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On the impact of regularization in data-driven predictive control

Valentina Breschi, Alessandro Chiuso, Marco Fabris, Simone Formentin

TL;DR

The paper addresses how two regularization penalties affect data-driven predictive control via $γ$-DDPC under finite data and noise. It presents two regularization forms, $\Psi_{\gamma_2}$ and $\Psi_u$, and proves an asymptotic equivalence between them when training input excitation is white, while showing the equivalence breaks with colored data, guiding when to apply each penalty. Through linear and nonlinear case studies, it demonstrates that regularizing $\gamma_2$ reduces predictor variance and improves closed-loop performance in non-white data, and that online tuning of $β_3$ can be highly effective, with joint offline optimization sometimes outperforming single-parameter tuning. The results yield practical tuning guidelines for end users and show that $γ$-DDPC can be competitive with model-based controllers, even in nonlinear settings, under noisy data conditions.

Abstract

Model predictive control (MPC) is a control strategy widely used in industrial applications. However, its implementation typically requires a mathematical model of the system being controlled, which can be a time-consuming and expensive task. Data-driven predictive control (DDPC) methods offer an alternative approach that does not require an explicit mathematical model, but instead optimize the control policy directly from data. In this paper, we study the impact of two different regularization penalties on the closed-loop performance of a recently introduced data-driven method called $γ$-DDPC. Moreover, we discuss the tuning of the related coefficients in different data and noise scenarios, to provide some guidelines for the end user.

On the impact of regularization in data-driven predictive control

TL;DR

The paper addresses how two regularization penalties affect data-driven predictive control via -DDPC under finite data and noise. It presents two regularization forms, and , and proves an asymptotic equivalence between them when training input excitation is white, while showing the equivalence breaks with colored data, guiding when to apply each penalty. Through linear and nonlinear case studies, it demonstrates that regularizing reduces predictor variance and improves closed-loop performance in non-white data, and that online tuning of can be highly effective, with joint offline optimization sometimes outperforming single-parameter tuning. The results yield practical tuning guidelines for end users and show that -DDPC can be competitive with model-based controllers, even in nonlinear settings, under noisy data conditions.

Abstract

Model predictive control (MPC) is a control strategy widely used in industrial applications. However, its implementation typically requires a mathematical model of the system being controlled, which can be a time-consuming and expensive task. Data-driven predictive control (DDPC) methods offer an alternative approach that does not require an explicit mathematical model, but instead optimize the control policy directly from data. In this paper, we study the impact of two different regularization penalties on the closed-loop performance of a recently introduced data-driven method called -DDPC. Moreover, we discuss the tuning of the related coefficients in different data and noise scenarios, to provide some guidelines for the end user.
Paper Structure (8 sections, 1 theorem, 30 equations, 3 figures)

This paper contains 8 sections, 1 theorem, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem setting
  3. The role of regularization
  4. Theoretical analysis
  5. Experimental analysis
  6. Benchmark LTI system
  7. Wheel slip control problem
  8. Concluding remarks and future directions

Key Result

Theorem 1

If the training input sequence $u(t)$ in the Hankel matrices $U_F$ and $U_P$ is (zero mean) white with variance $\sigma^2 I$, the regularization terms $\Psi_{\gamma_2}$ in eq:reg1 and $\Psi_{u}$ in eq:reg2 are asymptotically (in $N$) equivalent up to a rescaling of the weight $\beta_2$.

Figures (3)

  • Figure 1: (a): comparison between the Kalman-filter-based oracle performance $J_{OR}$ and the minimum cost realizations $\widehat{J}^{N_{data}}_{n_s,rg}$ for $\Sigma_{L}$ over $100$ Monte Carlo runs; (b): Optimal performance under the constraint $\beta_2=0$; (c): distribution of the corresponding minimizers $(\beta_2^\star,\beta_3^\star)$.
  • Figure 2: (a): training data set employed for all the $\gamma$-DDPC experiments on the wheel slip control problem. (b): comparison of the performance indexes obtained with different $\gamma$-DDPC strategies (bar and hat notation indicating offline and online approaches respectively) and a model-based oracle. The subscript $a \in \{0, 2,3,23\}$ on $J$ refers to the regularization scheme (respectively: no regularization, $\beta_2$, $\beta_3$ or both); (c): input/output tracking obtained from an MPC-based oracle. Mean (line) and $1.95$ times the standard deviation (shaded area) of the closed-loop input/output trajectories; the reference input and output are indicated with black dashed lines.
  • Figure 3: For all diagrams: mean (line) and $1.95$ times the standard deviation (shaded area) of the closed-loop input/output trajectories; the reference input and output are indicated with black dashed lines. (a): $\gamma$-DDPC with no regularization; (b)-(c): offline regularization strategies employing $\bar{\beta}_2$ and $\bar{\beta}_3$ separately; (d): offline regularization strategies employing $\bar{\beta}_2$ and $\bar{\beta}_3$ jointly; (e)-(f): online regularization strategies employing $\hat{\beta}_2$ and $\hat{\beta}_3$ separately.

Theorems & Definitions (2)

  • Theorem 1
  • proof