Efficient Computation of Reachable Sets of Linear Time-Invariant Systems with Inputs

Abstract

Abstract. This work is concerned with the problem of computing the set of reachable states for linear time-invariant systems with bounded inputs. Our main contribution is a novel algorithm which improves significantly the computational complexity of reachability analysis. Algorithms to compute over and under-approximations of the reachable sets are proposed as well. These algorithms are not subject to the wrapping effect and therefore our approximations are tight. We show that these approximations are useful in the context of hybrid systems verification and control synthesis. The performance of a prototype implementation of the algorithm confirms its qualities and gives hope for scaling up verification technology for continuous and hybrid systems.

Zonotope/Hyperplane Intersection for Hybrid Systems Reachability Analysis

“…Reachability analysis has been a major research issue in hybrid systems over the past decade [1][2][3][4][5][6][7][8][9]. This research has been motivated by the fact that a successful reachability analysis makes it possible to extend approaches, initially developed in the field of computer science for discrete systems, for analysis and control of hybrid systems [10][11][12][13].…”

“…The next step was to improve the scalability of these approaches in order to be able to handle larger hybrid systems. Various scalable approaches have been proposed for the reachability analysis of continuous (essentially linear) systems based on classes of polytopes such as hyper-rectangles [6] and zonotopes [7,8], or on ellipsoids [9]. However, in order to use these techniques for the analysis of hybrid systems, we must also deal with the discrete transitions in a satisfactory (i.e.…”

“…The reachable set is approximated using zonotopes. The reachability analysis of the continuous dynamics is processed using the algorithm presented in [8]. We handle discrete transitions of the hybrid systems by proposing two new algorithms for computing tight over-approximations of the intersection of a zonotope with a hyperplane.…”

“…The paper is organized as follows. In section 2, we present briefly the algorithm for reachability analysis of linear systems proposed in [8] and discuss the needs for its extension to hybrid systems reachability. Section 3 is the main contribution of the paper, we first show that the problem of computing a tight over-approximation of the intersection of a zonotope with a hyperplane can be reduced to the problem of computing the intersection of a two dimensional zonotope with a line.…”

Flow* 1.2: More Effective to Play with Hybrid Systems

“…Here, we describe an efficient Taylor model-based method which also has a good scalability. It can be viewed as a combination of the methods described in [15] and [22].…”