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

TL;DR

The paper develops a frequency-domain Willems' fundamental lemma (WFL) for linear time-invariant systems and extends it to collect-and-combine multiple frequency-domain data sets. It then builds FreePC, a frequency-domain data-driven predictive control scheme that is equivalent to the time-domain DeePC under noise-free conditions, while offering practical advantages for closed-loop data collection and noisy measurements. The work demonstrates applications including FRF-based simulation, frequency-response evaluation, and frequency-domain LQR, highlighting that data-driven control can leverage non-parametric FRF data without converting to state-space models. The findings provide a bridge between classical frequency-domain identification and modern data-driven control, with numerical case studies illustrating robustness to noise and benefits in computational efficiency. Overall, the approach enables direct control and analysis from frequency-domain data for potentially unstable plants and noisy measurements, broadening the toolbox for data-driven control practitioners.

Abstract

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 analysis and control, and the readily-available techniques and extensive expertise for non-parametric frequency-domain identification in academia and industry. In addition, we generalize our results to allow multiple frequency-domain data sets to be carefully combined 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. We prove that, under appropriate conditions, the new frequency-domain data-driven predictive control (FreePC) scheme is equivalent to the corresponding DeePC scheme, and we demonstrate the benefits of FreePC and the use of frequency-domain data in a numerical case study. These benefits include the ability to collect data in closed loop with a pre-stabilizing controller, dealing with noisy data, without increasing computational complexity, and intuitively visualizing the uncertainty in the frequency-domain data. In addition, we further showcase the potential of our frequency-domain Willems' fundamental lemma in applications to data-driven simulation, and the linear-quadratic regulator (LQR) problem. Finally, we show that our results can be used to evaluate the transfer function of the system at a desired frequency based on a finite amount of frequency-domain data.

Figures (5)

• Figure 1: Data-driven simulation of an unstable batch reactor using noise-free data. We see that the simulated outputs $y_1$ () and $y_2$ () closely resemble the true response ().
• Figure 2: Closed-loop measurement setup.
• Figure 3: Data-driven simulation of an unstable batch reactor using noisy data. We see that, using the data based on measuring $p=10$ periods, there is a significant error between the simulated output trajectories $y_1$ () and $y_2$ () and the true response (). However, using the data based on measuring $p=50$ periods, leads to significantly more accurate simulations for $y_1$ () and $y_2$ ().
• Figure 4: Estimated FRF of the system () using $p=2$ () and $p=50$ periods () with their respective $99\%$ confidence intervals (/ ).
• Figure 5: Simulation, with constraints (), of FreePC using data with $2$ () and $50$ () periods, and an MPC benchmark ().

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Lemma 1
• Definition 3
• Example 1: Incorporating FRF measurements
• Remark 1
• Definition 4
• Theorem 1
• Definition 5
• Theorem 2