A categorical approach to open and interconnected dynamical systems

TL;DR

This paper introduces a categorical graphical framework for discrete linear time-invariant dynamical systems, extending signal flow diagrams to include infinite past and future streams, and provides a new structural controllability characterization.

Contribution

It develops a sound and complete graphical theory for such systems using props, incorporating non-controllable systems and extending previous categorical approaches.

Findings

Graphical syntax closely related to classical signal flow diagrams

Extended semantics include non-controllable systems

Structural controllability characterization

Abstract

We develop a sound and complete graphical theory for discrete linear time-invariant dynamical systems. The graphical syntax, as in previous work, is closely related to the classical notion of signal flow diagrams, differently from previous work, these are understood as multi-input multi-output transducers that process streams with an \emph{infinite past} as well as an infinite future. This extended semantics features non-controllable systems, and we develop a novel, structural characterisation of controllability. Our approach is formalised through the theory of props, extending the work of Bonchi, Zanasi and the third author.