Formal composition of hybrid systems

TL;DR

This paper introduces a category-theoretic framework for the formal, compositional synthesis of hybrid systems, enabling hierarchical, sequential, and parallel compositions through spans, pushouts, and products.

Contribution

It develops a unified categorical approach to hybrid system composition, including hierarchical, sequential, and parallel methods, formalized via spans, pushouts, and double categories.

Findings

Framework models hierarchical hybrid systems as spans.

Sequential composition uses pushouts of directed systems.

Parallel composition is represented by categorical products.

Abstract

We develop a compositional framework for formal synthesis of hybrid systems using the language of category theory. More specifically, we provide mutually compatible tools for hierarchical, sequential, and independent parallel composition. In our framework, hierarchies of hybrid systems correspond to template-anchor pairs, which we model as spans of subdividing and embedding semiconjugacies. Hierarchical composition of template-anchor pairs corresponds to the composition of spans via pullback. To model sequential composition, we introduce "directed hybrid systems," each of which flows from an initial subsystem to a final subsystem in a Conley-theoretic sense. Sequential composition of directed systems is given by a pushout of graph embeddings, rewriting the continuous dynamics of the overlapping subsystem to prioritize the second directed system. Independent parallel composition corresponds to a categorical product with respect to semiconjugacy. To formalize the compatibility of these three types of composition, we construct a vertically cartesian double category of hybrid systems where the vertical morphisms are semiconjugacies, and the horizontal morphisms are directed hybrid systems.