spectrum, and the variance spectrum, would not be white.

Another consideration comes up when we think of the difference between the
disturbing variable and the disturbance signal. Suppose we are looking at an
arm-position control system with a force disturbance, where the effect of
the disturbance is an acceleration, not a position.

This disturbance signal goes through two integrations before it becomes an
effect on position, so the controlled variable is affected in a way that
falls off at 12 db per octave (power) if the disturbance has a white
spectrum. If, on the other hand, you define the disturbance signal in terms
of position, then the applied force would have to _increase_ at 12 db/octave
to yield a white spectrum of position disturbance signals. The
highest-frequency disturbing forces would have to have the greatest
magnitudes, which does not seem realistic.

In nature, we generally find that the highest-frequency changes are also of
the lowest amplitude (tuned circuits aside). I think a reasonable spectrum
to assume _a priori_ is not white, but 1/f (for two integrations, 1/f^2)
with an upper limit at some low frequency.

Do you agree? If so, this would probably change the above table of numbers
considerably.

Best,

Bill P.

[Martin Taylor 970428

Bill Powers (970428.1738 MST)]

Martin Taylor 970428 13.45--

Some examples for a disturbance bandwidth of 5Hz:

Lag seconds 1-c(t)^2 best stability factor
0.01 .0082 122
0.02 .032 30.8
0.03 .072 13.9
0.05 .189 5.3
0.07 .343 2.9
0.1 .595 1.7

For longer lags, there's hardly any control at all.

This conclusion assumes that the control system would be ineffective over
the entire 5-Hz bandwidth, which isn't true, is it?

No, I think you understnad what's happening, but misunderstand the basis
of the calculation.

Below a certain
frequency, control would be good, and the effective gain would be high. It's
only as the disturbance frequency components approach the point where the
phase shift begins to exceed 90 degrees that we begin to see larger errors.
So the error spectrum would not be 5 Hz, but some band between X and 5 Hz.
The error spectrum, and the variance spectrum, would not be white.

No, the error spectrum would indeed not be white. Nor the variance spectrum,
if by that you mean the spectrum of the controlled variable qi.

Your statement about what is happening agrees with what I think is happening.
The components that are slow compared to the lag are well compensated,
but the faster components are not. When the lag is very short, _all_ the
components are "slow". As the lag increases, more and more of the
components of the disturbance signal are "fast" and more poorly compensated,
leading to poorer overall correlation between the output signal and
the disturbance signal. When the lag is 0.1 sec, that's half a cycle at
the top of the disturbance band.

There's another point here, though, which may get lost. The point is that
it really doesn't depend on the disturbance signal of 5 Hz bandwidth being
from DC up to 5 Hz. It could be from 500 to 505 Hz, and the numbers would
still apply to the best possible control (a human presumably couldn't
do it, but might if the control loop included a heterodyne stage:-)

In nature, we generally find that the highest-frequency changes are also of
the lowest amplitude (tuned circuits aside). I think a reasonable spectrum
to assume _a priori_ is not white, but 1/f (for two integrations, 1/f^2)
with an upper limit at some low frequency.

Do you agree? If so, this would probably change the above table of numbers
considerably.

Yes, I agree, and I agree that it would change those numbers. If you can,
off the top of your head, provide the inverse Fourier transform for a
1/f or a 1/f^2 spectrum, that will be the autocorrelation function,
from which you can produce a table corresponding to the one I gave.
I used a white noise because I had a form for the autocorrelation function
for white noise. I'm far too rusty at the math to compute the inverse
transforms from scratch and I don't have a table for them.