⚡ electrical engineering

Physics-Informed Neural Networks for Nonlinear Output Regulation

This paper proposes a physics-informed neural network (PINN) framework that directly approximates the solution to the regulator equations for nonlinear output regulation, enabling real-time, data-free inference of steady-state manifolds and feedforward inputs that generalize across varying exosystem parameters, as demonstrated by a helicopter synchronization task.

Original authors: Sebastiano Mengozzi, Giovanni B. Esposito, Michelangelo Bin, Andrea Acquaviva, Andrea Bartolini, Lorenzo Marconi

Published 2026-04-29

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CC BY 4.0

Original authors: Sebastiano Mengozzi, Giovanni B. Esposito, Michelangelo Bin, Andrea Acquaviva, Andrea Bartolini, Lorenzo Marconi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to land a helicopter on a floating platform in the middle of the ocean. The platform isn't just sitting still; it's bobbing up and down with the waves in a rhythmic, predictable pattern. Your goal is to keep the helicopter hovering perfectly above that moving spot, matching its every rise and fall, so that the distance between them is always zero.

This is a classic problem in control theory called Output Regulation. For simple machines, we have good math tools to solve this. But for complex, non-linear machines like helicopters (where pushing the stick a little bit doesn't always result in a little bit of movement), the math gets incredibly messy. It involves solving a set of complex equations known as Regulator Equations. Traditionally, solving these is like trying to find a needle in a haystack made of other needles; it's slow, difficult, and often impossible to do perfectly.

The New Approach: The "Physics-Savvy" AI

The authors of this paper propose a clever new way to solve these messy equations using Physics-Informed Neural Networks (PINNs).

Think of a standard AI as a student who learns by memorizing flashcards. If you show it 1,000 pictures of a cat, it learns what a cat looks like. But if you show it a picture of a dog, it might get confused.

This new approach is different. Instead of memorizing data, the AI is given the rulebook of physics (the equations that describe how the helicopter moves) and asked to learn the solution directly from those rules.

The Analogy: Imagine you want to learn how to balance a broom on your hand. A traditional method might involve recording thousands of videos of people balancing brooms and trying to copy them. The PINN method is like giving the student the laws of gravity and friction and saying, "Figure out the perfect hand movements that must happen to keep this broom balanced, based on these laws alone."

How It Works

The Goal: The researchers wanted to find a "magic map." This map would tell the helicopter exactly what to do (how much thrust to use, how much to tilt) for any possible movement of the floating platform.
The Training: They didn't feed the AI pre-calculated answers. Instead, they let the AI guess the answers and then checked: "Does your guess satisfy the laws of physics?" If the guess was wrong, the AI adjusted itself. It did this millions of times until it found a solution that perfectly satisfied the physics equations.
The Result: The AI learned a "universal operator." Instead of learning how to land on a platform moving at one specific speed, it learned the general rule for landing on a platform moving at any speed or starting from any position.

The Helicopter Test

To prove this worked, they tested it on a simulation of a helicopter landing on a platform that bobs up and down like a sine wave.

The Test: They trained the AI on a few specific wave patterns. Then, they tested it on wave patterns it had never seen before (different speeds, different starting heights).
The Outcome: The AI-controlled helicopter performed remarkably well. Even when the platform moved in ways the AI hadn't practiced, the helicopter stayed perfectly aligned with the target. The error (the distance between the helicopter and the platform) was tiny—so small it was barely visible on a graph.

Why This Matters

The paper claims this is a breakthrough because:

It's Fast: Once the AI is trained, it can calculate the perfect control moves in real-time, which is fast enough to run on actual hardware (like a drone's computer).
It's General: It doesn't just solve one specific problem; it learns the family of solutions. If the wave pattern changes slightly, the AI already knows how to handle it.
No Data Needed: It doesn't need a massive dataset of "successful landings" to learn. It learns from the math of physics itself.

The Catch

The paper notes one major limitation: This method assumes the helicopter "knows" exactly where the platform is and how it's moving at every single moment (this is called "full information"). In the real world, sensors might be imperfect or delayed, which the authors admit is a challenge for future work.

In summary: The researchers built an AI that learned to solve complex math problems by understanding the laws of physics rather than memorizing data. They proved it could guide a helicopter to land perfectly on a moving target, even when the target moved in ways the AI had never seen before.

1. Problem Statement

The paper addresses the nonlinear output regulation problem for systems where both the plant state ( $x$ ) and the exosystem state ( $w$ ) are known (full-information setting). The goal is to design a feedback control law such that the system output asymptotically tracks a reference trajectory or rejects disturbances generated by an exosystem.

Theoretical Challenge: Solving this problem requires finding two mappings: a steady-state state manifold $\pi(w)$ and a feedforward input $c(w)$ . These mappings must satisfy the Regulator Equations (REs), a system of Partial Differential Equations (PDEs) coupled with algebraic constraints (Isidori-Byrnes equations).
Limitations of Existing Methods:
- Analytical: Exact solutions are rarely available for nonlinear systems.
- Approximation (Galerkin/Taylor): These methods scale poorly with nonlinearity, are valid only in small neighborhoods of the origin, or require complex high-order expansions.
- Previous Neural Network Approaches: Earlier NN-based solutions relied on data fitting with shallow models and lacked the ability to generalize across different system parameters or initial conditions without retraining.

2. Methodology

The authors propose a Physics-Informed Neural Network (PINN) framework to solve the Regulator Equations directly, bypassing the need for labeled datasets or precomputed trajectories.

Core Concept

Instead of learning a time-evolution trajectory, the network learns a steady-state operator that maps exosystem states $w$ directly to the required plant states $\pi(w)$ and control inputs $c(w)$ .

Architecture and Training

Network Structure: A fully connected neural network ( $n_\xi$ ) with four hidden layers and $\approx 8 \times 10^4$ parameters. It takes the exosystem state $w$ as input and outputs approximations $\hat{\pi}(w)$ and $\hat{c}(w)$ .
Loss Function: The training is unsupervised. The loss function $J$ $J$ minimizes the residuals of the Regulator Equations and enforces boundary conditions:
$J = J_{PDE} + \lambda \cdot J_{BC}$
- $J_{PDE}$ : The mean squared error of the PDE residuals ( $L_s\hat{\pi} - f(\hat{\pi}, \hat{c}, w)$ ) and algebraic constraints ( $h(\hat{\pi}, w)$ ).
- $J_{BC}$ : A penalty term for boundary conditions (e.g., actuator limits).
- Automatic Differentiation: Used to compute Lie derivatives ( $L_s\pi$ ) with respect to the network input $w$ , allowing the network to "understand" the physics of the exosystem dynamics.
Generalization Strategy: The network is trained by sampling a family of dynamical systems (varying exosystem initial conditions and parameters like frequency $\Omega$ ). This forces the network to learn a general solution operator rather than a specific instance.

3. Case Study: Helicopter Vertical Landing

To validate the approach, the authors applied the method to a nonlinear benchmark: synchronizing a helicopter's vertical dynamics with a harmonically oscillating platform.

System Dynamics: Modeled as a rigid body with underactuated, multi-input nonlinear dynamics (main rotor thrust, tail rotor thrust, and rotor tilts).
Exosystem: A harmonic oscillator generating a vertical sinusoidal reference with unknown frequency and amplitude.
Regulator Equations: The problem was reduced to solving for steady-state roll/pitch angles ( $\pi_\phi, \pi_\theta$ ) and a specific control input ( $c_b$ ) that satisfy the coupled PDEs derived from Newton-Euler equations.
Constraints: The solution must respect actuator limits (e.g., $|u_b| \leq 0.3491$ ).

4. Key Results

The framework was tested on a consumer GPU with a 30-second simulation horizon.

High-Fidelity Reconstruction: The PINN successfully reconstructed the zero-error manifold with high accuracy. The learned mappings ( $\pi(w), c(w)$ ) correctly captured the oscillatory behavior induced by the exosystem.
Generalization:
- The controller maintained low tracking error across a wide range of exosystem frequencies and initial conditions, including configurations not seen during training (out-of-distribution).
- In "in-distribution" and "out-of-distribution" scenarios, the residual tracking error was approximately two orders of magnitude smaller than the reference signal.
Trim Condition Recovery: When the exosystem state was zero (no disturbance), the network correctly recovered the known "trim" condition (steady hover inputs) for the helicopter, validating the physical consistency of the solution.
Performance Metrics:
- Training: Completed in a few minutes.
- Inference: Lightweight forward pass suitable for real-time embedded deployment.
- Error: The median absolute vertical tracking error was $\approx 0.031$ m, with most errors being negligible. Divergence occurred only in extreme, highly nonlinear regions (high frequency/amplitude), attributed to the feedback stabilizer design rather than the PINN's inability to solve the REs.

5. Significance and Contributions

Data-Efficient Solving: Demonstrates that complex nonlinear PDEs in control theory can be solved without massive datasets, relying solely on the mathematical structure of the system (physics-informed).
Operator Learning: The approach shifts from solving a single instance to learning a solution operator that generalizes across a family of systems, making it robust to parameter variations.
Real-Time Feasibility: The method bridges the gap between theoretical nonlinear control (which is often computationally prohibitive) and practical implementation, offering a computationally efficient path to real-time control on embedded hardware.
Broad Applicability: While tested on a helicopter, the methodology is applicable to any nonlinear system where the output regulation problem admits a solution, provided the Regulator Equations can be formulated.

6. Limitations and Future Work

Full-Information Assumption: The current method assumes the exosystem state $w$ is available for feedback. Future work aims to extend this to partial-information settings (estimating $w$ ).
Feedback Stabilization: The paper focuses on solving the REs (feedforward). The stability of the closed-loop system relies on a separate feedback gain, which was tuned empirically. Future work could integrate advanced stabilizing feedback designs to enlarge the feasible operating region.
Plant Uncertainty: Extending the framework to handle families of plants with parametric uncertainty (randomizing plant parameters during training) is a proposed direction.