Delay Independent Safe Control with Neural Networks: Positive Lur'e Certificates for Risk Aware Autonomy

TL;DR

The paper addresses safety certification for autonomous systems with learning-enabled controllers operating under time delays and interval uncertainty. It introduces a local FFNN sector bound and a positivity-based, delay-independent certificate that leverages Metzler/positive-system structure to certify local exponential stability in NN-in-the-loop Lur'e systems, across three risk configurations. The framework is complemented by an IQC-based verification baseline for benchmarking, showing orders-of-magnitude speedups and certification in regimes where SDP-based methods fail. This work enables scalable, potentially real-time safety guarantees for risk-aware autonomous systems in networked environments.

Abstract

We present a risk-aware safety certification method for autonomous, learning enabled control systems. Focusing on two realistic risks, state/input delays and interval matrix uncertainty, we model the neural network (NN) controller with local sector bounds and exploit positivity structure to derive linear, delay-independent certificates that guarantee local exponential stability across admissible uncertainties. To benchmark performance, we adopt and implement a state-of-the-art IQC NN verification pipeline. On representative cases, our positivity-based tests run orders of magnitude faster than SDP-based IQC while certifying regimes the latter cannot-providing scalable safety guarantees that complement risk-aware control.

Figures (6)

• Figure 1: Sector relaxations of $\phi(\nu)=\tanh(\nu)$: (i) $\beta\nu \le \tanh(\nu) \le \alpha\nu: \quad\beta=\tanh(\overline{\nu})/\overline{\nu}, \quad \alpha=\tanh(\underline{\nu})/\underline{\nu}$. (ii) $\beta\nu \le \tanh(\nu) \le \alpha\nu$$\quad \beta=\tanh(\overline{\nu})/\overline{\nu}$, $\quad \alpha=\tanh(\underline{\nu})/\underline{\nu}$ . (iii) $-\beta|\nu| \le \tanh(\nu) \le \alpha|\nu|$ with $\beta=\alpha=\tanh'(0)=1$.
• Figure 2: Linear fractional transformation (LFT) model of the NN-feedback-loop with delay and interval uncetainty.
• Figure 3: Using \ref{['eq:localsectorbound']}, the local sector bounds over the $\Gamma-$set: $y\in[0,4.5]$ are calculated as $\gamma_1=-3$ and $\gamma_2=-2.44$. As demonstrated $\gamma_1 y\le NN(y) \le \gamma_2 y$.
• Figure 4: Delay-free interval case with a trained NN controller. Four random plants are sampled from $[\underline A_0,\overline A_0]$ and $[\underline A_1,\overline A_1]$. For each plant, $100$ random initial conditions consistant with $\Gamma-$set $(Cx_0\in[0,4.5])$ are simulated; thin lines show $\|x(t)\|_2$ and the bold line marks a representative (median) trajectory. Subplot titles report the proportion converging to zero.
• Figure 5: Delay-only case with the trained NN controller. Four tiles correspond to $\tau\in\{0.2,2,8,16\}$ s. For each delay, $100$ random constant histories $x(t)\equiv x_0$ on $[-\tau,0]$ with $x_0\!\sim\!\mathrm{Unif}([-1.5,1.5]^3)$ are simulated. Thin curves plot $\|x(t)\|_2$; the bold curve marks a representative (median) trajectory. Subplot titles report the proportion converging to zero.

Theorems & Definitions (15)

• Definition 1
• Lemma 1
• Lemma 2
• Definition 2: $\Gamma-$Sector Bounded Function
• Theorem 1: Local Sector Bound for FFNN
• Lemma 3
• proof
• Theorem 2
• proof
• Theorem 3