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Distributionally Robust Frequency Response Estimation Using Non-Steady-State Frequency-Domain Data

Stefan Damjanovic, John W. Simpson-Porco, Klaske van Heusden

arXiv:2610.09395v1eess.SYcs.SY

TL;DR

Estimating frequency responses from noisy non-steady-state data is challenging because transients can bias estimates, while obtaining finite-sample guarantees under stochastic noise is difficult. The paper develops a Wasserstein DRO estimator for this setting. Its case study reports better performance than an averaged SVD estimator under heavy-tailed noise, but worse performance under light-tailed noise.

  • Problem

    Finite-sample guarantees for FRF estimation with stochastic measurement noise remain challenging, and non-steady-state data can introduce transient bias or require long experiments.

    Source: §I · p. 1

  • Method

    The paper formulates noisy non-steady-state FRF estimation as Wasserstein DRO over a weighted ambiguity set and derives an estimator by minimizing a convex upper bound.

    Source: §I · p. 2, §IV · p. 5

  • Results

    The DRO estimator substantially improved performance across several metrics over an averaged SVD-based estimator under heavy-tailed noise, but underperformed it under light-tailed noise.

    Source: §VI · pp. 8–9

  • Takeaways & Limitations

    The case study supports the estimator's potential for limiting large identification errors when repeated measurements contain extreme perturbations.

    Source: §VI · p. 9

    Optimal calibration of frequency-dependent weights, the calibration-versus-identification repetition tradeoff, and performance under varying initial conditions remain for future study.

    Source: §VI · p. 9

Abstract

from arXiv · show

This paper develops a distributionally robust optimization (DRO) framework for identifying the frequency response function (FRF) of an unknown linear time-invariant system from repeated noisy non-steady-state input-output measurements. Building on a recent frequency-domain extension of Willems' fundamental lemma, we formulate FRF estimation as a DRO problem, wherein robustness is desired with respect to the unknown probability distribution generating the noisy data. The proposed estimator computes FRF estimates by minimizing a convex upper bound on the DRO cost and allows for frequency-dependent calibration of the Wasserstein metric; the tractable upper bound consists of an empirical fit and a regularizer that penalizes the largest weighted sensitivity to spectral perturbations. Under suitable assumptions, we prove that the FRF estimation error is upper bounded by the optimal value of the DRO problem. The method is validated via a simple numerical study, which demonstrates improved performance under severely heavy-tailed noise.

I. INTRODUCTION

Noisy non-steady-state data complicate FRF estimation: ignoring transients can bias estimates, while waiting for them to decay can make experiments prohibitively long. The paper develops a Wasserstein DRO approach for this setting and builds on frequency-domain methods that account for finite-window transients.

I. INTRODUCTION

  • Ignoring transients can bias FRF estimates, while waiting for them to decay may require prohibitively long experiments, especially for slow systems.The introduction identifies the lack of an approach that combines noisy non-steady-state data, no preprocessing, and out-of-sample guarantees for unknown noise distributions.Source: §I · p. 1
  • Earlier noisy-data approaches use SVD truncation or least-squares-inspired projection heuristics, with guarantees absent in some methods and bounded-noise assumptions in another.Large perturbations, including heavy-tailed outliers, can distort singular values and directions used by SVD truncation.Source: §I · p. 1
  • The paper develops a Wasserstein DRO framework for noisy non-steady-state IO data, with weights that encode confidence across experiments, frequency bins, and output channels.Its estimator minimizes a convex upper bound on a worst-case frequency-domain residual and provides finite data-length error bounds for unknown noise distributions.Source: §I · p. 2

II. IDENTIFICATION WITH NON-STEADY-STATE FREQUENCY-DOMAIN DATA

  • The frequency-domain approach estimates FRF values on a frequency grid from non-steady-state data, where unknown initial and terminal states generally leave a nonzero transient.The paper follows the formulation of for this task; its finite-window spectra account for boundary-state mismatch.Source: §II · p. 2, A. Non-Steady-State Frequency Domain… · pp. 2–3

B. Multiple Datasets and Collective Excitation

Multiple frequency-domain datasets can jointly supply input excitation while retaining experiment-specific finite-window transients. Under collective excitation and input-rank conditions, the method recovers the FRF exactly in the noise-free case.

B. Multiple Datasets and Collective Excitation

  • For MIMO systems, multiple input signals may be needed to excite every input direction, and each finite-window dataset can have a different transient.The datasets may, for example, come from multi-sine experiments.Source: B. Multiple Datasets and Collective… · p. 3
  • Collective persistent excitation of order L means the aggregated block data matrix has full row rank; necessarily, E(2M −1) ≥ nsL.The definition uses the collection of frequency-domain data sequences across datasets.Source: B. Multiple Datasets and Collective… · p. 3

C. Noise-Free Recovery of the FRF

  • With a controllable system and augmented inputs collectively persistently exciting of order L + nx, the recovery equations have a unique solution and recover H(zm) under Assumption 1.This noise-free result applies at excited frequencies zm not in eig(A), and the recovery does not require identifying the system matrices or knowing the initial condition.Source: C. Noise-Free Recovery of the FRF · p. 3

III. SETUP FOR ROBUST FRF ESTIMATION: UNCERTAINTY MODELLING AND WASSERSTEIN UNCERTAINTY SETS

The robust-estimation setup treats repeated noisy output spectra as samples from an unknown distribution, with random initial conditions also affecting the noise-free response. It specifies moment and sampling assumptions while allowing dependence among spectral uncertainty coordinates within a repetition.

III. SETUP FOR ROBUST FRF ESTIMATION: UNCERTAINTY MODELLING AND WASSERSTEIN UNCERTAINTY SETS

  • The noise-free output spectrum can vary across repetitions because initial states vary, while additive measurement noise corrupts the output spectra.The setup generalizes the noise-free recovery mechanism to account for an unknown noise distribution and random initial conditions.Source: §III · p. 3, A. Modelling of Noisy Data · pp. 3–4

A. Modelling of Noisy Data

  • The setup assumes random initial conditions and zero-mean additive spectral errors have finite first moments, with repetitions that are independent and identically distributed.The uncertainty coordinates within a repetition need not be independent, so they may be dependent across experiments, frequency bins, output channels, and real and imaginary parts.Source: A. Modelling of Noisy Data · p. 4
  • Each repetition stacks the noise-affected output spectra into a sample vector, and the resulting samples are independent and identically distributed realizations of the unknown law.The unknown law is approximated using a subset of the repeated experiments.Source: A. Modelling of Noisy Data · pp. 3–4

B. Weighted Wasserstein Uncertainty Modelling

The uncertainty model centers a frequency-weighted Wasserstein ambiguity set on the empirical distribution from identification repetitions. Its weighting shapes the tolerated variability, while its transport geometry represents some dependencies but omits cross-frequency and cross-dataset dependencies.

B. Weighted Wasserstein Uncertainty Modelling

  • The empirical law from identification repetitions may differ from the true out-of-sample law when data are finite, so the model considers laws in a frequency-weighted Wasserstein ball around it.The repetitions are partitioned into calibration and identification sets; the calibration set is used to calibrate frequency-dependent weights.Source: B. Weighted Wasserstein Uncertainty… · p. 4
  • The weighting matrix shapes the Wasserstein geometry so tolerated variability can differ across experiments and frequencies.The metric uses block-diagonal weights for experiment-frequency pairs.Source: B. Weighted Wasserstein Uncertainty… · pp. 4–5
  • The transport geometry represents cross-output and real-imaginary dependencies within blocks, but does not explicitly incorporate cross-frequency or cross-dataset dependencies.This structure is adopted to reduce block-weighting-estimation requirements and preserve computational tractability.Source: B. Weighted Wasserstein Uncertainty… · p. 5

IV. MAIN RESULTS: DISTRIBUTIONALLY ROBUST FRF ESTIMATION

The estimator handles noisy non-steady-state frequency-domain data by optimizing against a weighted Wasserstein ambiguity set around the empirical output-spectrum law. It minimizes a convex upper bound that combines empirical residual fit with a penalty for weighted sensitivity to spectral perturbations.

IV. MAIN RESULTS: DISTRIBUTIONALLY ROBUST FRF ESTIMATION

  • The method models the unknown noise-generating law as an element of a weighted Wasserstein ball around the empirical law of noisy output spectra.Source: §IV · p. 5

A. Formulation of DR FRF Estimator

B. FRF Error Analysis

Under stated assumptions and when the true noise law lies in the Wasserstein ball, the paper bounds FRF estimation error using the optimal value of its estimator. The numerical study tests the estimator against an averaging-based baseline under noise with varying tail heaviness.

B. FRF Error Analysis

  • When the true law lies in the ambiguity set and the stated assumptions hold, the FRF estimation error is bounded by the estimator’s optimal value.Source: B. FRF Error Analysis · p. 6
  • The bound is not a consistency result: its optimal value generally does not tend to zero as ε approaches zero.Source: B. FRF Error Analysis · p. 6
  • For light-tailed laws satisfying an exponential-moment condition, a sufficient radius gives ambiguity-set containment with probability at least 1 − β; the study instead tunes ε empirically.Source: B. FRF Error Analysis · p. 6

V. NUMERICAL CASE STUDY

  • The case study compares the DRO estimator with an averaging-based baseline and examines how the comparison changes from light-tailed to heavy-tailed noise.Source: §V · p. 6

A. Plant and Experimental Data

  • The study uses a fourth-order SISO benchmark with two strong resonances, seven repeated experiments, and FRF evaluations at excited odd frequencies.Source: A. Plant and Experimental Data · p. 6

B. Noise Models

The numerical study varies frequency-dependent noise amplitude and tail heaviness, then evaluates the DRO estimator across Wasserstein radii. Under Cauchy noise, a small positive radius reduces error relative to no regularization, while excessive regularization worsens performance.

2) Frequency-Dependent Noise Distributions:

1) Frequency-Dependent Noise Scaling:

2) Frequency-Dependent Noise Distributions:

  • Noise scales are normalized to match the Gaussian coordinate’s marginal median absolute deviation, allowing tail behavior to be compared without variance normalization for laws with infinite second moments.Source: 2) Frequency-Dependent Noise… · p. 7

C. Implementation of DRO Estimator and Comparison

D. Statistical Comparison and Metrics

The comparisons show that DR's advantage is concentrated in severely heavy-tailed noise, while averaged-SVD performs better under light-tailed noise. Under Cauchy noise, frequency-wise FRF errors also favor DR across all evaluated frequencies, with one phase-error exception.

D. Statistical Comparison and Metrics

  • Evaluation used additional R noise seeds disjoint from Wasserstein-radius selection and reported mean relative RMSE, median, and CVaR, the mean of the upper 10% of trial errors.For the DRO method, ε was fixed at ε⋆ during this evaluation.Source: D. Statistical Comparison and Metrics · p. 7
  • For Cauchy noise, mean RMSE was 297.26% for averaged-SVD versus 40.25% for DR; averaged-SVD had lower mean, median, and CVaR under Gaussian and Student-t ν∈{∞,5,3}.Figure 2 compares these metrics across noise distributions at fixed Q = 7.Source: D. Statistical Comparison and Metrics · p. 7, Figure 2 · p. 8
  • Mean RMSE crosses near ν≈2; there, both methods are similar in mean, while DRO has higher median but lower CVaR.For Cauchy noise, averaging does not reduce noise because the average of iid real noise coordinates has the same distribution as one coordinate.Source: D. Statistical Comparison and Metrics · pp. 7–8
  • Under Cauchy noise, DR had lower FRF error at all 14 evaluated frequencies and lower phase error at 13, except ˆωm = 25π/28.DR estimation errors spike around resonance peak frequencies, reflecting sensitivity in that region.Source: D. Statistical Comparison and Metrics · p. 8, Figure 3 · p. 8

VI. CONCLUSION

  • The convex DRO estimator combines empirical residual fitting with a weighted sensitivity penalty, connecting spectral uncertainty to regularization.Source: §VI · p. 8
  • Future work will assess optimal frequency-dependent Wasserstein-weight calibration, the calibration-versus-identification repetition tradeoff, and performance under varying initial conditions.Source: §VI · p. 9

APPENDIX

The appendix proves an FRF estimation-error bound under stated assumptions and supplies proof details for the DRO cost bound. It also reports a regularization trade-off in the numerical tests.

APPENDIX

  • Theorem 1’s proof identifies L(ˆθ_m) as an induced weighted norm and uses it to establish that the cost is Lipschitz under the weighted ground metric.The proof also states that the upper bound is attained when Ξ = R^nξ and the cost is convex.Source: APPENDIX · p. 10
  • The proof invokes Corollary 1’s uniqueness statement and identifies λ_m := σmin(U(E)_m) as positive for each m ∈ M_ex.Source: APPENDIX · p. 11
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