9 research outputs found
Under-Approximate Reachability Analysis for a Class of Linear Uncertain Systems
Under-approximations of reachable sets and tubes have been receiving growing
research attention due to their important roles in control synthesis and
verification. Available under-approximation methods applicable to
continuous-time linear systems typically assume the ability to compute
transition matrices and their integrals exactly, which is not feasible in
general, and/or suffer from high computational costs. In this note, we attempt
to overcome these drawbacks for a class of linear time-invariant (LTI) systems,
where we propose a novel method to under-approximate finite-time forward
reachable sets and tubes, utilizing approximations of the matrix exponential
and its integral. In particular, we consider the class of continuous-time LTI
systems with an identity input matrix and uncertain initial and input values
belonging to full dimensional sets that are affine transformations of closed
unit balls. The proposed method yields computationally efficient
under-approximations of reachable sets and tubes, when implemented using
zonotopes, with first-order convergence guarantees in the sense of the
Hausdorff distance. To illustrate its performance, we implement our approach in
three numerical examples, where linear systems of dimensions ranging between 2
and 200 are considered
Fully-Automated Verification of Linear Systems Using Inner- and Outer-Approximations of Reachable Sets
Reachability analysis is a formal method to guarantee safety of dynamical
systems under the influence of uncertainties. A major bottleneck of all
reachability algorithms is the requirement to adequately tune certain algorithm
parameters such as the time step size, which requires expert knowledge. In this
work, we solve this issue with a fully-automated reachability algorithm that
tunes all algorithm parameters internally such that the reachable set enclosure
satisfies a user-defined accuracy in terms of distance to the exact reachable
set. Knowing the distance to the exact reachable set, an inner-approximation of
the reachable set can be efficiently extracted from the outer-approximation
using the Minkowski difference. Finally, we propose a novel verification
algorithm that automatically refines the accuracy of the outer- and
inner-approximation until specifications given by time-varying safe and unsafe
sets can either be verified or falsified. The numerical evaluation demonstrates
that our verification algorithm successfully verifies or falsifies benchmarks
from different domains without any requirement for manual tuning.Comment: 16 page
Avoiding geometric intersection operations in reachability analysis of hybrid systems
Although a growing number of dynamical systems studied in various fields are hybrid in nature, the verification of prop-erties, such as stability, safety, etc., is still a challenging problem. Reachability analysis is one of the promising meth-ods for hybrid system verification, which together with all other verification techniques faces the challenge of making the analysis scale with respect to the number of continuous state variables. The bottleneck of many reachability analysis techniques for hybrid systems is the geometrically computed intersection with guard sets. In this work, we replace the in-tersection operation by a nonlinear mapping onto the guard, which is not only numerically stable, but also scalable, mak-ing it possible to verify systems which were previously out of reach. The approach can be applied to the fairly common class of hybrid systems with piecewise continuous solutions, guard sets modeled as halfspaces, and urgent semantics, i.e. discrete transitions are immediately taken when enabled by guard sets. We demonstrate the usefulness of the new ap-proach by a mechanical system with backlash which has 101 continuous state variables
Reachability analysis of continuous-time piecewise affine systems
This paper proposes an algorithm for the characterization of reachable sets of states for continuous-time piecewise affine systems. Given a model of the system and a bounded set of possible initial states, the algorithm employs an LMI approach to compute both upper and lower bounds on reachable regions. Rather than performing computations in the state-space, this method uses impact maps to find the reachable sets on the switching surfaces of the system. This tool can then be used to deduce safety and performance results about the system