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

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.