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In article <1992Nov8.010847.24207@math.ucla.edu>
barry@arnold.math.ucla.edu (Barry Merriman) writes: 

Barry>  Step control is usually based on simple trial-and-error, plus
Barry>  an assumed asymptotic form for the time step error term. 
Barry>
Barry>  Does applying control theory to such a simple situation really
Barry>  lead to algorithms that are more _efficient_ for typical 
Barry>  problems?

The relation between the stepsize and the error may be quite different
than the normally assumed asymptotic form. Part of this "deviation"
has structure and can be modeled. By exploiting these models one
arrive at a stepsize selection rule that results in smoother stepsize
(and error) sequences as well as fewer stepsize rejections than the
standard rule. The gain in efficiency is moderate (typically 0-20%
depending on the type of problem). More important is the improved
robustness resulting from more often making a correct choice on the
next stepsize. In case of an implicit integration method the smoother
stepsize sequence also makes it possible to use slightly different
(and often more efficient) strategies for the updating of the
iteration matrix in the equation solver.

The use of control theory in this context is fairly simplistic. It
takes a dynamic model to accurately describe the stepsize-error
relation. In addition, the effect from the differential equation on
the stepsize-error relation can be viewed (and modeled) as an external
disturbance. The use of the dynamic process and disturbance models in
the design of the stepsize selection rule is facilitated with some
knowledge of feedback control.  Finally, some parameters in the
model depend on the integration method. One would like the stepsize
selection rule to work well for a large class of integration method,
and hence the design has to be made considering a range of model
parameter values. Again, some knowledge of basic feedback control design
principles come handy.

Kjell
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    mail: Kjell Gustafsson, Department of Computer Science MJH309
          Stanford University, Stanford CA 94305, USA
  e-mail: kjell@sccm.stanford.edu
   phone: (415) 723-4269 (voice), (415) 723-2411 (fax)