Robust Fundamental Lemma for Data-driven Control

Abstract: The fundamental lemma by Willems and coauthors facilitates a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays an important role in data-driven simulation and control. Under the hood, the fundamental lemma works by applying a persistently exciting input to the system. This ensures that the Hankel matrix of resulting input/output data has the "right" rank, meaning that its columns span the entire subspace of trajectories. However, such binary rank conditions are known to be fragile in the sense that a small additive noise could already cause the Hankel matrix to have full rank. Therefore, in this extended abstract we present a robust version of the fundamental lemma. The idea behind the approach is to guarantee certain lower bounds on the singular values of the data Hankel matrix, rather than mere rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of a deeper input Hankel matrix is sufficiently large. This inspires a new quantitative and robust notion of persistency of excitation. The relevance of the result for data-driven control will also be highlighted through comparing the predictive control performance for varying degrees of persistently exciting data.