PhasePlane:

Path: Common/Control

% Implements a phase plane controller. 

   The optimality condition is

   J = T + k*integral[0,T]( |u| dt ) 

   where u is the control and T is the final time.

   The solution consists of two switching curves. When k is inf you
   get the time-optimal solution with one switching curve. If k > 0
   you will get two switching curves. The former is known as a
   "bang-bang" controller and the latter a "bang-zero-bang" controller.

   The deadband separates the switching curves at the origin by db.
   The rate limit sets u = 0 if the state is within the control.

   Type PhasePlane for a demo.
--------------------------------------------------------------------------
   Form:
   u = PhasePlane( v, x, k, db, rl )
--------------------------------------------------------------------------

   ------
   Inputs
   ------
   v        (1,1)     Rate
   x        (1,1)     Position
   k        (1,1)     Weight on time
   db       (1,1)     Deadband
   rl       (1,1)     Rate limit

   -------
   Outputs
   -------
   u        (1,1)     Control [-1;0;1]

--------------------------------------------------------------------------
   Reference: Athans, M. and P. L. Falb. (1966.) Optimal Control
              Lincoln Laboratory Publication. 
--------------------------------------------------------------------------

Children:

Common: General/DeBlankLT
Common: Graphics/NewFig
Common: Graphics/PltStyle
Common: Graphics/TitleS
Common: Graphics/XLabelS
Common: Graphics/YLabelS

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