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

1. Problem Statement

The paper addresses the fundamental challenge of ensuring certifiably safe operation for autonomous systems operating under simultaneous model and environment uncertainties.

• Model Uncertainty: Arises from imperfect knowledge of system dynamics (epistemic uncertainty), such as unmodeled nonlinearities or parameter variations.
• Environment Uncertainty: Arises from exogenous random disturbances and unknown states of dynamic obstacles (aleatoric uncertainty).
• The Core Conflict:
  • Robust Control (e.g., $L_1$-AC): Guarantees stability against bounded model uncertainty but often relies on worst-case analysis, leading to overly conservative behavior.
  • Data-Driven/Probabilistic Control (e.g., DR-MPC): Offers better performance by analyzing distributions but struggles with safety guarantees when the true distribution is unknown or shifts. Specifically, standard Distributionally Robust Model Predictive Control (DR-MPC) requires constructing "ambiguity sets" (sets of possible distributions). While environment distributions can be estimated from data, system state distributions change over time, and at any single time step, only a single sample (the current state) is available. This makes it impossible to empirically construct a valid ambiguity set for the system dynamics, leaving safety guarantees unproven.

2. Methodology

The authors propose a hierarchical framework that integrates Distributionally Robust Model Predictive Control (DR-MPC) with $L_1$-Adaptive Control ($L_1$-AC) to close the loop between planning and adaptation.

A. System Formulation

• Dynamics: The system is modeled as a stochastic nonlinear system with matched uncertainties (drift and diffusion terms entering through the control channel).
• Safety Specification: Safety is defined via a Chance Constraint (CC): $P(Y_t \in \mathcal{C}) \leq \beta$, where $Y_t = (X_t, Z_t)$ is the lifted state (system + environment) and $\mathcal{C}$ is the unsafe set.
• Distributionally Robust CC (DR-CC): To handle distributional uncertainty, the constraint is enforced over a Wasserstein ambiguity set $\mathcal{B}_q(\bar{Y}_t, \rho_y)$, ensuring the worst-case probability of violation is bounded by $\beta$.

B. The Hierarchical Architecture

1. Upper Layer: DR-MPC Planner

   • Generates a nominal trajectory based on a nominal model.
   • Enforces DR-CCs over a joint Wasserstein ambiguity set for the system and environment.
   • The ambiguity radius $\rho_y$ is the sum of the system radius ($\rho_x$) and environment radius ($\rho_z$).

2. Lower Layer: $L_1$-Adaptive Controller ($L_1$-AC)

   • Tracks the nominal trajectory generated by the planner.
   • Compensates for unknown matched uncertainties in real-time.
   • Key Innovation: It provides online distributional certificates. Theorem 1 proves that the $L_1$-AC guarantees the true state distribution remains within a Wasserstein ball of radius $\rho_x$ around the nominal distribution, without requiring distribution samples.

3. Environment Handling

   • The environment uncertainty (e.g., obstacle positions) is handled via finite-sample concentration bounds (Lemma 1).
   • An empirical ambiguity radius $\rho_z$ is derived from $N$ independent samples of the environment state.

C. Tractable Reformulation

The DR-MPC problem involves intractable infinite-dimensional constraints. The authors use Wasserstein duality to reformulate the DR-CC:

• The worst-case probability constraint is converted into a Conditional Value-at-Risk (CVaR) constraint under the nominal distribution.
• The CVaR constraint is approximated using Monte Carlo sampling (Algorithm 1), making the problem solvable in real-time.
• The distance to the unsafe set is computed using geometric properties (e.g., for circular or polytopic obstacles), allowing for efficient convex formulations.

3. Key Contributions

1. Hierarchical Framework: A novel coupling of DR-MPC and $L_1$-AC where the adaptive controller provides the necessary "system-side" ambiguity certificate, enabling the planner to function without system distribution samples.
2. Closed-Loop Certification: A theoretical proof (Theorem 2) that the combination of the $L_1$-AC certificate and empirical environment bounds guarantees stagewise safety for the true closed-loop system under simultaneous uncertainties.
3. Tractable Implementation: A geometry-agnostic reformulation of DR-CCs using CVaR and Wasserstein duality, enabling a sample-based, computationally efficient implementation.

4. Experimental Results

The framework was evaluated via simulations on a 2D robot navigating dynamic obstacles with unmodeled nonlinear dynamics. Three methods were compared:

1. DR-MPC (Nominal): Plans with environment uncertainty but uses the nominal model (no adaptation).
2. DR-MPC (True Dynamics): Applies the nominal controller to the true uncertain system (no adaptation).
3. DR-MPC + $L_1$ (Proposed): The full hierarchical framework.

Key Findings (Table 1):

• Failure Rates:
  • DR-MPC (True Dynamics) failed catastrophically in several scenarios (e.g., 86% failure in the "Circle" scenario) due to model mismatch causing the system to deviate from the planned safe trajectory.
  • DR-MPC (Nominal) had 0% failure but suffered from high tracking costs and conservative behavior.
  • Proposed Method achieved 0% failure across all tested scenarios (Figure-eight, Circle, Lissajous, Spiral), effectively eliminating collisions.
• Performance: The proposed method maintained tracking performance close to the nominal baseline while ensuring safety, demonstrating that the $L_1$ layer successfully compensated for model uncertainty without sacrificing the efficiency of the distributional planner.

5. Significance

This work bridges a critical gap in autonomous control theory:

• Solving the "Single Sample" Problem: It resolves the theoretical limitation of DR-MPC where system state distributions cannot be sampled in real-time. By using $L_1$-AC to provide a guaranteed bound on the distributional deviation, it allows for rigorous safety certification without needing historical data of the system's state distribution.
• Unified Safety: It provides a unified approach to handle both epistemic uncertainty (via $L_1$-AC) and aleatoric uncertainty (via DR-MPC), which is essential for deploying autonomous systems in complex, real-world environments.
• Practical Applicability: The use of sample-based CVaR reformulations makes the approach computationally feasible for real-time implementation, moving distributionally robust control from theoretical abstraction to practical application.

In conclusion, the paper demonstrates that integrating adaptive control with distributionally robust planning creates a robust, certifiable, and high-performance control architecture capable of operating safely in the face of simultaneous model and environmental uncertainties.