Efficient Geometric Operations on Convex Polyhedra, with an Application to Reachability Analysis of Hybrid Systems

Abstract

We present a novel representation class for closed convex polyhedra, where a closed convex polyhedron is represented in terms of an orthogonal projection of a higher dimensional \(\mathcal{H}\)-polyhedron. The idea is to treat the pair of the projection and the polyhedron symbolically rather than to compute the actual described polyhedron. We call this representation a symbolic orthogonal projection, or a sop, for short. We show that fundamental geometrical operations, like affine transformations, intersections, Minkowski sums, and convex hulls, can be performed by simple block matrix operations on the representation. Due to the underlying \(\mathcal{H}\)-polyhedron, linear programs can be used to evaluate sops, e.g. the usage of template matrices easily yields tight over-approximations. Beyond this, we will also show how linear programming can be used to improve these over-approximations by adding additional supporting half-spaces, or even facet-defining half-spaces, to the over-approximation. On the other hand, we will also discuss some drawbacks of this representation, e.g. the lack of an efficient method to decide whether one sop is a subset of another sop, or the monotonic growth of the representation size under geometric operations. The second part deals with reachability analysis of hybrid systems with continuous dynamics described by linear differential inclusions and arbitrary affine maps for discrete updates. The invariants, guards, and sets of reachable states are given as convex polyhedra. First, we present a purely sop-based reachability algorithm where various geometric operations are performed exactly. Due to the monotonic growth of the representation size of the sops, this algorithm is not suited for practical applications. Then we propose a combination of the sop-based algorithm with a support function based reachability algorithm. Accompanied by some simple examples, we show that this combination results in an efficient reachability algorithm of better accuracy than pure support function based algorithm. We also discuss the current limitations of the proposed techniques and chart a path forward which might help us to solve linear programs over huge sops.