A Refinement Calculus for Hybrid Systems

System-level design for hybrid systems is complex and error-prone. To ensure correctness, formal methods are usually considered, and have been successfully applied in practice. Refinement for discrete systems is well-known, while little work has been done for hybrid systems. In this paper, we first present an envelope semantics for hybrid systems, where control and physical devices are separately described. Then, we relate the semantics to hybrid CSP (a well-known compositional modelling language for hybrid systems) by a Galois connection, to show its reasonableness. Based on this, we define a set of refinement rules that refine an abstract specification to a lower-level implementation. In the end, several examples are provided to show our methodology. Moreover, in our methodology classical refinement calculus is reused, and the soundness is provided by a monotonicity law.