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Polyhedral Control Design: Theory and Methods

Boris Houska, Matthias A. Müller, Mario E. Villanueva

TL;DR

Polyhedral computing provides a scalable framework to represent and optimize high-dimensional convex sets for constrained linear control systems. The authors unify polyhedral geometry with control design by formulating invariant, reachable, and robust constructs via convex optimization, including control invariant sets, CLFs, and MPC. They survey a broad spectrum of methods—from polyhedron representations and support functions to advanced Tube MPC and min–max MPC variants—clarifying how convex formulations can yield tractable, scalable control solutions. The work underscores the practical impact of polyhedral methods for large-scale, data-driven, robust control while outlining key computational challenges and avenues for future research.

Abstract

In this article, we survey the primary research on polyhedral computing methods for constrained linear control systems. Our focus is on the modeling power of convex optimization, featured to design set-based robust and optimal controllers. In detail, we review the state-of-the-art techniques for computing geometric structures such as robust control invariant polytopes. Moreover, we survey recent methods for constructing control Lyapunov functions with polyhedral epigraphs as well as the extensive literature on robust model predictive control. The article concludes with a discussion of both the complexity and potential of polyhedral computing methods that rely on large-scale convex optimization.

Polyhedral Control Design: Theory and Methods

TL;DR

Polyhedral computing provides a scalable framework to represent and optimize high-dimensional convex sets for constrained linear control systems. The authors unify polyhedral geometry with control design by formulating invariant, reachable, and robust constructs via convex optimization, including control invariant sets, CLFs, and MPC. They survey a broad spectrum of methods—from polyhedron representations and support functions to advanced Tube MPC and min–max MPC variants—clarifying how convex formulations can yield tractable, scalable control solutions. The work underscores the practical impact of polyhedral methods for large-scale, data-driven, robust control while outlining key computational challenges and avenues for future research.

Abstract

In this article, we survey the primary research on polyhedral computing methods for constrained linear control systems. Our focus is on the modeling power of convex optimization, featured to design set-based robust and optimal controllers. In detail, we review the state-of-the-art techniques for computing geometric structures such as robust control invariant polytopes. Moreover, we survey recent methods for constructing control Lyapunov functions with polyhedral epigraphs as well as the extensive literature on robust model predictive control. The article concludes with a discussion of both the complexity and potential of polyhedral computing methods that rely on large-scale convex optimization.

Paper Structure

This paper contains 29 sections, 95 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Polyhedral Computing
  3. Polyhedra
  4. Representation
  5. Polytopes
  6. Linear Programming
  7. Faces
  8. Support Functions
  9. Complexity
  10. Simplicity
  11. Polarity
  12. Enumeration
  13. Algebra
  14. Configurations
  15. Curvature
  16. ...and 14 more sections

Figures (1)

  • Figure 1: This article is structured into two key sections. Section \ref{['sec::PolyhedralComputing']} provides a comprehensive survey of pivotal findings in polytopic computing. We delve into the intricate relationship between the geometric structure and computational complexity of parametric polyhedra, fostering a profound understanding of these concepts. Building on this insight, Section \ref{['sec::PolyhedralControl']} explores modern polytopic control design methodologies. This section concludes with an in-depth discussion on modern polyhedral design approaches through convex optimization, encompassing techniques for computing invariant sets, control Lyapunov functions (CLFs), and robust model predictive control.