Combinatorial Vector Fields for Piecewise Affine Control Systems

This paper is intended to be a continuation of Habets and van Schuppen (2004) and Habets, Collins and van Schuppen (2006), which address the control problem for piecewise-affine systems on an arbitrary polytope or a family of these. Our work deals with the underlying combinatorics of the underlying discrete system. Motivated by Forman (1998), on the triangulated state space we define a combinatorial vector field, which indicates for a given face the future simplex. In the suggested definition we allow nondeterminacy in form of splitting and merging of solution trajectories. The combinatorial vector field gives rise to combinatorial counterparts of the flow lines. The main result is then an algorithm for synthesis of supervisory control.