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On the equivalence of direct and indirect data-driven predictive control approaches

Per Mattsson, Fabio Bonassi, Valentina Breschi, Thomas B. Schön

TL;DR

The paper addresses the question of how direct data-driven predictive control (DDPC) methods relate to indirect, model-based approaches. It shows that several direct DDPC methods are equivalent to an indirect predictor-based formulation that includes a slack term, and connects these ideas to SPC and γ-DDPC. The main result provides an explicit equivalence to an optimization problem with penalties on the regression input and the slack, clarifying conditions under which direct methods replicate indirect predictors and how data size affects performance. The findings offer a principled framework for analyzing DDPC methods, explain observed fragilities with fixed regularization in large data regimes, and suggest avenues for tuning and causal variants. Overall, the work deepens understanding of DDPC’s flexibility and limitations in finite-data, noisy settings.

Abstract

Recently, several direct Data-Driven Predictive Control (DDPC) methods have been proposed, advocating the possibility of designing predictive controllers from historical input-output trajectories without the need to identify a model. In this work, we show that these approaches are equivalent to an indirect approach. Reformulating the direct methods in terms of estimated parameters and covariance matrices allows us to give new insights into how they work in comparison with, for example, Subspace Predictive Control (SPC). In particular, we show that for unconstrained problems the direct methods are equivalent to SPC with a reduced weight on the tracking cost. Via a numerical experiment, motivated by the reformulation, we also illustrate why the performance of direct DDPC methods with fixed regularization tends to degrade as the number of training samples increases.

On the equivalence of direct and indirect data-driven predictive control approaches

TL;DR

The paper addresses the question of how direct data-driven predictive control (DDPC) methods relate to indirect, model-based approaches. It shows that several direct DDPC methods are equivalent to an indirect predictor-based formulation that includes a slack term, and connects these ideas to SPC and γ-DDPC. The main result provides an explicit equivalence to an optimization problem with penalties on the regression input and the slack, clarifying conditions under which direct methods replicate indirect predictors and how data size affects performance. The findings offer a principled framework for analyzing DDPC methods, explain observed fragilities with fixed regularization in large data regimes, and suggest avenues for tuning and causal variants. Overall, the work deepens understanding of DDPC’s flexibility and limitations in finite-data, noisy settings.

Abstract

Recently, several direct Data-Driven Predictive Control (DDPC) methods have been proposed, advocating the possibility of designing predictive controllers from historical input-output trajectories without the need to identify a model. In this work, we show that these approaches are equivalent to an indirect approach. Reformulating the direct methods in terms of estimated parameters and covariance matrices allows us to give new insights into how they work in comparison with, for example, Subspace Predictive Control (SPC). In particular, we show that for unconstrained problems the direct methods are equivalent to SPC with a reduced weight on the tracking cost. Via a numerical experiment, motivated by the reformulation, we also illustrate why the performance of direct DDPC methods with fixed regularization tends to degrade as the number of training samples increases.
Paper Structure (18 sections, 3 theorems, 33 equations, 2 figures)

This paper contains 18 sections, 3 theorems, 33 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem setting
  4. Background
  5. Multi-step prediction model
  6. Causal multi-step prediction model
  7. Direct methods
  8. $\gamma$-DDPC formulation
  9. The main result
  10. Implications
  11. The role of the slack variable
  12. The role of the past horizon $\rho$
  13. The role of the training data
  14. Illustrative example
  15. Conclusions
  16. ...and 3 more sections

Key Result

theorem 1

If eq:deepc is feasible, it has the same optimal $\textrm{\bfseries u}$ and $\hat{\textrm{\bfseries y}}$ as eq:result with $\widehat{\Theta} = Y\Phi^\dagger$ and Furthermore, if Assumption assumption holds then eq:result and eq:gddpc have the same optimal $\textrm{\bfseries u}$ and $\hat{\textrm{\bfseries y}}$ if $\lambda_1 = \beta_2$ and $\lambda_2 = \beta_3$.

Figures (2)

  • Figure 1: Mean squared value of the slack $\Delta \hat{\textrm{\bfseries y}}$ used by DeePC$_{\lambda_2}$ for different values of $\lambda_2$ (blue: $1$, purple: $10$, green: $100$, orange: $1000$).
  • Figure 2: Cost criterion \ref{['eq:perfcriterion']} (top) and dissimilarity from the oracle \ref{['eq:oracledist']} (bottom) achieved by the DeePC$_{\lambda_2}$ (solid lines, $\lambda_2$ color-coded as in Figure \ref{['fig:slack']}), SPC (black-dotted line), and C-SPC (red dashed-dotted line).

Theorems & Definitions (13)

  • Remark 1
  • Remark 2: SPC
  • Remark 3
  • Remark 4
  • theorem 1
  • proof
  • Remark 5
  • Remark 6
  • Remark 7
  • Corollary 1
  • ...and 3 more