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Feasibility Analysis and Regularity Characterization of Distributionally Robust Safe Stabilizing Controllers

Pol Mestres, Kehan Long, Nikolay Atanasov, Jorge Cortés

TL;DR

This work addresses safe stabilization of control-affine systems under distributional uncertainty by formulating distributionally robust control barrier and Lyapunov constraints as a convex SOCP, solvable from finite samples. It develops a necessary feasibility condition and two sufficient conditions with different data- and sample-dependence, analyzes their computational complexity, and demonstrates substantial speedups over direct SOCP solve-time. A key result shows the resulting controller mapping is point-Lipschitz under mild conditions, ensuring continuity and existence of the closed-loop solution. Simulations on a unicycle model validate the theoretical feasibility guarantees and highlight the practical efficiency of the proposed checks for online use.

Abstract

This paper studies the well-posedness and regularity of safe stabilizing optimization-based controllers for control-affine systems in the presence of model uncertainty. When the system dynamics contain unknown parameters, a finite set of samples can be used to formulate distributionally robust versions of control barrier function and control Lyapunov function constraints. Control synthesis with such distributionally robust constraints can be achieved by solving a (convex) second-order cone program (SOCP). We provide one necessary and two sufficient conditions to check the feasibility of such optimization problems, characterize their computational complexity and numerically show that they are significantly faster to check than direct use of SOCP solvers. Finally, we also analyze the regularity of the resulting control laws.

Feasibility Analysis and Regularity Characterization of Distributionally Robust Safe Stabilizing Controllers

TL;DR

This work addresses safe stabilization of control-affine systems under distributional uncertainty by formulating distributionally robust control barrier and Lyapunov constraints as a convex SOCP, solvable from finite samples. It develops a necessary feasibility condition and two sufficient conditions with different data- and sample-dependence, analyzes their computational complexity, and demonstrates substantial speedups over direct SOCP solve-time. A key result shows the resulting controller mapping is point-Lipschitz under mild conditions, ensuring continuity and existence of the closed-loop solution. Simulations on a unicycle model validate the theoretical feasibility guarantees and highlight the practical efficiency of the proposed checks for online use.

Abstract

This paper studies the well-posedness and regularity of safe stabilizing optimization-based controllers for control-affine systems in the presence of model uncertainty. When the system dynamics contain unknown parameters, a finite set of samples can be used to formulate distributionally robust versions of control barrier function and control Lyapunov function constraints. Control synthesis with such distributionally robust constraints can be achieved by solving a (convex) second-order cone program (SOCP). We provide one necessary and two sufficient conditions to check the feasibility of such optimization problems, characterize their computational complexity and numerically show that they are significantly faster to check than direct use of SOCP solvers. Finally, we also analyze the regularity of the resulting control laws.
Paper Structure (9 sections, 4 theorems, 25 equations, 2 figures, 1 table)

This paper contains 9 sections, 4 theorems, 25 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distributionally Robust Chance Constrained Programs
  4. Distributionally Robust Safety and Stability
  5. Problem Statement
  6. Feasibility Analysis
  7. Regularity Analysis
  8. Simulations
  9. Conclusions

Key Result

Proposition IV.1

(Necessary condition for feasibility of DRO-SOCP): Let $\epsilon\in (0,\frac{1}{N}]$ and $r>0$. For $x\in\mathbb{R}^n$, let for $l\in[M]$ and $i\in[N]$. Let $\bar{\lambda}_{l,i}(x)$ be the minimum eigenvalue of $\bar{F}_{l,i}(x)$ and suppose $\bar{Q}_{l}(x)\bar{Q}_{l}(x)^T$ is invertible for all $l\in[M]$. If eq:dro-clf-cbf-socp is feasible, then for each $l\in[M]$, there exists $i\in[N]$ such th

Figures (2)

  • Figure 1: (a): Time complexity comparison between necessary condition verification (cf. Proposition \ref{['prop:nec-cond-feasibility-1-constraint']}) and SOCP solver along the robot trajectory. The label "undetermined" means that the necessary condition is met, from which we may not know if the problem is feasible or not. The label "precision" represents the ratio of instances where the necessary condition indicated infeasibility against the total number of instances where the SOCP was actually infeasible. (b): Time complexity of necessary condition verification and SOCP solver with increasing uncertainty samples (constraints). (c): Log-scaled time complexity comparison of two sufficient conditions (cf. Proposition \ref{['prop:suff-cond-one-constr-data']} and Proposition \ref{['prop:suff-cond-feas-dro-socp']}) with the SOCP solver along the robot trajectory.
  • Figure 2: Time complexity comparison of necessary (cf. Proposition \ref{['prop:nec-cond-feasibility-1-constraint']}) and sufficient (cf. Proposition \ref{['prop:suff-cond-feas-dro-socp']}) conditions.

Theorems & Definitions (14)

  • Remark II.1
  • Remark II.2
  • Proposition IV.1
  • proof
  • Proposition IV.2
  • proof
  • Remark IV.3
  • Proposition IV.4
  • proof
  • Remark IV.5
  • ...and 4 more