Physics-Informed Neural Networks for Nonlinear Output Regulation

TL;DR

The paper tackles the full-information nonlinear output regulation problem by formulating the regulator equations $L_s\pi(w)=f(\pi(w),c(w),w)$ and $0=h(\pi(w),w)$ and solving them with a physics-informed neural network (PINN). The proposed approach learns a steady-state operator that maps exosystem states $w$ to the corresponding plant state and feedforward input $(\pi(w),c(w))$ by minimizing residuals, without requiring labeled data or trajectories, and generalizes across exosystem variations. The method is demonstrated on a nonlinear helicopter vertical-landing benchmark, where the PINN reconstructs the zero-error manifold with high fidelity and sustains regulation for unseen exosystem configurations, enabling real-time inference on embedded platforms. This work integrates the theoretical regulator structure with learning-based solvers, offering a data-efficient pathway to nonlinear output regulation across a broad class of systems that admit regulator solutions.

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 $π(w)$ and a feedforward input $c(w)$ that render such manifold invariant. The pair $(π(w), c(w))$ 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 $π(w)$ 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.

Figures (9)

• Figure 1: Loss landscape of the trained PINN evaluated over pairs of $(w_1, w_2)$, including both seen and unseen states. The surface is colored according to the loss value.
• Figure 2: Control structure for the vertical landing problem. The exosystem $\dot{\omega}$ generates the disturbance acting on the plant $\dot{x}$, while the goal is to maintain $e=0$ through the feedforward action and steady plant states produced by the PINN.
• Figure 3: Grid of experiments over different exosystem configurations. The $x$-axis reports the initial condition $w_1(0)$, while the $y$-axis reports values of $\Omega$. Each cell is colored by the mean absolute vertical tracking error over a $30$ s experiment. White dots highlight training configurations, while crosses indicate simulations that diverged due to instability and/or infeasibility.
• Figure 4: Histogram of the vertical tracking error over all grid experiments. The mean and median values are indicated by red and black dashed vertical lines, respectively.
• Figure 5: Trajectories of the exosystem reference signal and the vertical tracking error over time in the exosystem configuration $(w_1(0),\Omega) = (5,1)$, which was seen during training. The bottom plot shows a $100\times$ magnification of the error trajectory to highlight the residual ripple.