Harnessing Uncertainty for a Separation Principle in Direct Data-Driven Predictive Control

TL;DR

The paper develops a unified stochastic framework for direct data-driven predictive control (DDPC) centered on the Final Control Error ($FCE$), proving a separation principle that decouples predictor estimation and uncertainty from the control optimization. It shows that the $FCE$ decomposes into a certainty-equivalence control term and a regularization term accounting for data-driven uncertainty, enabling a tuning-free, probabilistic controller design. The framework encompasses and clarifies the relationship to existing DDPC methods such as DeePC and $\gamma$-DDPC, showing these as special cases or suboptimal implementations relative to full conditioning on data. Numerical experiments on a linear time-invariant benchmark demonstrate that the proposed $FCE$-based controller often outperforms or matches state-of-the-art methods without needing regularization parameter tuning, highlighting practical benefits for noise-tolerant and real-time data-driven control. The approach also supports incorporating priors on the system to enhance robustness and opens avenues for stability guarantees in future work.

Abstract

Model Predictive Control (MPC) is a powerful method for complex system regulation, but its reliance on an accurate model poses many limitations in real-world applications. Data-driven predictive control (DDPC) aims at overcoming this limitation, by relying on historical data to provide information on the plant to be controlled. In this work, we present a unified stochastic framework for direct DDPC, where control actions are obtained by optimizing the Final Control Error (FCE), which is directly computed from available data only and automatically weighs the impact of uncertainty on the control objective. Our framework allows us to establish a separation principle for Predictive Control, elucidating the role that predictive models and their uncertainty play in DDPC. Moreover, it generalizes existing DDPC methods, like regularized Data-enabled Predictive Control (DeePC) and $γ$-DDPC, providing a path toward noise-tolerant data-based control with rigorous optimality guarantees. The theoretical investigation is complemented by a series of experiments (code available on GitHub: https://github.com/marcofabris92/a-separation-principle-in-d3pc), revealing that the proposed method consistently outperforms or, at worst, matches existing techniques without requiring tuning regularization parameters as other methods do.

Figures (6)

• Figure 1: (a): Distributions of the estimated truncation rule $\hat{\rho}$ over $100$ Monte Carlo runs obtained for the Setup 1. (b)-(d): Distributions of the closed-loop performance index (see \ref{['eq:closed-loopperfindex']}) over 100 Monte Carlo runs. Note that the over-lined indexes are used when offline tuning approaches are exploited, whereas indexes with hats refer to the performance attained with online (and, thus, practically feasible) tuning strategies.
• Figure 2: Output tracking (solid black line) over $100$ Monte Carlo runs w.r.t. $a=DeePC$ (red tones) and $a=FCE$ (blue tones). (a)-(b) The 3 worst output trajectories in terms of instantaneous output tracking error adopting the same criterion to draw whiskers in MATLAB$^{\text{\textregistered}}$ boxplots (solid red and green lines). (c)-(d) Median realization in terms of instantaneous output tracking error (dashed red and blue lines); confidence intervals ($\pm 1.96$ std. dev., red and blue shaded areas).
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Theorems & Definitions (19)

• Definition 1
• Remark 1: Alternative formulations
• Remark 2: On the role of \ref{['eq:predictor']}
• Remark 3: On the choice of $\rho$
• Proposition 1
• Theorem 1
• Remark 4
• Lemma 1
• Remark 5: Truncation in a Bayesian setup
• Theorem 2: FCE with non-informative prior