Enclosing the behavior of a hybrid automaton up to and beyond a Zeno point

Abstract

Even simple hybrid automata like the classic bouncing ball can exhibit Zeno behavior. The existence of this type of behavior has so far forced a large class of simulators to either ignore some events or risk looping indefinitely. This in turn forces modelers to either insert ad-hoc restrictions to circumvent Zeno behavior or to abandon hybrid automata. To address this problem, we take a fresh look at event detection and localization. A key insight that emerges from this investigation is that an enclosure for a given time interval can be valid independent of the occurrence of a given event. Such an event can then even occur an unbounded number of times. This insight makes it possible to handle some types of Zeno behavior. If the post-Zeno state is defined explicitly in the given model of the hybrid automaton, the computed enclosure covers the corresponding trajectory that starts from the Zeno point through a restarted evolution. ©2015 The Authors. Published by Elsevier Ltd.<p>A preliminary version of this paper was published in the proceedings of CPSNA 2013. This work was supported by the US National Science Foundation, awards NSF-CPS-1136099/1136104, the Swedish Knowledge Foundation (KK) and the Center for Research on Embedded Systems (CERES) grant number 20100314, and EPSRC grant number EP/C01037X/1.</p

Bracketing the solutions of an ordinary differential equation with uncertain initial conditions

“…In a near future, we plan to extend our approach to dealing with hybrid systems [11] when jumps could occur with respect to some guard conditions.…”

Accurate rigorous simulation should be possible for good designs

“…More relevant to the present work is that the implementation supports an enclosure simulation semantics intended to produce rigorous over-approximations (guaranteed upper and lower bounds) for all simulations [10]. This is the semantics that we would like Acumen, ultimately, to provide as the primary semantics.…”

“…For example, this is the case when there is Zeno behavior. Acumen's method for dealing with Zeno behavior is described elsewhere [10]. However, it is possible that a guard condition is not decidable even in the absence of Zeno behavior.…”

“…A final and important type of system that is useful for evaluating the extent to which an implementation realizes the proposed principle is Zeno systems [10]. Zeno behavior is known to be a phenomena unique to hybrid systems that naturally arises in models of mechanical systems, control systems, and others.…”

“…However, the ball would have to bounce an unbounded number of times before reaching this point. In previous work [10] we were able to show that it is possible to simulate such models past the Zeno point using a fixed-point computation on enclosures during simulation. However, such enclosures are not necessarily converging past the Zeno point.…”