Category theory for cyber-physical systems

The Cyber-Physical Systems (CPS) comprise interacting digital, analog, physical, and human components engineered for function through integrated physics and logic. The CPS Initiative within the US National Institute of Standards and Technology has programs for the foundations, applications and testbed science of CPS that exhibit logical, physical and logical-physical interactions. CPS have become the means by which a large variety of industrial and commercial system are automated. The logical and physical domains each have established mathematical methods for modeling and assurance. Examples include a variety of formal logical systems for logical interactions and algebraic systems of ODEs for physical interactions. However, we lack the mathematics for describing arbitrary compositions or schema with alternating logical and physical interactions, i.e., cyber-physical system interactions.
 
We introduce a simple category-theoretic, functorial definition of cyber-physical systems using functors from the symmetric monoidal categories to a category denoted CyPhy that contains all the interaction types essential to CPS. This functorial representation induces a cyber-physical systems structure on the pre-image of such a functors and, in doing so, provides a representation of CPS as well as a means of projecting to the logical and physical subsystems. Morphisms in this category are commuting diagrams of these functors. Other applications of Category Theory to the representation of ‘hybrid’ dynamical systems, the performance metrics on and formal assurance for CPS will be discussed.