Refinement and Proof Based Development of Systems Characterized by Continuous Functions

The specification of cyber-physical systems usually relies on continuous functions over dense real numbers whereas their implementation is discrete. Proving the correctness of the discrete implementation with respect to the continuous specification remains a challenge in the presence of dense real numbers. In this paper, we propose a refinement-based formal method, relying on Event-B, for such developments. We illustrate our proposal with the development of a simple stability controller for a generic plant model. The continuous function that models the system behavior is refined as a discrete model of the same kind preserving stability expressed as a safety invariants of the continuous model. The obtained discrete model uses discrete time instants modeled on $$\mathbb {N}$$, whereas the continuous model is based on dense time on $$\mathbb {R}$$. The Rodin Platform, together with the Theory plug-in handling the Real datatype and its properties supported the whole developments and proofs.