An algebra of hybrid systems

Abstract

Hybrid systems are heterogeneous systems characterised by the interaction of discrete and continuous dynamics. We present a trajectory-based algebraic model for describing hybrid systems; the trajectories used are closely related to streams. The algebra is based on left quantales and left semirings and provides a new application for these algebraic structures. We show that hybrid automata, which are probably the standard tool for describing hybrid systems, can conveniently be embedded into our algebra. Moreover we point out some important advantages of the algebraic approach. In particular, we show how to handle Zeno eects, which are excluded by most other authors. The development of the theory is illustrated by a running example and a larger case study.

Deriving real-time action systems with multiple time bands using algebraic reasoning

“…Certain types of interval predicate are useful for compositional proofs, where the predicates may be moved in and out of the chop and iteration operators. Similar properties for compositionality have been observed by Höfner and Möller [36] and also by von Karger [47,48].…”

“…The following lemmas allow one to fold/unfold infinite and finite iterations and to prove iterative definitions via induction [6,22,36]. For an interval predicate p, we let p + = p ; p * and p ω+ = p ; p ω denote the finite and possibly infinite positive iterations of p, respectively.…”

“…We may also define operators 2 and 3 on interval predicates, where 2p.∆ and 3p.∆ hold iff p holds in all subintervals and some subinterval of ∆, respectively. Like [36,44,52], we define these operators in terms of chop as follows.…”

“…To prove (36), when the controller switches from executing (on1-false-b) to (idle-noton1) water 1 is at least empty 1 + acc.water 1 .TB pump and water 2 are at most full 2 − acc.water 2 .TB pump . Note that the accuracy is with respect to the pump time band as opposed to the previous properties that were with respect to the time band of the controller.…”

“…However, unlike Back and von Wright, we address real-time issues by basing our semantics on an algebra for interval predicates [22,35,36]. This allows one to improve uniformity across the model by enabling one to use interval predicates to express system requirements as well as program behaviour.…”