Understand
Control theory uses `signal-flow diagrams' to describe processes where real-valued functions of time are added, multiplied by scalars, differentiated and integrated, duplicated and deleted.
- These diagrams can be seen as string diagrams for the PROP FinRel_k, the strict version of the category of finite-dimensional vector spaces over the field of rational functions k = R(s) and linear relations, where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces.
- Control processes are also described by controllability and observability---whether the input can drive the process to any state, and whether any state can be determined from later outputs.
- For any field k we give a presentation of FinRel_k in terms of generators of the free PROP of signal-flow diagrams together with the equations that give FinRel_k its structure.