[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