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Stability Mechanisms for Predictive Safety Filters

Elias Milios, Kim Peter Wabersich, Felix Berkel, Lukas Schwenkel

TL;DR

The proposed framework extends wellknown stability results from model predictive control (MPC) theory while supporting commonly used design techniques, and is demonstrated using an automotive advanced driver assistance scenario, involving a reference trajectory stabilization problem.

Abstract

Predictive safety filters enable the integration of potentially unsafe learning-based control approaches and humans into safety-critical systems. In addition to simple constraint satisfaction, many control problems involve additional stability requirements that may vary depending on the specific use case or environmental context. In this work, we address this problem by augmenting predictive safety filters with stability guarantees, ranging from bounded convergence to uniform asymptotic stability. The proposed framework extends well-known stability results from model predictive control (MPC) theory while supporting commonly used design techniques. As a result, straightforward extensions to dynamic trajectory tracking problems can be easily adapted, as outlined in this article. The practicality of the framework is demonstrated using an automotive advanced driver assistance scenario, involving a reference trajectory stabilization problem.

Stability Mechanisms for Predictive Safety Filters

TL;DR

The proposed framework extends wellknown stability results from model predictive control (MPC) theory while supporting commonly used design techniques, and is demonstrated using an automotive advanced driver assistance scenario, involving a reference trajectory stabilization problem.

Abstract

Predictive safety filters enable the integration of potentially unsafe learning-based control approaches and humans into safety-critical systems. In addition to simple constraint satisfaction, many control problems involve additional stability requirements that may vary depending on the specific use case or environmental context. In this work, we address this problem by augmenting predictive safety filters with stability guarantees, ranging from bounded convergence to uniform asymptotic stability. The proposed framework extends well-known stability results from model predictive control (MPC) theory while supporting commonly used design techniques. As a result, straightforward extensions to dynamic trajectory tracking problems can be easily adapted, as outlined in this article. The practicality of the framework is demonstrated using an automotive advanced driver assistance scenario, involving a reference trajectory stabilization problem.
Paper Structure (10 sections, 5 theorems, 42 equations, 2 figures, 1 algorithm)

This paper contains 10 sections, 5 theorems, 42 equations, 2 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries and Problem Statement
  3. Stabilizing Predictive Safety Filters
  4. Theoretical Results and Design
  5. Bounded convergence
  6. Uniform asymptotic stability
  7. Performance
  8. Extension to Trajectory Stabilization
  9. Numerical Example
  10. Conclusion

Key Result

Theorem 1

Let Assumptions ass:cont_stagecost - ass:init_stab_bound hold and suppose that $\mathcal{X}$ contains a neighborhood of the origin. Then application of Algorithm algo:nom_stabfilter_nom with $\mathbb{P} := \mathbb{P}(k)$ yields bounded convergence according to Definition defi:stability, (1) with res

Figures (2)

  • Figure 1: Illustration of the stability mechanisms considered in this work. Left: Safety in terms of constraint satisfaction. Center: Safe stability in terms of a bounded and converging closed-loop state evolution, which satisfies the constraint. Right: Safe stability in terms of an uniformly asymptotically stable closed-loop state evolution, which satisfies the constraints.
  • Figure 2: Numerical simulation of the stable safety filter to enhance a human driver with safety and stability guarantees during, e.g., obstacle avoidance. Left: Lateral position resulting from the desired input (black dotted) and the filtered closed-loop input (solid colorized), where the colors indicate the value of the current closed-loop stability cost. The reference is displayed by the black dashed line. Center: Desired steering angle (black dotted) and filtered closed-loop steering angle (solid colorized), where the colors indicate the value of the current input matching error. Right: Closed-loop performance resulting from the desired input (black dotted) and from the filtered closed-loop input (solid colorized), where the colors indicate the current value of the relaxation parameter. The stability bound $H_\mathrm{B}$ is displayed by the black dashed line.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 1
  • Proposition 2
  • proof
  • Remark 1
  • Definition 2
  • ...and 2 more