[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.