The Bias of Subspace-based Data-Driven Predictive Control

TL;DR

This paper analyzes and quantifies the bias in subspace-based Data-Driven Predictive Control methods caused by closed-loop data collection and proposes that Transient Predictive Control avoids these biases, improving tracking.

Contribution

It provides a theoretical analysis of biases in subspace-based DDPC methods and compares them with Transient Predictive Control, highlighting the sources of poor performance.

Findings

Closed-loop data introduces Subspace Bias in DDPC.

DeePC and $oldsymbol{ extgamma}$-DDPC exhibit both Subspace and Optimism Bias.

Transient Predictive Control does not suffer from these biases.

Abstract

This paper quantifies and addresses the bias of subspace-based Data-Driven Predictive Control (DDPC) for linear, time-invariant (LTI) systems. The primary focus is the bias that arises when the training data is gathered with a feedback controller in closed-loop with the system. First, the closed-loop bias of Subspace Predictive Control is quantified using the training data innovations. Next, the bias of direct, subspace-based DDPC methods DeePC and $\gamma$-DDPC is shown to consist of two parts--the Subspace Bias, which arises from closed-loop data, and an Optimism Bias, which arises from DeePC/$\gamma$-DDPC's "optimistic" adjustment of the output trajectory. We show that, unlike subspace-based DDPC methods, Transient Predictive Control does not suffer from Subspace Bias or Optimism Bias. Double integrator experiments demonstrate that Subspace and Optimism Bias are responsible for poor reference tracking by the subspace-based DDPC methods.