⚡ electrical engineering

Data-Driven Frequency-Selective Output Regulation of Nonlinear Systems under Almost Periodic Exosignals

This paper proposes a data-driven frequency-selective output regulation framework for unknown nonlinear systems driven by almost periodic exosignals, which utilizes a p-copy internal model and a noise-robust semidefinite program to eliminate specific Fourier-Bohr error components and bound residual energy without requiring system identification or exosignal amplitude/phase measurements.

Original authors: Yifei Li, Wenjie Liu, Gang Wang, Lihua Xie

Published 2026-06-12

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CC BY 4.0

Original authors: Yifei Li, Wenjie Liu, Gang Wang, Lihua Xie

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 keep a drone hovering perfectly still in the air. But there's a catch: the drone is carrying a heavy package hanging from a rope. Because of physics, that package swings back and forth like a pendulum. This swinging creates a rhythmic, wobbly force that pushes the drone up, down, and tilts it.

Your goal is to tell the drone's motors exactly how to push back to cancel out these wobbles and keep the drone steady.

The Old Way: "Know Everything to Fix Everything"

Traditionally, to solve this, engineers would need to build a perfect mathematical map of the drone, the rope, the wind, and the swinging package. They would have to solve incredibly complex equations (called "Nonlinear Regulator Equations") to figure out exactly how to counteract the swing.

The Problem: In the real world, you rarely know the exact weight of the package, the exact length of the rope, or the exact aerodynamics of the drone. If your map is even slightly wrong, the math fails, and the drone crashes or wobbles uncontrollably.

The New Way: "Learn from the Rhythm, Not the Map"

This paper proposes a smarter, "data-driven" approach. Instead of trying to map the entire drone and rope system, the researchers teach the drone to listen to the rhythm of the problem.

Here is how they do it, using a few simple analogies:

1. The "Tuning Fork" (The Internal Model)

Imagine the swinging package is like a musical note being played over and over. The drone doesn't need to know why the package is swinging or how heavy it is. It just needs to know the frequency (the speed) of the swing.

The researchers built a special "tuning fork" inside the drone's computer. This tuning fork is programmed to vibrate at the exact same speed as the swinging package (and any other known rhythmic disturbances).

How it works: When the package swings, the tuning fork inside the drone starts "humming" in sync. The drone uses this hum to predict exactly when the next push will come and pushes back before the wobble gets bad. It cancels out the specific rhythm of the swing.

2. The "Black Box" Learner (Data-Driven Design)

Usually, to tune this "tuning fork" correctly, you need to know the drone's engine specs. But this paper says, "We don't need to know the engine specs!"

Instead, they fly the drone around, let it get pushed by the wind and the swinging package, and record what happened. They take this raw data (like a video of the drone's movements) and run it through a special mathematical filter (a "Semidefinite Program").

The Magic: This filter figures out the perfect controller settings directly from the data, without ever needing to identify the drone's weight or the rope's length. It's like learning to ride a bike by feeling the balance, rather than studying the physics of gears and chains.

3. The "Noise-Canceling" Goal

The researchers admit they can't make the drone perfectly still 100% of the time. There might be tiny, random jitters or weird frequencies they didn't program the tuning fork to catch.

The Promise: Their method guarantees that the specific, rhythmic wobbles (the ones matching the package swing) will disappear completely. The remaining tiny wobbles are mathematically proven to be very small and harmless. It's like noise-canceling headphones: they might not block out a sudden shout, but they will completely silence the constant hum of an airplane engine.

The Result

They tested this on a computer simulation and a physics-based robot environment (a drone with a hanging weight).

Without their method: The drone wobbled wildly when the package swung.
With their method: The drone stayed remarkably steady. The rhythmic shaking caused by the swinging package was almost entirely eliminated, even though the computer didn't know the exact weight of the package or the length of the rope.

In Summary

This paper presents a way to control complex, wobbly machines (like drones carrying swinging loads) without needing a perfect manual or a complete understanding of the machine's physics. By embedding a "rhythm detector" (the internal model) and learning directly from trial-and-error data, the system can cancel out specific, annoying vibrations and keep the machine stable, even in the face of unknown disturbances.

Technical Summary: Data-Driven Frequency-Selective Output Regulation of Nonlinear Systems under Almost Periodic Exosignals

Problem Formulation
This paper addresses the output regulation problem for a class of unknown continuous-time nonlinear systems driven by almost periodic exosignals. The plant dynamics are assumed to be linearly parameterized over a prescribed dictionary of continuously differentiable basis functions, yet all coefficient matrices (system, input, output, and exosignal channels) remain unknown. Furthermore, the exosignal amplitudes and phases are unmeasurable.

In classical nonlinear output regulation, achieving exact asymptotic regulation requires solving nonlinear regulator equations (NOREs), a task that is generally intractable even with known models and impossible in data-driven settings without identification. While existing data-driven methods often rely on periodic signals or restrictive structural assumptions, this work targets a broader class of exosignals where frequencies are rationally independent, resulting in almost periodic (but not periodic) trajectories. The objective is formulated as frequency-selective regulation: rather than driving the error to zero, the controller must eliminate the Fourier–Bohr coefficients of the steady-state regulation error at the specific frequencies embedded in the exosystem, while explicitly bounding the energy of the residual error components.

Methodology
The proposed solution integrates the internal model principle with direct data-driven control synthesis, avoiding explicit system identification.

Internal Model Structure: A $p$ -copy internal model is embedded within a dynamic controller. This model reproduces the constant mode and the selected sinusoidal modes corresponding to the known exosystem frequencies. The controller state $z$ evolves according to $\dot{z} = \Phi z + \Gamma e$ , where $\Phi$ and $\Gamma$ are constructed to embed the specific frequency set $\{0, \sigma_1, \dots, \sigma_s\}$ .
Augmented System Representation: The plant and the internal model are combined into an augmented nonlinear system. The unknown dynamics of this augmented system are represented directly using offline measured input-state-error data.
Data-Driven Synthesis via SDP: A robust semidefinite program (SDP) is derived to synthesize the feedback gain matrix $K$ directly from noisy data. This formulation utilizes a "noise-robust" constraint that accounts for approximation errors and measurement noise via a bounded disturbance set. The SDP enforces an incremental dissipation inequality, which guarantees that the closed-loop vector field is exponentially contractive on a prescribed operating set.
Stability and Regulation Analysis:
- Contraction Theory: The exponential contraction of the closed-loop system ensures the existence, uniqueness, and exponential attractivity of a bounded entire solution.
- Almost Periodicity: Since the exosignal is almost periodic and the system is input-to-state convergent, the unique attracting steady-state trajectory is proven to be almost periodic.
- Fourier–Bohr Analysis: By analyzing the steady-state internal model dynamics, it is shown that the boundedness of the internal model state forces the Fourier–Bohr coefficients of the regulation error at the embedded frequencies to vanish.
- Residual Energy Bound: For the unmodeled spectral components, a Parseval-type argument establishes a time-averaged energy bound. This bound depends on the derivative of the error and the smallest magnitude of the unmodeled frequencies.

Key Contributions

Problem Formulation: The paper formulates a frequency-selective data-driven output regulation problem for nonlinear systems under finite-spectrum almost periodic exosignals, characterizing regulation via the vanishing of specific Fourier–Bohr coefficients rather than exact asymptotic convergence.
Direct Synthesis Method: A direct synthesis method is developed for an internal-model-based nonlinear regulator. It constructs a feedback gain via a robust SDP without identifying system matrices or measuring exosignal amplitudes/phases.
Theoretical Guarantees: The work provides a complete steady-state analysis proving that the closed-loop system admits a unique, exponentially attracting, almost periodic trajectory. It rigorously demonstrates the elimination of embedded-frequency error components and provides a quantitative bound on the residual error energy.
Validation: The approach is validated through numerical simulations and physics-based experiments in a gym-pybullet-drones environment using a quadrotor with a cable-suspended payload.

Results
Numerical and physics-based simulations on a quadrotor carrying a suspended payload demonstrate the effectiveness of the proposed controller. In the presence of payload-induced oscillatory disturbances (which generate almost periodic signals due to unknown initial conditions), the proposed controller successfully drives the altitude and pitch angle errors to a small neighborhood of the origin.
Comparative experiments show that a data-driven controller without the internal model fails to suppress these oscillatory disturbances, resulting in significant steady-state errors. In contrast, the proposed internal-model-based controller substantially attenuates these oscillations. The results confirm that the controller eliminates the dominant frequency components of the error while maintaining bounded residual oscillations consistent with the theoretical energy bounds.

Significance
The paper claims significance by bridging the gap between data-driven control and nonlinear output regulation for general almost periodic signals. Unlike prior works that are limited to linear systems, periodic signals, or require local linearization, this approach handles nonlinear systems with unknown parameters and rationally independent frequencies directly from data. By leveraging contraction theory and Fourier–Bohr analysis, it offers a rigorous alternative to solving NOREs, providing explicit performance guarantees (vanishing coefficients and energy bounds) in a setting where exact regulation is typically unattainable. The method is particularly relevant for applications like aerial transportation with suspended loads, where disturbances are inherently oscillatory and model parameters are uncertain.