Iterative computation of polyhedral invariants sets for polynomial dynamical systems
Résumé
This paper deals with the computation of polyhedral positive invariant sets for polynomial dynamical systems. A positive invariant set is a subset of the state-space such that if the initial state of the system belongs to this set, then the state of the system remains inside the set for all future time instances. In this work, we present a procedure that constructs an invariant set, iteratively, starting from an initial polyhedron that forms a “guess” at the invariant. At each iterative step, our procedure attempts to prove that the given polyhedron is a positive invariant by setting up a non-linear optimization problem for each facet of the current polyhedron. This is relaxed to a linear program through the use of the blossoming principle for polynomials. If the current iterate fails to be invariant, we attempt to use local sensitivity analysis using the primal-dual solutions of the linear program to push its faces outwards/inwards in a bid to make it invariant. Doing so, however, keeps the face normals of the iterates fixed for all steps. In this paper, we generalize the process to vary the normal vectors as well as the offsets for the individual faces. Doing so, makes the procedure completely general, but at the same time increases its complexity. Nevertheless, we demonstrate that the new approach allows our procedure to recover from a poor choice of templates initially to yield better invariants.