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Insights into the explainability of Lasso-based DeePC for nonlinear systems

Gianluca Giacomelli, Simone Formentin, Victor G. Lopez, Matthias A. Müller, Valentina Breschi

TL;DR

This work investigates the explainability of data-driven predictive control for nonlinear systems by examining DeePC with Lasso regularization. It derives an explicit, albeit non-prioritizing, solution showing the selector is piecewise affine in the initial condition, highlighting inherent unexplainability of Lasso-DeePC under generic data. The authors propose data-structuring strategies—Mosaic, explainable Hankel, and explainable Page matrices—to emphasize operating conditions and study their effect on performance and interpretability using a benchmark unbalanced-disk system. The findings reveal a fundamental trade-off: while data design can improve tracking, the Lasso-based DeePC decision remains difficult to explain purely from data, motivating future work on explainable-by-design controllers for nonlinear and LPV/PWA systems.

Abstract

Data-enabled Predictive Control (DeePC) has recently gained the spotlight as an easy-to-use control technique that allows for constraint handling while relying on raw data only. Initially proposed for linear time-invariant systems, several DeePC extensions are now available to cope with nonlinear systems. Nonetheless, these solutions mainly focus on ensuring the controller's effectiveness, overlooking the explainability of the final result. As a step toward explaining the outcome of DeePC for the control of nonlinear systems, in this paper, we focus on analyzing the earliest and simplest DeePC approach proposed to cope with nonlinearities in the controlled system, using a Lasso regularization. Our theoretical analysis highlights that the decisions undertaken by DeePC with Lasso regularization are unexplainable, as control actions are determined by data incoherent with the system's local behavior. This result is true even when the available input/output samples are grouped according to the different operating conditions explored during data collection. Our numerical study confirms these findings, highlighting the benefits of data grouping in terms of performance while showing that explainability remains a challenge in control design via DeePC.

Insights into the explainability of Lasso-based DeePC for nonlinear systems

TL;DR

This work investigates the explainability of data-driven predictive control for nonlinear systems by examining DeePC with Lasso regularization. It derives an explicit, albeit non-prioritizing, solution showing the selector is piecewise affine in the initial condition, highlighting inherent unexplainability of Lasso-DeePC under generic data. The authors propose data-structuring strategies—Mosaic, explainable Hankel, and explainable Page matrices—to emphasize operating conditions and study their effect on performance and interpretability using a benchmark unbalanced-disk system. The findings reveal a fundamental trade-off: while data design can improve tracking, the Lasso-based DeePC decision remains difficult to explain purely from data, motivating future work on explainable-by-design controllers for nonlinear and LPV/PWA systems.

Abstract

Data-enabled Predictive Control (DeePC) has recently gained the spotlight as an easy-to-use control technique that allows for constraint handling while relying on raw data only. Initially proposed for linear time-invariant systems, several DeePC extensions are now available to cope with nonlinear systems. Nonetheless, these solutions mainly focus on ensuring the controller's effectiveness, overlooking the explainability of the final result. As a step toward explaining the outcome of DeePC for the control of nonlinear systems, in this paper, we focus on analyzing the earliest and simplest DeePC approach proposed to cope with nonlinearities in the controlled system, using a Lasso regularization. Our theoretical analysis highlights that the decisions undertaken by DeePC with Lasso regularization are unexplainable, as control actions are determined by data incoherent with the system's local behavior. This result is true even when the available input/output samples are grouped according to the different operating conditions explored during data collection. Our numerical study confirms these findings, highlighting the benefits of data grouping in terms of performance while showing that explainability remains a challenge in control design via DeePC.

Paper Structure

This paper contains 12 sections, 2 theorems, 31 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Setting & Goal
  3. Conditions on the data & their implications
  4. Lasso-based DeePC: an explicit solution
  5. Explainability & Lasso-DeePC
  6. Strategies to highlight operating conditions in data
  7. Lasso-DeePC with grouped matrices & its explainability
  8. A benchmark case study: the unbalanced disk
  9. The importance of data collection
  10. The impact of $\lambda_g$
  11. Performance vs explainability
  12. Conclusions

Key Result

Proposition 1

Let $\lambda_g>0$ and $\lambda_2>0$, with $\lambda_2$ infinitesimally small. Under Assumption assump:indep, the explicit solution of eq:DeePC_lasso3 is a PWA law in the initial conditions $z_{\mathrm{ini}}$ (see eq:PWAlaw), where all the available data, the references and the weights characterizing

Figures (4)

  • Figure 1: Considered data structures: Hankel matrices comprising data collected (a) around a single operating point, (b) when “ rapidly” transitioning between two operating points multiple times, (c) transitioning from one operating point to the other only once.
  • Figure 2: Sensitivity to $\lambda_g$: performance indicators for the two OPs Hankel vs block Hankel for two different “ switching” references.
  • Figure 3: Tracking performance of Lasso-based DeePC: two OPs Hankel vs block Hankel for two different “ switching” references.
  • Figure 4: Values of the selector's component at different instants of the simulation horizon for two different “ switching” references.

Theorems & Definitions (9)

  • Definition 1: Persistency of excitation willems2005note
  • Definition 2: Explainable Lasso-based DeePC
  • Remark 1: The selector $g_{t}$
  • Definition 3: Explainability through local behaviors
  • Proposition 1: Explicit solution
  • proof
  • Corollary 1: Explicit solution & grouped matrices
  • proof
  • Remark 2