#
Categories in control

##
John C. Baez and Jason Erbele

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 FinVectk 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 FinVectk 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 FinRelk, where objects are still
finite-dimensional vector spaces but the morphisms are linear relations. We
also give a presentation for FinRelk. 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.

Keywords:
control theory, graphical calculus, Frobenius algebra, bialgebra,
dagger-compact
category, signal-flow diagram

2010 MSC:
18D10, 16T10

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 24, pp 836-881.

Published 2015-06-22.

http://www.tac.mta.ca/tac/volumes/30/24/30-24.pdf

TAC Home