Discrete Controls

[glen e. p. ropella]

I've been trying to digest something else sparked from that CS&E
issue... If anybody has any input that might correct my views, I'd
like to hear about it.

"Classical" or continuous control theory has the following as its
template problem:

   dy
   -- y(t) = f(y(t), u(t))   &  y(t0) = x
   dt

where u(t) is the control function and y(t) is the system state
function and y' is the dynamic.  I think we can paraphrase the goal
as: "design u(t) such that y(t0) leads to y(t1).


Now, "discrete" control theory might be characterized as an
attempt to solve the following problem:

  xi is-a-member-of X defined-as a set of interacting elements

  dij(xi) is-a-member-of D defined-as a set of dynamics for 
                                      those elements

  uij(xi) is-a-member-of U defined-as a set of control dynamics 
                                      for those elements

A question I have is: Does one *graft* the set U of control dynamics
onto the set D, or whether U is-is-a-subset-of D?  I would think that
in any completely artificial system (like that of a simulation), the
question is trivial because it doesn't matter.  But, in any natural or
physical system (including that of designing physically constrained
mechanical devices), the control dynamics has to be grafted onto
naturally occuring ones.

It seems like control of physical systems wherein continuous controls
are used, the separation is obvious: the system behaves according to
some y(t) and so we apply some force u(t) to make it do what we want
it to.  But, in true discrete systems (i.e.  not in discretized
continuous systems) the line is blurred, somewhat.

Does any of this make sense?  Or am I all turned around, here? [grin]

glen
-- 
{glen e. p. ropella <address@hidden>   |                                  }
{Episkopos 11~11                       |            Hail Eris!            }
{http://www.trail.com/~gepr/home.html  |               =><=               }


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