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From Raw Data to Safety: Reducing Conservatism by Set Expansion

Mohammad Bajelani, Klaske van Heusden

TL;DR

This work addresses safety for learning-based controllers by removing reliance on explicit models and deriving safe terminal sets directly from data via Willems' lemma. It introduces online and offline set-expansion techniques to enlarge data-driven safe sets, reducing conservatism even with short prediction horizons and unknown time delays, by operating in an augmented input-output framework. The approach demonstrates recursive feasibility and equivalence of online and offline results in simulations on a time-delay system, and investigates robustness to lag overestimation. The findings advance practical data-driven safety filters and point to future work on noise, disturbances, and integration with data-driven predictive control, with potential impact on reliable learning-enabled systems.

Abstract

In response to safety concerns associated with learning-based algorithms, safety filters have been proposed as a modular technique. Generally, these filters heavily rely on the system's model, which is contradictory if they are intended to enhance a data-driven or end-to-end learning solution. This paper extends our previous work, a purely Data-Driven Safety Filter (DDSF) based on Willems' lemma, to an extremely short-sighted and non-conservative solution. Specifically, we propose online and offline sample-based methods to expand the safe set of DDSF and reduce its conservatism. Since this method is defined in an input-output framework, it can systematically handle both unknown and time-delay LTI systems using only one single batch of data. To evaluate its performance, we apply the proposed method to a time-delay system under various settings. The simulation results validate the effectiveness of the set expansion algorithm in generating a notably large input-output safe set, resulting in safety filters that are not conservative, even with an extremely short prediction horizon.

From Raw Data to Safety: Reducing Conservatism by Set Expansion

TL;DR

This work addresses safety for learning-based controllers by removing reliance on explicit models and deriving safe terminal sets directly from data via Willems' lemma. It introduces online and offline set-expansion techniques to enlarge data-driven safe sets, reducing conservatism even with short prediction horizons and unknown time delays, by operating in an augmented input-output framework. The approach demonstrates recursive feasibility and equivalence of online and offline results in simulations on a time-delay system, and investigates robustness to lag overestimation. The findings advance practical data-driven safety filters and point to future work on noise, disturbances, and integration with data-driven predictive control, with potential impact on reliable learning-enabled systems.

Abstract

In response to safety concerns associated with learning-based algorithms, safety filters have been proposed as a modular technique. Generally, these filters heavily rely on the system's model, which is contradictory if they are intended to enhance a data-driven or end-to-end learning solution. This paper extends our previous work, a purely Data-Driven Safety Filter (DDSF) based on Willems' lemma, to an extremely short-sighted and non-conservative solution. Specifically, we propose online and offline sample-based methods to expand the safe set of DDSF and reduce its conservatism. Since this method is defined in an input-output framework, it can systematically handle both unknown and time-delay LTI systems using only one single batch of data. To evaluate its performance, we apply the proposed method to a time-delay system under various settings. The simulation results validate the effectiveness of the set expansion algorithm in generating a notably large input-output safe set, resulting in safety filters that are not conservative, even with an extremely short prediction horizon.
Paper Structure (7 sections, 2 theorems, 15 equations, 5 figures, 2 algorithms)

This paper contains 7 sections, 2 theorems, 15 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Data-Driven Safety Filter with Sampled Terminal Sets
  4. Simulation Example
  5. First study: Sampled safe sets vs Equilibrium point safe set
  6. Second study: Overestimation of system's lag
  7. Conclusion

Key Result

theorem 1

Let $u^d$ be persistently exciting of order $L+n$, and ${\{u^d_k},{y^d_k\}}_{k=0}^{k=N_{0}-1}$ a trajectory of $G$. Then, ${\{\bar{u}_k},{{\bar{y}_k}\}}_{k=0}^{k=N_{0}-1}$ is a trajectory of $G$ if and only if there exists $\alpha \in \mathbb{R}^{N_{0}-L+1}$ such that

Figures (5)

  • Figure 1: A visualization of the final set is shown. The solid black line shows the realized system trajectory. The dashed lines are the system's backup trajectories at time $t$. The blue polytopes show the convex hulls provided by extended states' realized trajectory. (A) The final set is the system's equilibrium point as proposed in bajelani2023data. (B) The terminal set is expanded by the algorithm (\ref{['algorithm: DDSF_online']}) at times $t-2$, $t-1$, and $t$.
  • Figure 2: First case study: Input-output trajectories of system (\ref{['example: equ']}) under the policy of (\ref{['eq:DDSF']}) for three different safe sets: the system's equilibrium point, subplots (A) and (D), the sampled safe set computed by the online algorithm (\ref{['algorithm: DDSF_online']}), subplots (B) and (E), and the sampled safe set computed by the offline algorithm (\ref{['algorithm: DDSF_offline']}), subplots (C) and (F).
  • Figure 3: First case study: The sampled terminal safe set at four different times $t=0 [s]$, $t=20 [s]$, $t=50 [s]$, $t=200 [s]$ computed by the online algorithm (\ref{['algorithm: DDSF_online']}), and the sampled terminal safe set (\ref{['algorithm: DDSF_online']}) at four different iterations $k=0$, $k=200$, $k=500$, $k=1000$ computed by the offline algorithm (\ref{['algorithm: DDSF_offline']}).
  • Figure 4: Second case study: Input-output trajectories of system (\ref{['example: equ']}) under the policy of (\ref{['eq:DDSF']}) for the exact and overestimated system's lags, $T_{\text{ini}}=2$ and $T_{\text{ini}}=3$, respectively.
  • Figure : Set Expansion (Online)

Theorems & Definitions (10)

  • definition 1: System's Lag
  • definition 2: Extended State berberich2021design
  • definition 3: Persistently Excitation berberich2020data
  • theorem 1: Fundamental Lemma berberich2020robust
  • Remark 1
  • definition 4: Invariant Set
  • definition 5: Equilibrium Point
  • Remark 2
  • theorem 2: Recursive Feasibility of DDSF with sampled safe set - online algorithm
  • Remark 3