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$\mathcal{L}_1$-Certified Distributionally Robust Planning for Safety-Constrained Adaptive Control

Astghik Hakobyan, Amaras Nazarians, Aditya Gahlawat, Naira Hovakimyan, Ilya Kolmanovsky

Abstract

Safe operation of autonomous systems requires robustness to both model uncertainty and uncertainty in the environment. We propose a hierarchical framework for stochastic nonlinear systems that integrates distributionally robust model predictive control (DR-MPC) with $\mathcal{L}_1$-adaptive control. The key idea is to use the $\mathcal{L}_1$ adaptive controller's online distributional certificates that bound the Wasserstein distance between nominal and true state distributions, thereby certifying the ambiguity sets used for planning without requiring distribution samples. Environment uncertainty is captured via data-driven ambiguity sets constructed from finite samples. These are incorporated into a DR-MPC planner enforcing distributionally robust chance constraints over a receding horizon. Using Wasserstein duality, the resulting problem admits tractable reformulations and a sample-based implementation. We show theoretically and via numerical experimentation that our framework ensures certifiable safety in the presence of simultaneous system and environment uncertainties.

$\mathcal{L}_1$-Certified Distributionally Robust Planning for Safety-Constrained Adaptive Control

Abstract

Safe operation of autonomous systems requires robustness to both model uncertainty and uncertainty in the environment. We propose a hierarchical framework for stochastic nonlinear systems that integrates distributionally robust model predictive control (DR-MPC) with -adaptive control. The key idea is to use the adaptive controller's online distributional certificates that bound the Wasserstein distance between nominal and true state distributions, thereby certifying the ambiguity sets used for planning without requiring distribution samples. Environment uncertainty is captured via data-driven ambiguity sets constructed from finite samples. These are incorporated into a DR-MPC planner enforcing distributionally robust chance constraints over a receding horizon. Using Wasserstein duality, the resulting problem admits tractable reformulations and a sample-based implementation. We show theoretically and via numerical experimentation that our framework ensures certifiable safety in the presence of simultaneous system and environment uncertainties.

Paper Structure

This paper contains 22 sections, 6 theorems, 54 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Uncertain System Dynamics
  5. Environment and Safety Specification
  6. Wasserstein Ambiguity Set
  7. Distributionally Robust Safety with $\mathcal{L}_1$ Certificate
  8. Distributionally Robust Planner
  9. Control Architecture
  10. Distributional Certificates and Closed-Loop Safety
  11. Tractable Reformulation of the DR-MPC Problem
  12. CVaR Reformulation of the DR-CC
  13. Sample-based DR-MPC Problem
  14. Experiments
  15. Experiment Setup
  16. ...and 7 more sections

Key Result

Theorem 1

Suppose ass:true_sysass:nom_stab hold, and let the control input be given by eqn:aug_cont. In addition, assume $\bar{\xi}_0 \sim \mathcal{P}_{2 p^\star}(\mathbb{R}^{n})$ for some $p^\star \in \mathbb{N}$, $p^\star\geq 1$. Then there exists a known constant $\rho_x>0$ such that $\mathbb X_t \in \math

Figures (2)

  • Figure 1: Overview of the proposed framework. A DR-MPC planner generates nominal trajectories satisfying safety constraints, which are tracked by a baseline controller augmented with an $\mathcal{L}_1$-adaptive term for fast uncertainty compensation. The adaptive layer certifies that the true state distribution remains within a prescribed Wasserstein tube around the nominal distribution.
  • Figure 2: Closed-loop trajectories for the figure-eight scenario at different times. Without adaptation, the controller deviates from the reference due to model mismatch, whereas the $\mathcal{L}_1$-augmented controller closely tracks the nominal trajectory and maintains safe obstacle avoidance.

Theorems & Definitions (11)

  • Remark 1
  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Remark 2
  • Lemma 2
  • Proposition 1
  • proof
  • proof
  • Lemma 3
  • ...and 1 more