[Martin Taylor 920720 16:15]
(Rick Marken 920719.1300)
Earlier today I replied to Rick Marken's claim that his spreadsheet model
refuted my intuitive claim that there could be hidden positive feedback loops
in the hierarchy. I now think I understand why he got his results, and have a
better feel for what is going on (Bill is right to warn against thinking one
can intuit what goes on in a hierarchic control system, and to argue for
modelling instead).
The example situation can be reduced from my earlier description. Here's a
simpler one that is easier to understand.
There is an ECS called Y whose output affects something in the outer world
called B that provides a sensory input b, which Y controls as the percept y(b).
There is another ECS called Z whose loop goes through B and separately through
D. Z controls for the percept z(b+d). Take B and D to represent gain functions
such that delta(b) = B delta (Oy + Oz) where Oy and Oz are the outputs of
the two ECSs, and delta(d) = D delta (Oz).
Even if the sign of D would lead to positive feedback, Z can be stable if the
sign of B is correct and B>D. Z can move b more with a given output than it
moves d. But if Y is acting, REGARDLESS OF THE DIRECTION IN WHICH Y TRIES
TO MOVE B, B becomes less susceptible to the actions of Z. The algebraic
expressions get horribly complex, but we went through this a while back with
one of Rick's studies that showed it to happen. So if the gain of Y is large
enough with respect to the gain of Z, then Z can go unstable.
It is the effective gain of Y as applied to B that is important in this, not
the direction in which Y moves B, cooperatively or in conflict with Z, just
as Rick says. There can still be hidden positive feedback links in the
hierarchy, but they can become manifest only when some gain changes, not
just when some reference level shifts.
My Degree of Freedom argument (920701 0240) suggests that gains must be
changeable, even if it is only in the binary sense that some ECS turns off to
allow another to take control (please don't complain about the sloppy wording
here; good wording takes pages). So the potential problem of masked positive
feedback still exists in the hierarchy.
Models Rule, OK!
Martin