# Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictors

## TL;DR

This paper introduces a neural operator-based predictor feedback control method for nonlinear systems with unknown delays, providing stability guarantees and demonstrating significant speed improvements over traditional methods.

## Contribution

It presents a novel neural operator approximation approach for predictor feedback in nonlinear systems with delays, including stability analysis and practical validation.

## Key findings

- Achieves 15x speedup over traditional numerical predictor methods.
- Provides semi-global practical stability guarantees.
- Validates approach on a biological system example.

## Abstract

In this work, we propose a rigorous method for implementing predictor feedback controllers in nonlinear systems with unknown and arbitrarily long actuator delays. To address the analytically intractable nature of the predictor, we approximate it using a learned neural operator mapping. This mapping is trained once, offline, and then deployed online, leveraging the fast inference capabilities of neural networks. We provide a theoretical stability analysis based on the universal approximation theorem of neural operators and the transport partial differential equation (PDE) representation of the delay. We then prove, via a Lyapunov-Krasovskii functional, semi-global practical convergence of the dynamical system dependent on the approximation error of the predictor and delay bounds. Finally, we validate our theoretical results using a biological activator/repressor system, demonstrating speedups of 15 times compared to traditional numerical methods.