The Innovation Null Space of the Kalman Predictor: A Stochastic Perspective for DeePC

TL;DR

This paper identifies an optimal null space for the decision variable in Willems' lemma under noisy conditions, linking it to the Kalman predictor and unifying various data-driven control methods.

Contribution

It introduces the concept of the innovation null space for the decision variable g, connecting Willems' lemma with Kalman filtering in noisy systems.

Findings

Optimal null space for g is the null space of the innovation Hankel matrix.

Predictors with g in this null space approximate the Kalman predictor.

Existing methods like DeePC, instrumental-variable, and ARX relate to this null space concept.

Abstract

Willems' fundamental lemma uses a key decision variable $g$ to combine measured input-output data and describe trajectories of a linear time-invariant system. In this paper, we ask: what is a good choice for this vector $g$ when the system is affected by noise? For a linear system with Gaussian noise, we show that there exists an optimal subspace for this decision variable $g$, which is the null space of the innovation Hankel matrix. If the decision vector lies in this null space, the resulting predictor gets closer to the Kalman predictor. To show this, we use a result that we refer to as the Kalman Filter Fundamental Lemma (KFFL), which applies Willems' lemma to the Kalman predictor. This viewpoint also explains several existing data-driven predictive control methods: regularized DeePC schemes act as soft versions of the innovation null-space constraint, instrumental-variable methods enforce it by construction, and ARX-based approaches explicitly estimate this innovation null space.