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Implicit predictors in regularized data-driven predictive control

Manuel Klädtke, Moritz Schulze Darup

TL;DR

This work defines implicit predictors as the predictive behavior implicitly attributed to a data-driven predictive control (DPC) scheme, even when an explicit predictor constraint is not enforced. It analyzes how finite 2-norm regularization and projection-based regularization shape the implicit predictor in regularized DPC, deriving closed-form expressions that relate DPC predictions to the SPC predictor and showing how constraints induce a multiparametric quadratic program that yields a piecewise affine predictor. The key finding is that, under full-row-rank data and typical regularizers, the implicit predictor lies on a subspace between the SPC predictor and zero prediction, with the tilt controlled by the regularization weight; constraints can push the predictor toward feasible regions via a PWA map. This perspective provides a theoretical lens to diagnose, compare, and potentially design DPC schemes by matching their implicit predictors to known system properties and desired predictive behavior, and it suggests avenues for stability analyses and new regularization strategies that exploit the implicit-predictor structure.

Abstract

We introduce the notion of implicit predictors, which characterize the input-(state)-output prediction behavior underlying a predictive control scheme, even if it is not explicitly enforced as an equality constraint (as in traditional model or subspace predictive control). To demonstrate this concept, we derive and analyze implicit predictors for some basic data-driven predictive control (DPC) schemes, which offers a new perspective on this popular approach that may form the basis for modified DPC schemes and further theoretical insights.

Implicit predictors in regularized data-driven predictive control

TL;DR

This work defines implicit predictors as the predictive behavior implicitly attributed to a data-driven predictive control (DPC) scheme, even when an explicit predictor constraint is not enforced. It analyzes how finite 2-norm regularization and projection-based regularization shape the implicit predictor in regularized DPC, deriving closed-form expressions that relate DPC predictions to the SPC predictor and showing how constraints induce a multiparametric quadratic program that yields a piecewise affine predictor. The key finding is that, under full-row-rank data and typical regularizers, the implicit predictor lies on a subspace between the SPC predictor and zero prediction, with the tilt controlled by the regularization weight; constraints can push the predictor toward feasible regions via a PWA map. This perspective provides a theoretical lens to diagnose, compare, and potentially design DPC schemes by matching their implicit predictors to known system properties and desired predictive behavior, and it suggests avenues for stability analyses and new regularization strategies that exploit the implicit-predictor structure.

Abstract

We introduce the notion of implicit predictors, which characterize the input-(state)-output prediction behavior underlying a predictive control scheme, even if it is not explicitly enforced as an equality constraint (as in traditional model or subspace predictive control). To demonstrate this concept, we derive and analyze implicit predictors for some basic data-driven predictive control (DPC) schemes, which offers a new perspective on this popular approach that may form the basis for modified DPC schemes and further theoretical insights.
Paper Structure (10 sections, 5 theorems, 27 equations, 2 figures)

This paper contains 10 sections, 5 theorems, 27 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem statement and preliminaries
  3. DPC and its relation to MPC
  4. The role of predictors and SPC
  5. Implicit predictors in regularized DPC
  6. (Squared) 2-norm regularization
  7. Projection-based (squared) 2-norm regularization
  8. Predictions affected by (output) constraints
  9. A remark on the numerical example
  10. Conclusions and Outlook

Key Result

Lemma 1

The SPC predictor eq:SPC_predictor is an implicit predictor for the unregularized DPC problem eq:DPC with exact trajectory data generated by an LTI system.

Figures (2)

  • Figure 1: The implicit predictor (grey) is equal for both analyzed unconstrained DPC schemes and its structure is given by a subspace that "tilts" between $\hat{\mathrm{y}}_\text{DPC}(\xi, \mathrm{u}_f)=0$ and the SPC predictor \ref{['eq:SPC_predictor']} depending on $\lambda_a$. The parametric DPC solutions $(\xi, \mathrm{u}_f^\ast(\xi), \mathrm{y}_f^\ast(\xi))$ for the different regularizations (green/orange) evolve on this subspace, confirming its validity.
  • Figure 2: The implicit predictor (grey) for the analyzed DPC schemes with (output) constraints is a PWA function. For increasing $\lambda_a$, the parametric DPC solutions (orange/green, see Fig. \ref{['fig:unconstrainedDPC']} for a legend) tend to stay on the segment that matches the unconstrained predictor (if possible).

Theorems & Definitions (13)

  • Definition 1
  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 3 more