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Safe Feedback Optimization through Control Barrier Functions

Giannis Delimpaltadakis, Pol Mestres, Jorge Cortés, W. P. M. H. Heemels

TL;DR

This work tackles enforcing state constraints in continuous-time feedback optimization by integrating safe gradient flows with high-order control-barrier functions. The proposed SGF framework solves an online QP to minimally adjust the gradient flow, guaranteeing forward invariance of the safe set and preserving the equivalence between equilibria and optimization critical points. It provides conditions for feasibility, existence, and uniqueness of the closed-loop, as well as local (and in convex cases global) asymptotic convergence to optima, including regularization to interior points when global optima lie on the boundary. Simulations in convex and non-convex settings demonstrate constraint satisfaction during transients and convergence to the optimizer, highlighting practical robustness and limitations. The approach offers a principled path to safe, continuous-time optimization in dynamical systems with broad potential applications in power, traffic, and networked control.

Abstract

Feedback optimization refers to a class of methods that steer a control system to a steady state that solves an optimization problem. Despite tremendous progress on the topic, an important problem remains open: enforcing state constraints at all times. The difficulty in addressing it lies on mediating between the safety enforcement and the closed-loop stability, and ensuring the equivalence between closed-loop equilibria and the optimization problem's critical points. In this work, we present a feedback-optimization method that enforces state constraints at all times employing high-order control-barrier functions. We provide several results on the proposed controller dynamics, including well-posedness, safety guarantees, equivalence between equilibria and critical points, and local and global (in certain convex cases) asymptotic stability of optima. Various simulations illustrate our results.

Safe Feedback Optimization through Control Barrier Functions

TL;DR

This work tackles enforcing state constraints in continuous-time feedback optimization by integrating safe gradient flows with high-order control-barrier functions. The proposed SGF framework solves an online QP to minimally adjust the gradient flow, guaranteeing forward invariance of the safe set and preserving the equivalence between equilibria and optimization critical points. It provides conditions for feasibility, existence, and uniqueness of the closed-loop, as well as local (and in convex cases global) asymptotic convergence to optima, including regularization to interior points when global optima lie on the boundary. Simulations in convex and non-convex settings demonstrate constraint satisfaction during transients and convergence to the optimizer, highlighting practical robustness and limitations. The approach offers a principled path to safe, continuous-time optimization in dynamical systems with broad potential applications in power, traffic, and networked control.

Abstract

Feedback optimization refers to a class of methods that steer a control system to a steady state that solves an optimization problem. Despite tremendous progress on the topic, an important problem remains open: enforcing state constraints at all times. The difficulty in addressing it lies on mediating between the safety enforcement and the closed-loop stability, and ensuring the equivalence between closed-loop equilibria and the optimization problem's critical points. In this work, we present a feedback-optimization method that enforces state constraints at all times employing high-order control-barrier functions. We provide several results on the proposed controller dynamics, including well-posedness, safety guarantees, equivalence between equilibria and critical points, and local and global (in certain convex cases) asymptotic stability of optima. Various simulations illustrate our results.
Paper Structure (18 sections, 16 theorems, 94 equations, 5 figures)

This paper contains 18 sections, 16 theorems, 94 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Feedback optimization with state constraints
  4. Well-posedness and safety results
  5. Well-posedness
  6. Safety
  7. Critical points, equilibria and convergence to optima
  8. Relationship between critical points and equilibria
  9. Asymptotic stability of optima in the interior of safe set
  10. Simulations
  11. Convex case
  12. Non-convex case
  13. Discussion on item \ref{['it:weird-assumption']} of Theorem \ref{['thm:global_convergence']}
  14. Undesirable equilibria, optimum at the boundary and regularization
  15. Conclusions
  16. ...and 3 more sections

Key Result

Proposition IV.1

Let Assumption assum:differentiability, regularity and relative degree hold. Assume that, for some compact set ${\mathcal{C}}_r \subseteq \mathcal{S}_r$, for any $(x,u) \in {\mathcal{C}}_r$, there exists $q_{(x,u)}\in\mathbb{R}^m$ such that: Then, there exist $\alpha_f, \gamma_f > 0$ such that, for all $\alpha > \alpha_f$, $\gamma > \gamma_f$, the QP in eq:sgf_QP is feasible for any $(x,u)\in{\ma

Figures (5)

  • Figure 1: Implementation of \ref{['eq:closed_loop_sgf']} in a convex feedback optimization problem (cf. Section \ref{['sec:convex-case']}): (a) trajectories from different initial conditions, (b) evolution of $h$ and $b$ along the trajectory with initial condition at $x_0 = (1.55, -0.25), u_0 = (0, 0)$, and (c) comparison with the approach in YC-LC-JC-EDA:23-csl.
  • Figure 2: Implementation of \ref{['eq:closed_loop_sgf']} in a non-convex feedback optimization problem (cf. Section \ref{['sec:not-convex']}): (a) trajectories from different initial conditions, (b) evolution of $h$ and $b$ along the trajectory with initial condition at $x_0 = (1.55, -0.25), u_0 = (0, 0)$, and (c) comparison with the approach in YC-LC-JC-EDA:23-csl.
  • Figure 3: (top) Evolution of trajectories under \ref{['eq:closed_loop_sgf']} for the example in Section \ref{['sec:discussion-assumption-global-convergence']}, showing global asymptotic stability of the unique minimizer even when item \ref{['it:weird-assumption']} of Theorem \ref{['thm:global_convergence']} is not satisfied. (bottom left) Phase portrait of the corresponding closed-loop vector field. (bottom right) Zoomed in version of said phase portrait.
  • Figure 4: Evolution of various trajectories under \ref{['eq:closed_loop_sgf']} for the example in Section \ref{['sec:undesirable-eq']}. The point $(1,0)$ is a locally asymptotically stable equilibrium which does not correspond to the steady state-optimizer of \ref{['eq:steady-state_opti_problem']}.
  • Figure 5: Evolution of various trajectories under \ref{['eq:closed_loop_sgf']} for the example in Section \ref{['sec:undesirable-eq']} with the regularization term in Proposition \ref{['prop:regularization']}. The point $(0.48,0.48)$ is an asymptotically stable equilibrium and a steady state-optimizer of the regularized version of \ref{['eq:steady-state_opti_problem']}.

Theorems & Definitions (33)

  • Remark 1
  • Remark 2: Enforcing state constraints
  • Proposition IV.1: Feasibility
  • Proposition IV.2: Existence and uniqueness of solutions of \ref{['eq:closed_loop_sgf']}
  • Proposition IV.3
  • Proposition IV.4: Safety
  • Remark 3: Compactness
  • Proposition IV.5
  • Proposition V.1: Critical points of \ref{['eq:steady-state_opti_problem']} - equilibria of \ref{['eq:closed_loop_sgf']}
  • Proposition V.2: Regularization of the feedback optimization problem
  • ...and 23 more