From Bill Powers (980906.1734 MDT)]
Rupert Young (980906.2112 BST)--
Bill, as a precursor to reproducing your armac algorithm I've gone back to
bascis to investigate standard control systems. Being Vensim-less I've done
this on my own system and have recorded these simulations in a ps and html
document at
http://www.ee.surrey.ac.uk/cgi-bin/R.Young/pct/pct.html
(though the html version doesn't seem to working properly yet)
I looked at the html version and it worked fine.
This looks very good. I presume that you were using the usual control
system with the output function being a leaky integrator:
qo := qo + (gain*error - qo)/slow.
An important thing to develop in your analysis is the relationship between
gain and loop delay. For any delay, you can have any gain you like (such as
1000), if you pick the right slowing factor. For each gain and delay, there
will be one slowing factor that will give the quickest possible control
without overshoot.
Using higher gains will make the resistance to disturbances almost perfect,
although because of the lag there is a limit to how rapidly a changing
disturbance can be counteracted. A good test-disturbance is a step
function, which is zero for the first quarter or so of the plot and some
constant value afterward. By varying the slowing factor you can find the
value at which the overshoot just disappears while the rise time of the
opposing output is as fast as possible. Can you work out the general
relationship among gain, lag, and slowing factor for the fastest response
without overshoot? Martin Taylor showed that my derivation in the Psych Rev
article was a special case for a one-iteration lag. Can you find,
empirically, the general case, or derive it analytically? This is not a
test question to which I know the answer -- I'm really curious about what
you would find.
I wanted to see what happens when a transport lag is introduced and have a
section on that. I'd be particularly interested in any comments you have on
this section and my conclusion that control is possible only if the response
is slow enough, that the longer the lag the slower must be the response.
You're perfectly correct. There is a neat way, using Laplace transforms, to
show that the perfect compensator for a transport lag is a pure integrator.
I don't remember it at the moment, but perhaps you could look it up.
Best,
Bill P.