Event-B Refinement for Continuous Behaviours Approximation

Hybrid systems are systems that integrate both discrete and continuous behaviours. The hybrid nature of such systems renders them difficult to model and verify in a single formal method. One of the key point when modelling these continuous features is the richness of the behaviours they may exhibit. In practice, continuous dynamics are expressed using complex differential equations, and are often difficult to handle during the implementation and validation process. To overcome this issue, controller designers use approximation allowing to substitute dynamics that have a close behaviour. Despite that it is based on sound, exact mathematics, this operation is rarely rigorous, and is performed prior to controller design, making it implicit in the resulting system. In this paper, we propose a general formalised approach to approximation. It relies on the definition of a Galois connection, and refinement is used to embed it, explicitly, into a high-level development operation, associated to particular correctness constraints and useful properties. Two types of usage for approximation are presented and discussed in the light of existing cases studies, as to showcase their particularities on the modelling and proving sides.