Categories in Control

TL;DR

This paper models control systems using string diagrams within symmetric monoidal categories, providing presentations for categories of finite-dimensional vector spaces and linear relations, and connecting to structures like Frobenius algebras and the ZX-calculus.

Contribution

It offers a categorical framework for control theory diagrams, presenting finite-dimensional vector spaces and linear relations as symmetric monoidal categories with explicit generators and relations.

Findings

Categories FinVect_k and FinRel_k are presented via generators and relations.

Signal-flow diagrams correspond to string diagrams in these categories.

Frobenius structures on the 1-dimensional vector space relate to known algebraic structures.

Abstract

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 symmetric monoidal category FinVect_k of finite-dimensional vector spaces over the field of rational functions k = R(s), where the variable s acts as differentiation and the monoidal structure is direct sum rather than the usual tensor product of vector spaces. For any field k we give a presentation of FinVect_k in terms of the generators used in signal flow diagrams. A broader class of signal-flow diagrams also includes "caps" and "cups" to model feedback. We show these diagrams can be seen as string diagrams for the symmetric monoidal category FinRel_k, where objects are still finite-dimensional vector spaces but the morphisms are linear relations. We also give a presentation for FinRel_k. The relations say, among other things, that the 1-dimensional vector space k has two special commutative dagger-Frobenius structures, such that the multiplication and unit of either one and the comultiplication and counit of the other fit together to form a bimonoid. This sort of structure, but with tensor product replacing direct sum, is familiar from the "ZX-calculus" obeyed by a finite-dimensional Hilbert space with two mutually unbiased bases.