PaperlayerView exact revision on arXiv

Source-linked AI summary

Data-driven design of steady-state feedforward inputs for nonlinear systems under partial measurement

Sathya Aswath Govind Raju, Berk Altiner, Zongxuan Sun, Arunava Banerjee, Rajasree Sarkar, Kenneth Kim, Chol-Bum Mike Kweon

arXiv:2609.09450v1eess.SYcs.SY

TL;DR

Nonlinear output regulation is difficult when plant dynamics are unknown and measurements are partial, while model-based design requires an accurate plant description and regulator-equation solutions. This paper learns a frequency-domain feedforward generator from input-output data; simulations demonstrate recovery of the analytical harmonic structure in a friction example and accurate tracking on an electrohydraulic actuator.

  • Problem

    Existing nonlinear regulation methods often assume known nonlinear structures, full-state measurements, or a prespecified internal model, leaving feedforward-generator design for unknown plants under partial measurement unaddressed.

    Source: §1 · p. 2

  • Method

    The framework learns input-output behavior, represents histories in the frequency domain, and trains a neural network to generate steady-state feedforward inputs from reference amplitude and phase.

    Source: §1 · pp. 2–3

  • Results

    The friction example recovered the analytical regulator-equation solution's harmonic structure, while the electrohydraulic actuator tracked multi-harmonic and square-wave references accurately.

    Source: §5 · p. 12

  • Takeaways & Limitations

    The method is presented as a data-driven realization of the steady-state generator that avoids prior plant-model knowledge and explicit regulator-equation solutions.

    Source: §5 · pp. 11–12

    Training-data requirements grow with retained harmonics and can become prohibitive for high-bandwidth tracking; incorporating feedback into the feedforward design remains open.

    Source: §5 · p. 13

Abstract

from arXiv · show

Designing trajectory tracking controllers for nonlinear systems remains a significant challenge, traditionally requiring precise mathematical models and complex analytical derivations. While the Internal Model Principle (IMP) provides a robust theoretical foundation for such problems, its application is often hindered by model uncertainty and the inherent complexity of nonlinear controller synthesis. This work proposes a practical data-driven control framework that bypasses the need for an explicit first-principles model by utilizing raw input-output data. By integrating fundamental results from IMP theory with nonlinear system analysis, the proposed approach improves design tractability. The framework's efficacy is validated through numerical simulations on two distinct nonlinear platforms: a mechanical load with a nonlinear friction term and an electrohydraulic actuator system.

1 INTRODUCTION

The introduction frames nonlinear output regulation as difficult because internal models must capture plant-induced frequency distortions, while existing approaches often rely on explicit models or restrictive system structures. The paper proposes a data-driven feedforward generator for unknown nonlinear systems using partial input-output measurements.

1 INTRODUCTION

  • Nonlinear plant distortions create additional harmonics for single-frequency references and intermodulation components for multi-frequency references, which internal models must reproduce for regulation.The internal model acts as a steady-state generator of the ideal feedforward input that drives regulation error to zero when properly initialized.Source: §1 · p. 2
  • Existing nonlinear regulation methods often assume known nonlinear basis functions, full-state measurements, or a prespecified internal-model structure, limiting their fit to partially measured, poorly characterized plants.The stated gap is data-driven design of the steady-state feedforward generator under unknown plant dynamics and partial measurement.Source: §1 · p. 2
  • The proposed framework is organized into data-driven modeling and steady-state feedforward-generator design stages, with the modeling stage following.The introduction also previews numerical results and the paper's conclusion.Source: §1 · pp. 2–3
  • The framework learns input-output behavior from data, transforms time-domain histories into frequency-domain features, and trains a neural network to generate feedforward inputs from reference amplitude and phase.The frequency-domain representation is used to address the dimensionality burden of high-rate time-domain histories and parameterize feedforward inputs.Source: §1 · p. 3
  • The paper positions its approach as a data-driven realization of classical internal-model design, recovering finite-bandwidth spectral structure without a plant model or regulator-equation solution.Classical polynomial regulator solutions produce harmonics at integer multiples of the exosystem's fundamental frequency, with amplitudes determined by plant nonlinearity.Source: §1 · p. 3

2 PRELIMINARIES AND PROBLEM DEFINITION

The preliminaries introduce a nonlinear-system setting for output regulation, where an exogenous signal supplies both the tracking reference and disturbance. They then state the controller-based regulation formulation.

2.1 Output Regulation for Nonlinear Systems

  • The nonlinear-system setup distinguishes state, control input, and regulated output, while the exogenous signal represents both the reference to track and disturbance to reject.The exogenous signal is generated by an exosystem.Source: §2.1 · p. 3
  • The output-regulation problem is posed as designing a controller of a specified form that satisfies the stated regulation properties.The supplied passage introduces those properties but does not specify them.Source: §2.1 · p. 3

1. The unforced closed-loop system

The section focuses on the steady-state feedforward input required for output regulation and reviews how its structure relates to internal models. It motivates identifying that input from data instead of solving difficult regulator equations for an explicit nonlinear plant model.

1. The unforced closed-loop system

  • When c(v) has a polynomial representation in exosystem states, a finite-dimensional internal model can be constructed, and the steady-state input has a finite harmonic representation.For a linear exosystem with purely imaginary eigenvalues, fundamental frequencies follow exosystem dynamics; higher harmonics reflect nonlinear combinations, with coefficients dependent on system parameters and exosystem initial amplitudes and phases.Source: §1 · p. 4
  • The paper focuses on designing the steady-state feedforward input c(v), the steady-state component of an internal-model-based controller.Under standard stabilizability and detectability assumptions, regulator-equation solutions define the invariant zero-error manifold through mappings x = π(v) and u = c(v).Source: §1 · p. 4

2.2 Problem definition

  • Model-based design requires an accurate structural plant description and solving nonlinear regulator equations is generally challenging; missing structural terms can omit corresponding harmonics.The section asks whether the ideal steady-state control input can instead be identified without an explicit first-principles plant model.Source: §2.2 · p. 4
  • The proposed framework uses frequency-domain equivalence results to design a data-driven steady-state feedforward generator.The approach is presented as addressing the problem without an explicit first-principles plant model.Source: §2.2 · p. 4

3 FREQUENCY-BASED STEADY-STATE GENERATOR DESIGN METHODOLOGY

The methodology learns a frequency-domain input-output model from experimental data, then uses it to design a steady-state feedforward generator that minimizes tracking error for reference signals. Harmonic parameterization connects the design to classical internal model theory while avoiding the need to solve regulator equations from a known plant model.

3.1 Model Generation

  • The model-generation stage uses experimental input-output data because the system dynamics and state dimension are unknown, and only partial state information is available.The data consist of input and output trajectories collected over experimental horizons.Source: §3.1 · pp. 4–5
  • High sampling rates make time-domain histories dimensionally costly, so the method transforms input-output data with a Fourier transform.For periodic inputs, nonlinear dynamics produce discrete frequency components, including harmonics and intermodulation components.Source: §3.1 · p. 5
  • Concentrating relevant dynamics into dominant harmonic coefficients reduces dimensionality and supports structured dataset generation.The frequency-domain representation aligns with the internal-model view of steady-state inputs as a finite set of components.Source: §3.1 · p. 5

3.2 Data-driven Steady-State Generator Design in Frequency Domain

  • The steady-state generator learns harmonic control coefficients that minimize tracking error for a class of reference signals.It maps reference amplitude and phase information to feedforward input components; its objective can weight error at selected frequencies.Source: §3.2 · p. 6
  • A cascaded generator network and pretrained forward model map reference parameters to harmonic coefficients, with an output constraint layer enforcing imposed limits.The forward model represents the learned plant mapping, while the generator produces the control coefficients.Source: §3.2 · p. 6
  • The learned generator is a data-driven realization of the ideal feedforward input, using harmonic structure without requiring the plant model or regulator equations.The harmonic basis remains useful for non-polynomial nonlinearities, whose content is concentrated in a finite set of bins up to truncation.Source: §3.2 · p. 6

4 NUMERICAL RESULTS

Numerical studies compare a data-driven steady-state feedforward design on a mechanical load with nonlinear friction and an electrohydraulic actuator. The design recovers dominant harmonic structure for the mechanical benchmark, while electrohydraulic tracking errors remain dependent on the number and range of allowed frequency components.

4 NUMERICAL RESULTS

  • The mechanical-load surrogate was trained on 1.28 million sampled inputs and achieved test MSE of 2.13 × 10^-4 deg2.The generator stage used 40,000 reference combinations and achieved test MSE of 2.18 × 10^-6 deg2.Source: §4.1 · p. 8
  • The mechanical analytic input requires 3 Hz and 5 Hz components beyond the 1 Hz reference fundamental, and the data-driven solution captures the required harmonic structure.The comparison retains frequency components up to 5 Hz.Source: Figure 2 · p. 8, §4.1 · p. 8
  • 0.55° versus 0.36° maximum tracking error: the data-driven input is close to the truncated analytical benchmark for the 10° mechanical-load reference.The data-driven error is 5.5% of the reference amplitude.Source: §4.1 · pp. 8–9, Figure 3 · p. 9

4.2 Electrohydraulic actuator system

  • For the electrohydraulic design, the inference constraint allows at most four harmonic components, clips their magnitudes to training ranges, and permits added harmonics to compensate for plant nonlinearities.The simulation-model outputs closely match the data-driven model outputs across the shown cases.Source: §4.2 · p. 11
  • 3.5 mm maximum steady-state time-domain tracking error was reported for one electrohydraulic case; the other cases reached 4.3 mm and errors below 1.5 mm below 30 Hz, respectively.The 4.3 mm and 3.5 mm errors are reported for t > 2 seconds; the square-wave case reports frequency-component errors below 1.5 mm under 30 Hz.Source: §4.2 · p. 11

5 CONCLUSION AND FUTURE WORKS

The paper presents a data-driven framework for designing steady-state feedforward generators for partially measured nonlinear systems with unknown dynamics. Simulations validate the framework on a mechanical load with nonlinear friction and an electrohydraulic actuator, while the conclusion identifies data requirements and feedback integration as limitations.

5 CONCLUSION AND FUTURE WORKS

  • The data-driven framework designs feedforward steady-state generators for partially measured nonlinear systems with unknown dynamics, without prior plant knowledge or explicit regulator-equation solutions.The method is presented as a data-driven realization of the steady-state generator.Source: §5 · pp. 11–12
  • The framework was validated on a nonlinear-friction mechanical load and an electrohydraulic actuator, with accurate EHA tracking demonstrated for multi-harmonic and square-wave references.The friction example recovers the harmonic structure of the analytical regulator-equation solution; the EHA tests illustrate application to more complex nonlinear dynamics.Source: Figure 5 · p. 12, Figure 6 · p. 12, §5 · p. 12
  • Training data requirements grow with the number of retained harmonics and can become prohibitive for high-bandwidth tracking; incorporating feedback remains an open issue.The authors identify data-efficient training, structured surrogate models, and feedback integration as future directions.Source: §5 · p. 13
Loading 2609.09450v1…