Source-linked AI summary
NL-FRA: A MATLAB package for nonlinear frequency response analysis
Rajintha Gunawardena, Zi-Qiang Lang, Fei He
TL;DR
Nonlinear systems can produce output frequencies absent from their inputs, while multidimensional GFRFs are difficult to interpret. NL-FRA implements data-driven least-squares NOFRF analysis from input–output data, integrating estimation, validation, visualization, and transmissibility analysis. Its scope is currently limited to SISO nonlinear systems.
Problem
Multidimensional GFRFs are computationally heavy and difficult to interpret, motivating a more accessible representation of nonlinear output spectra.
Source: Statement of need · p. 2Method
NL-FRA estimates NOFRFs from input–output data using a data-driven least-squares approach and provides an integrated analysis workflow.
Source: Statement of need · pp. 2–3, Features in NL-FRA · p. 3Results
NL-FRA includes NOFRF estimation, validation, visualization, and nonlinear transmissibility analysis, with examples covering band-limited and multi-tone inputs and transmissibility comparisons.
Source: Features in NL-FRA · p. 3, General band-limited input · p. 4, Multi-tone input · p. 6Takeaways & Limitations
The one-dimensional NOFRF representation supports analysis of higher-order spectral characteristics using data from suitable dynamic models, including neural networks.
Source: Statement of need · pp. 2–3NL-FRA currently supports SISO nonlinear systems; MISO and MIMO extensions are future work.
Source: Future work · p. 8
Abstract
from arXiv · showhide
Nonlinear Output Frequency Response Functions (NOFRFs) provide a one-dimensional frequency-domain representation of nonlinear dynamics, enabling direct decomposition of an output spectrum into contributions from different orders of nonlinearity. NL-FRA (NonLinear Frequency Response Analysis) is an open-source MATLAB package implementing a data-driven Least Squares (LS) method for estimating NOFRFs directly from system input-output data. It provides an integrated workflow for NOFRF estimation, validation, and visualisation, together with nonlinear transmissibility analysis. The package supports different types of inputs, i.e. general band-limited, multi-one and harmonic inputs, and includes routines for identifying the frequency ranges over which each nonlinear order contributes to the output. Furthermore, since the LS approach estimates NOFRFs directly from input-output data, it can be applied to experimental measurements or data generated using any suitable dynamic model of the system. The software facilitates interpretable nonlinear frequency-domain analysis for applications including fault diagnosis, condition and structural health monitoring, and biomedical engineering.
Statement of need
Nonlinear systems can generate output frequencies absent from their inputs, while their multidimensional GFRFs are computationally heavy and difficult to interpret. NL-FRA addresses this need with data-driven least-squares NOFRF estimation from input–output data and a one-dimensional representation of higher-order spectral characteristics.
Statement of need
- Nonlinear systems can generate harmonics, sub-harmonics, and intermodulation products absent from the input, unlike LTI systems whose output contains only input frequencies.These additional components arise through energy transfer between frequencies.Source: Statement of need · p. 1
- NOFRFs represent nth-order output responses as one-dimensional functions of frequency, making higher-order spectral characteristics more accessible than multidimensional, computationally heavy GFRFs.The nth-order GFRF describes the corresponding nonlinearity's spectral characteristics, and its output response is the nth-order OFRF.Source: Statement of need · p. 2
- NL-FRA focuses on data-driven least-squares NOFRF estimation, addressing the stated absence of an open-source package implementing either available evaluation method.The LS method estimates NOFRFs from input–output data that can come from a valid dynamic representation, including a neural network.Source: Statement of need · pp. 2–3
- The LS procedure scales the input by several factors, uses the corresponding output spectra and input compositions, and incorporates them into a matrix representation to estimate NOFRFs.The matrix can become sparse and ill-conditioned where input compositions vanish; the described approach forms a dense matrix containing only relevant terms and uses a pseudo-inverse.Source: Statement of need · p. 3
Features in NL-FRA
NL-FRA implements a workflow for NOFRF estimation, validation, and visualisation across standard probing inputs. It also supports nonlinear transmissibility analysis across nonlinear orders.
Features in NL-FRA
- NL-FRA provides a complete pipeline for NOFRF evaluation and validation.It includes plotting functions for NOFRF-based transmissibility analysis.Source: Features in NL-FRA · p. 3
- NL-FRA includes nonlinear transmissibility analysis across different orders of nonlinearity.Source: Features in NL-FRA · p. 3
- The package supports sinusoidal, general band-limited, and multi-tone probing inputs, with routines to determine each order’s contributing frequency set.The frequency sets are defined for nonlinear orders n = 1 to N.Source: Features in NL-FRA · p. 3
Examples
The examples demonstrate NL-FRA with a band-limited input and an identified NARX model, a multi-tone Duffing input, and sinusoidal transmissibility analysis. They show order-specific frequency functions and reconstruct nonlinear response contributions for comparison across excitation frequencies.
General band-limited input
- The band-limited example identifies and independently validates a polynomial NARX model, then evaluates its first four NOFRFs from simulated input–output data.Figures 1–5 show the response, input compositions, NOFRFs, and output frequency response functions.Source: General band-limited input · p. 4, Figure 1 · p. 4, Figure 2 · p. 4
Multi-tone input
- For the Duffing example, multi-tone probing evaluates NOFRFs at A = 1.4 using constant input scalings between 1.3 and 1.2.The routine determines discrete frequency components for each nonlinear order; Figures 6–7 show spectra, NOFRFs, and output frequency response functions.Source: Multi-tone input · p. 6, Figure 6 · p. 6, Figure 7 · p. 6
Transmissibility analysis
- At the test amplitude A = 1.4, the transmissibility workflow reconstructs the output spectrum and nonlinear-order contributions for comparison at the excitation frequency and third harmonic.The comparison is between actual and NOFRF-generated transmissibilities; Figure 9 shows the individual order contributions over the frequency sweep.Source: Transmissibility analysis · p. 7, Figure 8 · p. 7, Figure 9 · p. 7
Future work
NL-FRA currently supports SISO nonlinear systems. Planned work includes extending support to MISO and MIMO systems, evaluating GFRFs, and providing a Python implementation.
Future work
- NL-FRA currently supports only single-input single-output nonlinear systems.Source: Future work · p. 8
- Planned extensions add multi-input single-output and then multi-input multi-output system support.Source: Future work · p. 8
- Other planned developments are procedures for evaluating GFRFs and an open-source Python implementation.Source: Future work · p. 8