[Martin Taylor 2006.07.09.23.29]
[from Tracy Harms 2006;07,09.16:50]
My original post on this topic was in reply to Jim
It may be simultaneously true that a least
something corresponds simultaneously to a more
something. In the case of PCT I suppose less
error means more gain.
It's good to get back to the real thread!
Not only do I think Jim was getting caught up in an
illusion, akin to the behaviorist illusion, I think he
was trying to assert a strict (mathematical, absolute,
or formal) relationship between error and gain, which
failed to include recognition of the
finite-range-of-effectiveness which actual systems
necessarily entail. Rick seems to think that we can
assume that the interior of said range may be presumed
to be under discussion whenever the terms are used,
but this does not suffice for me. A formulaic
assertion is either true for all values, or it is
false. Saying that it is true when it is identifiably
false cannot be a thing of no importance.
I have to side with Rick on this one, thinking about how useful Newton's laws of gravity and motion are in the non-relativistic, non-quantum size and speed range in which we mostly operate.
However, re-reading the quote from Jim, I see something else that hasn't been addressed in this thread as far as I remember. Jim is looking (I think) at a system working in a real world, not a toy system of a single Elementary Control System (ECS) with a simple instantaneous connection representing the environmental feedback path. He's also looking at it from outside, but as an analyst, not as an observer. (The "Analyst's Viewpoint" allows access to internal parameters such as error, whereas the "Observer's Viewpoint" allows access only to effects the observed system has on its environment.)
If an analyst with access to the values of the disturbance but not of the system parameters sees less error than might have been anticipated, it's a reasonable inference that the loop gain is greater than had been assumed. That's the sense of Jim's way of putting it: not "more gain means less error", which would be a comment from the "Designer's Viewpoint" ("How much gain should I give it?"), but "less error means more gain" ("what IS the gain of this thing, anyway?).
If we are dealing in a real(-ish) world with complex environmental feedback paths, we can't really make that kind of assertion. "Gain" isn't a simple number (it isn't really, even in the trivial case), but a time function. The output value depends on the history of the error, not on its value at any particular moment (the trivial case usually shows the output function as an integrator, so even there this is true). Now consider what happens if we simply scale the gain function by multiplying it by some number.
There are frequency-dependent phase shifts embodied in the gain function. There are frequency-dependent phase shifts in the environmental feeback path, and maybe the feedback path even has reverberations or resonances. In those kinds of condition, even if the system doesn't oscillate, increasing the gain by a multiplicative constant can easily increase the error (ripples might take longer to die down even if they don't explode to infinity). The optimum gain, meaning the gain that gives least error on average, may be well below the gain that causes the system to blow up.
In the real world, it simply isn't true that the harder you try to keep something really stable the more stable it will be. Trying too hard is a very good way not to achieve your goal in a lot of situations -- such as playing golf!
Viewpoint does matter, but so does the proper application of the maths to the real world. Write the equations in terms of scalar numbers and you get simple results. Treat them as time functions, and it's more realistic, but much harder to be sure what's going to happen with any particular manipulation, such as (Designer's Viewpoint) increasing the gain. (I call it the Designer's Viewpoint, though I'm well aware that there may well be real-time influences on the gain of specific conrol systems; I believe I remember references to Tom Boubon having done some modelling of that kind, but I don't remember what.)
I hope that's not too muddled. I've had a rather exhausting day and my brain feels foggy. I may regret this in the morning, but I doubt I'll have time to post much tomorrow, so now or never.