[Paper Review] Data-driven predictive control in a stochastic setting: a unified framework

Valentina Breschi, Alessandro Chiuso+1 more|arXiv (Cornell University)|Mar 21, 2022

Advanced Control Systems Optimization5 citations

TL;DR

This paper proposes a unified framework for regularized data-driven predictive control (DDPC) in stochastic systems, introducing γ-DDPC—a two-stage optimization scheme that decouples initial condition fitting from performance optimization. The method improves robustness to noise and simplifies tuning by leveraging subspace identification, outperforming existing schemes in closed-loop performance and stability with reduced computational complexity.

ABSTRACT

Data-driven predictive control (DDPC) has been recently proposed as an effective alternative to traditional model-predictive control (MPC) for its unique features of being time-efficient and unbiased with respect to the oracle solution. Nonetheless, it has also been observed that noise may strongly jeopardize the final closed-loop performance since it affects both the data-based system representation and the control update computed from the online measurements. Recent studies have shown that regularization is potentially a successful tool to counteract the effect of noise. At the same time, regularization requires the tuning of a set of penalty terms, whose choice might be practically difficult without closed-loop experiments. In this paper, by means of subspace identification tools, we pursue a three-fold goal: $(i)$ we set up a unified framework for the existing regularized data-driven predictive control schemes for stochastic systems; $(ii)$ we introduce $γ$-DDPC, an efficient two-stage scheme that splits the optimization problem into two parts: fitting the initial conditions and optimizing the future performance, while guaranteeing constraint satisfaction; $(iii)$ we discuss the role of regularization for data-driven predictive control, providing new insight on $when$ and $how$ it should be applied. A benchmark numerical case study finally illustrates the performance of $γ$-DDPC, showing how controller design can be simplified in terms of tuning effort and computational complexity when benefiting from the insights coming from the subspace identification realm.

Motivation & Objective

• To unify existing regularized data-driven predictive control (DDPC) schemes under a single theoretical framework.
• To address the challenge of noise-induced performance degradation in data-driven control by formalizing the role of regularization.
• To propose γ-DDPC, a two-stage optimization approach that decouples initial condition estimation from future performance optimization.
• To simplify controller tuning and reduce computational complexity through insight from subspace identification.
• To validate the effectiveness of the proposed framework via a benchmark numerical case study.

Proposed method

• The framework is built on subspace identification tools to unify various regularized DDPC schemes, revealing their connection to Subspace Predictive Control (SPC).
• γ-DDPC decomposes the control problem into two sequential optimizations: fitting initial conditions and optimizing future control actions under constraints.
• Regularization is applied via penalty terms on the parameter vector and slack variables to handle noisy data and ensure constraint satisfaction.
• The method uses a data-based representation of system dynamics via the fundamental lemma, replacing model equations with data-driven constraints.
• Performance is evaluated using Monte Carlo simulations with varying regularization parameters (λ₁, λ₂), comparing against an oracle MPC solution.
• The approach is validated on a benchmark system with SNR = 18 dB, analyzing performance gaps and input variability across different tuning configurations.

Experimental results

Research questions

• RQ1 How can existing regularized DDPC schemes be unified under a single theoretical framework?
• RQ2 What is the role of regularization in improving closed-loop performance under stochastic disturbances?
• RQ3 How does the two-stage γ-DDPC formulation simplify tuning and improve robustness compared to existing DDPC methods?
• RQ4 What is the impact of different regularization parameters (λ₁, λ₂) on controller performance and stability?
• RQ5 Can the insights from subspace identification be leveraged to design a more interpretable and efficient data-driven controller?

Key findings

• γ-DDPC achieves closed-loop performance comparable to the oracle MPC, with a performance gap of less than 5% when regularization parameters are properly tuned.
• The scheme significantly reduces variability in control inputs and performance across Monte Carlo trials, outperforming [17] and [19] in terms of consistency.
• Higher values of λ₂ (regularization on the slack variables) and lower values of λ₁ (regularization on the parameter vector) lead to performance closer to the oracle, validating the theoretical insights.
• The use of slack variables combined with regularization leads to a control input sequence that closely matches the oracle MPC, especially when λ₂ is large.
• Even with optimal tuning, previous schemes like [17] and [19] show higher performance variability and worse average performance than γ-DDPC.
• The two-stage structure of γ-DDPC simplifies the interpretation of regularization effects, enabling more intuitive and efficient tuning.