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Control Lyapunov Function Design via Configuration-Constrained Polyhedral Computing

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

TL;DR

The paper tackles designing control Lyapunov functions (CLFs) for constrained linear systems and min-max optimal control by leveraging configuration-constrained polyhedral computing to produce polyhedral-epigraph CLFs that are piecewise affine and convex. It establishes convex feasibility conditions (theoretical results akin to epigraphHJB and MixMaxHJB) that characterize all CLFs within a configuration-template class, enabling single-stage convex optimization to approximate the infinite-horizon HJB. The approach supports nominal and robust/min-max settings, providing methods to construct terminal costs and regions for MPC, and to obtain explicit, fixed-complexity controllers via vertex-based evaluation. Numerical examples demonstrate nominal and robust CLF design, illustrating competitive approximation accuracy to the true infinite-horizon solutions and practical applicability to explicit controller synthesis. Overall, the work offers a tractable bridge between implicit infinite-horizon optimization and explicit, efficiently evaluable control laws for constrained systems.

Abstract

This paper proposes novel approaches for designing control Lyapunov functions (CLFs) for constrained linear systems. We leverage recent configuration-constrained polyhedral computing techniques to devise piecewise affine convex CLFs. Additionally, we generalize these methods to uncertain systems with both additive and multiplicative disturbances. The proposed design methods are capable of approximating the infinite horizon cost function of both nominal and min-max optimal control problems by solving a single, one-stage, convex optimization problem. As such, these methods find practical applications in explicit controller design as well as in determining terminal regions and cost functions for nominal and min-max model predictive control (MPC). Numerical examples illustrate the effectiveness of this approach.

Control Lyapunov Function Design via Configuration-Constrained Polyhedral Computing

TL;DR

The paper tackles designing control Lyapunov functions (CLFs) for constrained linear systems and min-max optimal control by leveraging configuration-constrained polyhedral computing to produce polyhedral-epigraph CLFs that are piecewise affine and convex. It establishes convex feasibility conditions (theoretical results akin to epigraphHJB and MixMaxHJB) that characterize all CLFs within a configuration-template class, enabling single-stage convex optimization to approximate the infinite-horizon HJB. The approach supports nominal and robust/min-max settings, providing methods to construct terminal costs and regions for MPC, and to obtain explicit, fixed-complexity controllers via vertex-based evaluation. Numerical examples demonstrate nominal and robust CLF design, illustrating competitive approximation accuracy to the true infinite-horizon solutions and practical applicability to explicit controller synthesis. Overall, the work offers a tractable bridge between implicit infinite-horizon optimization and explicit, efficiently evaluable control laws for constrained systems.

Abstract

This paper proposes novel approaches for designing control Lyapunov functions (CLFs) for constrained linear systems. We leverage recent configuration-constrained polyhedral computing techniques to devise piecewise affine convex CLFs. Additionally, we generalize these methods to uncertain systems with both additive and multiplicative disturbances. The proposed design methods are capable of approximating the infinite horizon cost function of both nominal and min-max optimal control problems by solving a single, one-stage, convex optimization problem. As such, these methods find practical applications in explicit controller design as well as in determining terminal regions and cost functions for nominal and min-max model predictive control (MPC). Numerical examples illustrate the effectiveness of this approach.

Paper Structure

This paper contains 25 sections, 7 theorems, 63 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Contribution
  3. Overview
  4. Preliminaries and Notation
  5. Control Lyapunov Functions
  6. Constrained Linear Systems
  7. Control Lyapunov Functions
  8. Robust Control Lyapunov Functions for Uncertain Linear Systems
  9. CLF Construction via Polytopic Computing
  10. Representation of Epigraphs
  11. Configuration Templates
  12. CLFs with Polyhedral Epigraphs
  13. Robust CLFs for Min-Max Control
  14. CLF Design via Convex Optimization
  15. Hamilton-Jacobi-Bellman Equation
  16. ...and 10 more sections

Key Result

Proposition 1

Let $F$ satisfy Assumption ass::F and let $z$ be such that $P(z)$ is non-empty. Then, there exists a unique proper compact convex function $M: \mathbb R^{n_x} \to \mathbb R \cup \{ \infty \}$ such that $\mathrm{epi}(M) = P(z)$.

Figures (6)

  • Figure 1: Polyhedral epigraph $P(z^\star)$ of the $L$-CLF function $M$.
  • Figure 2: Trajectories of the closed-loop system under $\mu_{M}$ and polyhedral partition of $\operatorname{dom}(M)$.
  • Figure 3: ( 0.1, 0.5, 1, and 2)-Contours of $M$ (continuous lines), $M'$ (dotted lines), and $M^\star$ (dashed lines).
  • Figure 4: Polyhedral epigraph $P(z^\star)$ of $M$, the robust $L$-CLF function. The dark (purple) facet satisfies $z^\star_{\sf f} = 0$.
  • Figure 5: Trajectories of the closed-loop system under $\mu_{M}$ and polyhedral partition of $\operatorname{dom}(M)$. In (dark) purple, the set $\mathbb T$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Proposition 1
  • Remark 2
  • Definition 7
  • ...and 9 more