Towards explainable data-driven predictive control with regularizations

TL;DR

Data-driven predictive control (DPC) predicts future trajectories from data rather than a model, leveraging Willems' Fundamental Lemma. The paper introduces two analysis tools—trajectory-specific effects of regularization and implicit predictors—to explain how regularization shapes predictions and to bridge DPC with model-based viewpoints. It extends these insights to common DPC modifications (affine systems, offset regularization, slack variables, terminal constraints) and provides general feasibility observations. The results offer a principled, modular framework for understanding regularization effects, improving explainability and reliability of DPC in noisy or nonlinear settings. The findings have practical implications for parameter tuning and the design of robust DPC schemes.

Abstract

Data-driven predictive control (DPC), using linear combinations of recorded trajectory data, has recently emerged as a popular alternative to traditional model predictive control (MPC). Without an explicitly enforced prediction model, the effects of commonly used regularization terms (and the resulting predictions) can be opaque. This opacity may lead to practical challenges, such as reliance on empirical tuning of regularization parameters based on closed-loop performance, and potentially misleading heuristic interpretations of norm-based regularizations. However, by examining the structure of the underlying optimal control problem (OCP), more precise and insightful interpretations of regularization effects can be derived. In this paper, we demonstrate how to analyze the predictive behavior of DPC through implicit predictors and the trajectory-specific effects of quadratic regularization. We further extend these results to cover typical DPC modifications, including DPC for affine systems, offset regularizations, slack variables, and terminal constraints. Additionally, we provide a simple but general result on (recursive) feasibility in DPC. This work aims to enhance the explainability and reliability of DPC by providing a deeper understanding of these regularization mechanisms.

Figures (4)

• Figure 1: Direct data-driven control schemes aim to design control directly from data. This is in contrast to the indirect (i.e., model-based) data-driven control design paradigm. Here, we aim for an indirect viewpoint (highlighted in blue) on the predictions made by direct schemes via implicit predictors.
• Figure 2: Visualization of the data matrix $\mathcal{D}$ for the low-dimensional state-space example in Section \ref{['sec:example']}. The green marks show data columns $\left(x_0^{(i)}, \mathrm{u}^{(i)}, \mathrm{x}^{(i)}\right)$ for (a) the ideal deterministic LTI setting and (b) the same data with added measurement noise. The gray subspaces depict $\mathcal{R}\left({\mathcal{D}}\right)$, showing $\mathrm{rank}\left({\mathcal{D}}\right)=\mathrm{rank}\left({Z}\right)=2$ in (a) and full rank $\mathcal{D}$ with $\mathrm{rank}\left({\mathcal{D}}\right)=3>\mathrm{rank}\left({Z}\right)=2$ in (b).
• Figure 3: Implicit predictor, optimal parametric solutions, and least-squares mappings for the DPC problem discussed in Section 2.3. (a-d) The optimal parametric DPC solutions $(x_0, \mathrm{u}^\ast(x_0), \mathrm{x}^\ast(x_0))$ for the different regularizations $h(a) = \lambda \|a\|_2^2$ (orange) and $h(a) = \lambda \|\Pi_\perp a\|_2^2$ (green) naturally evolve on the implicit predictor $\hat{\mathrm{x}}_\text{DPC}(x_0, \mathrm{u})$ (gray). (e) Visualization of the least-square solutions $\hat{\mathrm{x}}_\text{LS}(x_0, \mathrm{u})$ (gray) and $\hat{\mathrm{u}}_\text{LS}(x_0)$. The latter is shown via the tuple $(x_0, \hat{\mathrm{u}}_\text{LS}(x_0), \hat{\mathrm{x}}_\text{LS}(x_0, \hat{\mathrm{u}}_\text{LS}(x_0)))$ (orange).
• Figure 4: Implicit predictor $\hat{\mathrm{x}}_\text{DPC}(x_0, \mathrm{u})$ for the DPC example discussed in Section \ref{['sec:example']} with prediction horizon $N_f = 2$ and reference $\mathrm{x}_\text{ref} = 0.50.5^\top$. To deal with the higher dimensionality, the first (green) and second (orange) prediction step of $\hat{\mathrm{x}}_\text{DPC}(x_0, \mathrm{u})$ are visualized individually. Furthermore, we constrain the second input to $\mathrm{u}_2=0$ and only visualize the first input $\mathrm{u}_1$.

Theorems & Definitions (19)

• Remark 1
• Definition 1
• Corollary 1: klaedtke2024unifyingPreprint
• Remark 2
• Definition 2: KLAEDTKE2023
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 4