Quantifying the Safety of Trajectories using Peak-Minimizing Control

TL;DR

This work quantifies the safety of trajectories of a dynamical system by the perturbation intensity required to render a system unsafe (crash into the unsafe set) using polynomial optimization and the moment-Sum-of-Squares hierarchy.

Abstract

This work quantifies the safety of trajectories of a dynamical system by the perturbation intensity required to render a system unsafe (crash into the unsafe set). Computation of this measure of safety is posed as a peak-minimizing optimal control problem. Convergent lower bounds on the minimal peak value of controller effort are computed using polynomial optimization and the moment-Sum-of-Squares hierarchy. The crash-safety framework is extended towards data-driven safety analysis by measuring safety as the maximum amount of data corruption required to crash into the unsafe set.

Figures (9)

• Figure 1: Two trajectories with nearly the same distance but different crash-bounds
• Figure 2: Subvalue function for Flow system \ref{['eq:flow_w1']} between degrees $1..5$.
• Figure 3: Numerical optimal control yields worst-case $Q^* \approx 0.4639$ for the half-circle $X_u$
• Figure 4: Numerical optimal control yields $Q^* \approx 0.3232$ for the moon $X_u$
• Figure 5: Subvalue map for the moon \ref{['eq:crash_moon']} on the flow system \ref{['eq:flow_w1']}

Theorems & Definitions (33)

• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Remark 1
• Theorem 3.4
• proof
• Theorem 4.1