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

Tyler Hanks, Baike She, Matthew Hale, Evan Patterson, Matthew Klawonn, James Fairbanks

TL;DR

This paper introduces a category theoretic framework for constructing complex MPC problem formulations by composing subproblems, and constructs a monoidal category - called Para(Conv) - whose objects are Euclidean spaces and whose morphisms represent constrained convex optimization problems.

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.

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

TL;DR

This paper introduces a category theoretic framework for constructing complex MPC problem formulations by composing subproblems, and constructs a monoidal category - called Para(Conv) - whose objects are Euclidean spaces and whose morphisms represent constrained convex optimization problems.

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.
Paper Structure (15 sections, 4 theorems, 18 equations)

This paper contains 15 sections, 4 theorems, 18 equations.

Table of Contents

  1. Introduction
  2. Motivation and Problem Statements
  3. Multistage MPC Problems
  4. Problem Formulation
  5. Category Theory Background
  6. Categories of Convex Bifunctions
  7. Convex bifunctions
  8. The SMC of Convex Bifunctions
  9. Parameterized convex bifunctions
  10. MPC Optimization Problems are Morphisms
  11. Modeling initial and terminal constraints and costs
  12. Modeling constraints across timesteps
  13. Utilizing Hierarchical Composition
  14. Implementation
  15. Conclusion and Future Work

Key Result

Theorem IV.3

There is a category, Conv, in which

Theorems & Definitions (18)

  • Definition II.1
  • Example II.2
  • Definition III.1
  • Definition III.2
  • Definition IV.1
  • Remark IV.2
  • Theorem IV.3
  • Definition IV.4
  • Definition IV.5
  • Theorem IV.6
  • ...and 8 more