Academic Awards 2025 booklet

15 Algorithms to solve the control-to-facet problem for piecewise affine hybrid systems on polytopes with control limitations Many engineering and physical systems combine continuous and discrete dynamics. An example is a moving robot that interacts with logic-based controllers. These so-called hybrid systems can have unsafe states, so control is essential to guarantee safety. State constraints, such as speed or position restrictions, are modelled as a set of linear inequalities, which describe a polytope in geometry. The state polytope can be divided into triangles. This thesis studies how this triangulation can be used to construct piecewise-affine control laws that achieve a given control objective, while avoiding unsafe states. In practice, control power often is limited. This limitation is used in our advantage to develop a new algorithm that efficiently solves the control-to-facet problem by combining techniques from mathematics, computational geometry and control. Additionally, relaxed conditions for the existence of admissible control laws are proven that allows engineers to control more systems in practice. The algorithm and its conditions on correctness are not only presented intuitively and rigorously, but also visually and practically by a MATLAB implementation. Build the solution constructively, triangle per triangle, using the results of the previous subproblems. Intuition behind the relaxed conditions: if a trajectory leaves polytopes P1 and P2 in finite time and if it cannot go back and forth between polytopes, then it must leave the full polytope in finite time.