Safe & robust reachability analysis of hybrid systems

Abstract

Hybrid systems-more precisely, their mathematical models-can exhibit behaviors, like Zeno behaviors, that are absent in purely discrete or purely continuous systems. First, we observe that, in this context, the usual definition of reachability-namely, the reflexive and transitive closure of a transition relation-can be unsafe, ie, it may compute a proper subset of the set of states reachable in finite time from a set of initial states. Therefore, we propose safe reachability, which always computes a superset of the set of reachable states.Second, in safety analysis of hybrid and continuous systems, it is important to ensure that a reachability analysis is also robust wrt small perturbations to the set of initial states and to the system itself, since discrepancies between a system and its mathematical models are unavoidable. We show that, under certain conditions, the best Scott continuous approximation of an analysis A is also its best robust approximation. Finally, we exemplify the gap between the set of reachable states and the supersets computed by safe reachability and its best robust approximation.

System Analysis and Robustness

“…Robustness. In [13], we introduced robustness, a property of monotonic maps between complete lattices of (closed) subsets in metric spaces. Intuitively, robustness requires that small changes to the input I of a map F cause small changes to its output, where the definition of small relies on the metrics.…”

“…the restriction to closed subsets is due to indistinguishability of S and S. 3. Results about existence of best robust approximations [13] (Sec 4). Example 2 (Euclidean).…”

“…Definition 2 (Robustness [13]). Given A: Po(C(S 1 ), C(S 2 )) with S 1 and S 2 metric spaces, we say that:…”

Towards Physical Hybrid Systems

“…However, robustness is fundamentally different from our deductive verification approach. The analysis of robust systems often relies on a reachability analysis (see, e.g., [20]) which loses much of the logical precision present in deductive verification, e.g., decidability of differential equation invariants [26]. Further, by building on dL, which is a general purpose hybrid systems logic, we are able to handle a wide class of models, whereas tools like dReach [16] tend to be slightly more limited in scope.…”

“…The main thing we need to show (20) is that v 0 <100 ∀t>0 ((∀0≤s≤t 0≤v 0 +s≤100) → t + v 0 ≤100) , and this is a valid sequent because in particular when s = t, 0≤v 0 + t≤100. In a similar way, to close (17)…”