Physics-Informed Neural Networks for Nonlinear Output Regulation

TL;DR

This paper introduces a physics-informed neural network approach to solve regulator PDEs for nonlinear output regulation, enabling real-time, generalizable control solutions without precomputed data.

Contribution

It presents a novel PINN-based method to directly approximate regulator solutions, improving real-time inference and generalization in nonlinear output regulation tasks.

Findings

PINN accurately reconstructs the zero-error manifold.

Method generalizes across exosystem variations.

High regulation performance demonstrated on helicopter dynamics.

Abstract

This work addresses the full-information output regulation problem for nonlinear systems, assuming the states of both the plant and the exosystem are known. In this setting, perfect tracking or rejection is achieved by constructing a zero-regulation-error manifold $\pi(w)$ and a feedforward input $c(w)$ that render such manifold invariant. The pair $(\pi(w), c(w))$ is characterized by the regulator equations, i.e., a system of PDEs with an algebraic constraint. We focus on accurately solving the regulator equations introducing a physics-informed neural network (PINN) approach that directly approximates $\pi(w)$ and $c(w)$ by minimizing the residuals under boundary and feasibility conditions, without requiring precomputed trajectories or labeled data. The learned operator maps exosystem states to steady state plant states and inputs, enables real-time inference and, critically, generalizes across families of the exosystem with varying initial conditions and parameters. The framework is validated on a regulation task that synchronizes a helicopter's vertical dynamics with a harmonically oscillating platform. The resulting PINN-based solver reconstructs the zero-error manifold with high fidelity and sustains regulation performance under exosystem variations, highlighting the potential of learning-enabled solvers for nonlinear output regulation. The proposed approach is broadly applicable to nonlinear systems that admit a solution to the output regulation problem.