From a Frequency-Domain Willems' Lemma to Data-Driven Predictive Control

Willems’ fundamental lemma has recently received an impressive amount of attention from the data-driven control community. In this paper, we formulate a version of this celebrated result based on frequency-domain data. In doing so, we bridge the gap between recent developments in data-driven control, and the readily-available techniques and expertise for non-parametric frequency-domain identification. We also generalize our results to combine multiple frequency-domain data sets to form a sufficiently rich data set. Building on these results, we propose a data-driven predictive control scheme based on measured frequency-domain data of the plant. This novel scheme provides a frequency-domain counterpart of the well-known data-enabled predictive control scheme DeePC based on time-domain data. Under appropriate conditions, the new frequency-domain data-driven predictive control (FreePC) scheme is equivalent to the corresponding DeePC scheme. We demonstrate the benefits of FreePC and the use of frequency-domain data in several examples and a numerical case study, including the ability to collect data in closed loop, computational benefits, and intuitive visualization of the data.

💡 Research Summary

The paper introduces a frequency‑domain counterpart of Willems’ Fundamental Lemma (WFL) and leverages it to develop a data‑driven predictive control scheme called FreePC. Classical WFL states that, for a linear time‑invariant (LTI) system, a single persistently exciting (PE) time‑domain input‑output trajectory can generate all possible trajectories of a given length through linear combinations of its Hankel sub‑matrices. This result underpins the DeePC algorithm, which replaces an explicit model with a data‑based predictor. However, in many industrial settings, non‑parametric frequency‑response function (FRF) measurements are readily available, while time‑domain data must be synthesized from these measurements or from a parametric model, incurring modeling bias and extra effort.

The authors address this gap by formulating WFL directly in the frequency domain. They define complex‑valued input, output, and state spectra sampled on a (possibly equidistant) frequency grid. By arranging these spectra into a structured data matrix that respects the conjugate symmetry inherent to real‑valued systems, they prove that if the frequency‑domain data matrix satisfies a rank condition analogous to PE, then any finite‑horizon time‑domain trajectory can be exactly reconstructed as a linear combination of the stored spectral windows. Their proof works for both stable and unstable systems, provided the data are collected in closed loop with a pre‑stabilizing controller, thus extending earlier results that were limited to stable plants.

A further contribution is the treatment of multiple frequency‑domain data sets. For multi‑input multi‑output (MIMO) plants, separate FRF sweeps for each input‑output pair can be concatenated into a larger matrix. Even if individual sweeps are not PE, the combined set can meet the rank condition, enabling the use of fragmented or band‑limited measurements. This flexibility allows experiment designers to focus on frequency bands of interest, reducing measurement time and cost.

Building on the frequency‑domain WFL, the authors propose FreePC, a predictive control formulation that mirrors DeePC’s quadratic program but replaces the time‑domain Hankel matrices with the newly defined frequency‑domain data matrices. The decision variables include the control sequence, predicted outputs, and a coefficient vector g that weights the spectral basis functions. Constraints enforce that the predicted trajectories lie in the subspace spanned by the measured spectra. Regularization terms (λ_g, λ_σ) and slack variables handle measurement noise and possible model mismatch, just as in DeePC.

The paper demonstrates several key advantages of FreePC:

1. Closed‑loop data acquisition – FRFs can be measured while a stabilizing controller is in place, allowing safe identification of unstable plants.
2. Computational efficiency – Frequency‑domain data typically involve far fewer samples than time‑domain trajectories, leading to smaller matrices and faster quadratic‑program solves.
3. Uncertainty visualization – Because FRF measurements come with well‑established confidence intervals, the resulting data matrix can be visualized directly, giving practitioners intuitive insight into data quality.
4. Band‑selective control – Designers can weight specific frequency bands in the cost function, tailoring performance (e.g., robustness) where it matters most.

The authors validate FreePC through numerical case studies, including an unstable second‑order plant, an LQR‑type benchmark, and a scenario where only a limited set of frequency points is available. In each case, FreePC matches DeePC’s closed‑loop performance while requiring no time‑domain reconstruction and offering the aforementioned benefits. They also present a method to estimate the plant’s transfer function at arbitrary complex frequencies from a finite FRF dataset, extending the utility of the frequency‑domain WFL beyond control to system analysis.

In conclusion, the work bridges a longstanding divide between classical frequency‑domain system identification and modern data‑driven control. By establishing a rigorous frequency‑domain version of Willems’ lemma, it enables direct use of raw FRF data for predictive control, reduces experimental and computational overhead, and provides a transparent way to incorporate measurement uncertainty. This contribution is poised to influence both academic research on data‑driven control and practical industrial applications where non‑parametric frequency measurements are the norm.