Symbolic Orthogonal Projections: A New Polyhedral Representation for Reachability Analysis of Hybrid Systems

Abstract

This thesis deals with reachability analysis of linear hybrid systems. Special importance is given to the treatment of the necessary geometrical operations. In the first part, we introduce a new representation class for convex polyhedra, the symbolic orthogonal projections (sops). A sop encodes a polyhedron as an orthogonal projection of a higher-dimensional polyhedron. This representation is treated purely symbolically, in the sense that the actual computation of the projection is avoided. We show that fundamental geometrical operations, like affine transformations, intersections, Minkowski sums, and convex hulls, can be performed by block matrix operations on the representation. Compared to traditional representations, like half-space representations, vertex representations, or representations by support functions, this is a unique feature of sops. Linear programming helps us to evaluate sops. In the second part, we investigate the application of sops in reachability analysis of hybrid systems. It turns out that sops are well-suited for the discrete computations. Thereupon, we demonstrate also the applicability of sops for the continuous computation of reachability analysis. By a subtle parallel computation of a precise sop-based representation and a simplified representation, we tackle the problem of monotonic growing sizes of the sops. Additionally, we present experimental results which also show that sops allow an efficient and accurate computation of the reachable states.