Combinatorial hybrid systems

As initially suggested by E. Sontag [1], [2] it is possible to approximate an arbitrary nonlinear system by a set of piecewise linear systems. In this work we concentrate on how to control a system given by a set of piecewise linear systems defined on simplices. By using the results of L. Habets and J. van Schuppen [3] it is possible to find a controller for the system on each of the simplices thus guaranteeing that the system flow on the simplex only will leave the simplex through a subset of its faces. Motivated by R. Forman [4], 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 most concepts from dynamical systems, such as duals to vector fields, flow, flow lines, fixed points and Lyapunov functions. Finally it will be shown how this theory extends to switched dynamical systems and an algorithmic overview of how to do supervisory control will be shown towards the end.