Modeling Model Predictive Control: A Category Theoretic Framework for Multistage Control Problems

TL;DR

This paper introduces a category theoretic framework for modeling multistage model predictive control problems, enabling compositional construction and visualization of complex MPC formulations.

Contribution

It develops a monoidal category framework, Para(Conv), for constructing MPC problems through composition, with a diagrammatic syntax and software implementation in Julia.

Findings

Framework captures multistage MPC as sequential and parallel compositions.

Provides a diagrammatic syntax for visualization and modification.

Enables software implementation integrating with mathematical programming libraries.

Abstract

Model predictive control (MPC) is an optimal control technique which involves solving a sequence of constrained optimization problems across a given time horizon. In this paper, we introduce a category theoretic framework for constructing complex MPC problem formulations by composing subproblems. Specifically, we construct a monoidal category - called Para(Conv) - whose objects are Euclidean spaces and whose morphisms represent constrained convex optimization problems. We then show that the multistage structure of typical MPC problems arises from sequential composition in Para(Conv), while parallel composition can be used to model constraints across multiple stages of the prediction horizon. This framework comes equipped with a rigorous, diagrammatic syntax, allowing for easy visualization and modification of complex problems. Finally, we show how this framework allows a simple software realization in the Julia programming language by integrating with existing mathematical programming libraries to provide high-level, graphical abstractions for MPC.