Zonotope/Hyperplane Intersection for Hybrid Systems Reachability Analysis

Abstract

Abstract. In this paper, we are concerned with the problem of computing the reachable sets of hybrid systems with (possibly high dimensional) linear continuous dynamics and guards defined by switching hyperplanes. For the reachability analysis of the continuous dynamics, we use an efficient approximation algorithm based on zonotopes. In order to use this technique for the analysis of hybrid systems, we must also deal with the discrete transitions in a satisfactory (i.e. scalable and accurate) way. For that purpose, we need to approximate the intersection of the continuous reachable sets with the guards enabling the discrete transitions. The main contribution of this paper is a novel algorithm for computing efficiently a tight over-approximation of the intersection of (possibly high-order) zonotopes with a hyperplane. We show the accuracy and the scalability of our approach by considering two examples of reachability analysis of hybrid systems.

Compositional abstraction refinement for control synthesis under lasso-shaped specifications

“…[5], [25], [15], [11], [20], [24]. System (1) can also be described as a non-deterministic infinite transition system S = (X, U, −→) where…”

“…In most continuous systems, exact computation of the reachable sets as in [13], [27] is not possible. We thus rely on methods to efficiently compute over-approximations of the reachable sets (for a given finite time), using for example polytopes [5], oriented hyper-rectangles [25], ellipsoids [15], zonotopes [11], level sets [20] or the monotonicity property [24], which is considered in the examples of this paper. Other relevant works with similar objectives include: [22] which focuses on reach-avoid-stay control specifications and computes abstractions based on infinite-time reachability of neighbor states; and [21] which uses sets of finite prefixes to describe abstractions of infinite behaviors.…”

Time-Triggered Conversion of Guards for Reachability Analysis of Hybrid Automata

“…Frehse et al [30] cast the intersection operator as a convex minimization problem. Other research examines the problem of e ciently computing geometric intersections for particular choices of data structures [31,33,35,40].…”

Avoiding geometric intersection operations in reachability analysis of hybrid systems

“…Even reachable sets represented by polytopes (bounded polyhedra), which are closed under intersection with polyhedra, are not computationally efficient for guard intersection since partial intersections have to be unified by a convex hull to avoid a combinatorial explosion in the representation size and time required to compute the reachable set. Since the computation of the convex hull is itself costly and in many cases numerically unstable [5], alternative approaches have been proposed for unifying reachable sets with simpler representations, such as two-dimensional projections of zonotopes [14], bundles of parallelotopes (a special case of zonotopes) [2], and template polyhedra [11].…”