Transfer function fitting (TFF)¶

This page expands 2.4 Transfer function fitting from the SHM roadmap: fit a rational transfer function (e.g. \(H(s)=N(s)/D(s)\) or polynomial ratio in frequency) to measured FRF or input–output frequency-domain data, and extract modal frequency, damping, and mode shape from the poles and residues of the fitted model.

Tutorial video¶

Concept¶

Core idea — The system transfer function \(H(s)\) (with \(s\) the complex frequency) or its frequency-domain form \(H(\omega)\) can be written as a rational function (ratio of polynomials) in modal analysis: its poles correspond to natural frequency and damping, and its residues to mode shapes. Transfer function fitting is the process of fitting such a rational function to FRF or frequency-domain input–output data from tests, so as to obtain a parametric model and extract all modal parameters. In short: obtain FRF or frequency-domain data → choose rational form and fit → compute poles and residues → modal parameters.

Transfer function form — In continuous time the rational form in \(s\) is common; when data are in the frequency domain, \(j\omega\) is used. The Rational Fraction Polynomial (RFP) form is:

\[H(\omega) = \frac{N(\omega)}{D(\omega)} = \frac{\sum_{k=0}^{m} a_k (j\omega)^k}{\sum_{k=0}^{n} b_k (j\omega)^k}, \tag{1}\]

Notation for (1):

\(H(\omega)\): transfer function in frequency (i.e. FRF for SISO).
\(a_k, b_k\): numerator and denominator polynomial coefficients (real or complex).
\(m, n\): numerator and denominator orders; typically \(n \geq m\), with \(n\) on the order of twice the number of modes.
\(j\): imaginary unit.

Pole–residue form (partial fractions):

\[H(\omega) = \sum_{r=1}^{N_m} \frac{R_r}{j\omega - s_r} + \text{(optional higher-order terms)}, \tag{2}\]

Notation for (2):

\(N_m\): number of modes in the frequency band.
\(s_r\): pole of mode \(r\) (complex), \(s_r = -\zeta_r \omega_r \pm j\omega_r\sqrt{1-\zeta_r^2}\).
\(R_r\): residue of mode \(r\) (complex, related to mode shape).
\(\omega_r, \zeta_r\): undamped natural frequency and damping ratio.

Relation to FRF curve fitting — FRF curve fitting is the rational fitting of measured FRF in the frequency domain and is one instance of transfer function fitting; this page emphasises the transfer function view (rational \(H(s)/H(\omega)\), poles and residues) and covers the general process of obtaining a parametric modal model from FRF or frequency-domain input–output data.

Algorithm in brief¶

Data source — The fit is usually performed on:

Measured FRF: from input–output via H1/H2/Hv estimators (see FRF); or
Frequency-domain input–output: Fourier coefficients at each frequency, fitting \(Y(\omega)/U(\omega)\) or equivalent.

Rational function fitting — Fit \(\hat{H}(\omega_k)\) at discrete frequencies \(\omega_k\). The problem is nonlinear in the coefficients (or poles/residues). Common approaches:

Linear LS on numerator/denominator: Write (1) as \(D(\omega)H(\omega)=N(\omega)\), linearise in \(a_k,b_k\), solve LS; iterate with weighting if needed (e.g. Sanathanan–Koerner).
Nonlinear optimisation: Optimise poles \(s_r\) and residues \(R_r\) (or polynomial coefficients) directly, minimising \(\sum_k |H(\omega_k)-\hat{H}(\omega_k)|^2\) with Gauss–Newton, Levenberg–Marquardt, etc.
Orthogonal polynomials: Use an orthogonal basis (e.g. Forsythe) instead of monomials to improve conditioning, then find roots for poles.

Poles and residues — Poles \(s_r\) are the roots of the fitted denominator \(D\), or read from form (2). Residues \(R_r\) come from partial fraction expansion or the fitted model; for MIMO, residues form a matrix whose columns (or rows) give mode shapes.

Modal parameters — \(\omega_r = |s_r|\) (or solve for \(\omega_r,\zeta_r\) from real and imaginary parts of \(s_r\)); mode shapes from the residue matrix and output mapping.

Procedure (outline)¶

Data preparation: Obtain FRF or frequency-domain input–output data (known excitation required; see FRF estimation).
Band and order: Choose frequency band of interest; choose numerator/denominator orders \(m,n\) or number of modes \(N_m\), e.g. via stabilisation diagram (poles vs order) or AIC/BIC.
Rational fit: Fit (1) or (2) over the chosen band using linear LS (with iteration if needed), nonlinear optimisation, or orthogonal polynomials.
Pole extraction: Get poles \(s_r\) from denominator roots or from (2); drop unstable or clearly spurious poles.
Residues and mode shapes: Compute residues \(R_r\) at each pole; for multi-channel, form mode shapes from the residue matrix and normalise.
Validation: Compare fitted curve with measured FRF; check pole stability with order; compare mode shapes with reference (e.g. MAC) if available.

When to use and limitations¶

Use when — FRF or frequency-domain input–output data are available (e.g. impact or shaker test); a parametric modal model (poles, residues, or rational coefficients) is needed; classical modal analysis, model updating, or control design; frequency-domain, rational representation is preferred.

Limitations — Requires known excitation: not for output-only (ambient) data; controlled input and FRF (or frequency-domain I/O) estimation are needed. Order sensitivity: too low underfits, too high introduces spurious poles; stabilisation diagram or criteria help. Compute and numerics: rational fitting and root finding are non-trivial; high order can be ill-conditioned; orthogonal basis or partial fraction form help. Multi-channel: multiple FRFs or MIMO increase data and compute; fit channel-by-channel or use reduced MIMO.

Engineering practice: practical notes¶

Aspect	Notes
Data and FRF	Same as FRF: known excitation, H1/H2/Hv estimation, multiple averages for SNR; coherence for quality.
Rational form	RFP (1) is simple to implement; pole–residue (2) has clear physical meaning and can be numerically better; orthogonal polynomials improve conditioning.
Order and stabilisation	Try several orders, plot poles vs order; stable poles are physical modes, drifting or scattered ones are often spurious.
Fitting method	Linear LS + iteration is fast and easy; nonlinear optimisation is more accurate but needs initial guess; use linear solution as initial guess if needed.
Poles and residues	Use robust root finding (e.g. companion matrix eigenvalue); residues give relative mode shapes, absolute scaling needs mass or normalization.
Validation	Compare fitted and measured FRF (magnitude, phase); check poles with stabilisation diagram; use MAC etc. for mode shapes in multi-channel.

Edge and online computing¶

Suitability — Partially suited. Band and order can be limited to reduce compute; but FRF or frequency-domain data are still required (hence input measurement and estimation), and rational fitting plus root finding remain non-trivial on resource-limited devices.

Potential — Band and reduced order: Fit only the band and number of modes of interest to reduce order and compute. Pre-fitted deployment: Fit offline, deploy poles/residues or rational coefficients on edge for real-time FRF evaluation or modal filtering. Tiered: Edge does FRF estimation and coarse low-order fit for screening; suspect data uploaded for high-order fit and validation.

Challenges — FRF and data: Input measurement, buffer, CPSD/FFT. Fitting and roots: Nonlinear or iterative fitting and polynomial root finding are not light on MCUs. Real-time: Full fitting is usually offline; edge is better with pre-fitted models for coarse checks (e.g. frequency shift?), then upload for refinement.

Practical strategy — Use pre-fitted models on edge: do transfer function fitting offline, deploy poles/residues; edge only evaluates FRF or response. For online updates, use recursive or incremental fitting (e.g. recursive LS on polynomial coefficients) or band-limited updates; or a tiered setup with coarse fit on edge and refined fit in the cloud.

Relation to FRF curve fitting¶

Transfer function fitting (this page): Emphasises fitting a rational transfer function \(H(s)/H(\omega)\) to data (FRF or frequency-domain I/O) to obtain poles, residues, and modal parameters; applies to classical modal analysis, modelling, and control.
FRF curve fitting: Focuses on rational curve fitting of measured FRF; the procedure and algorithms are the same as here, and can be seen as transfer function fitting when the data source is FRF; see FRF.

The two are the same in mathematics and implementation: rational fitting plus pole/residue extraction; this page stresses the “transfer function” and general data source, the FRF page the “FRF estimation + curve fitting” workflow.