A Quantitative Notion of Persistency of Excitation and the Robust Fundamental Lemma

The fundamental lemma by Willems and coauthors enables a parameterization of all trajectories of a linear time-invariant system in terms of a single, measured one. This result plays a key role in data-driven simulation and control. The fundamental lemma relies on a persistently exciting input to the system to ensure 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. In this letter 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 qualitative rank conditions. This is achieved by designing the inputs of the experiment such that the minimum singular value of an input Hankel matrix is sufficiently large, inspiring a quantitative notion of persistency of excitation. We highlight the relevance of the result in a data-driven control case study by comparing the predictive control performance for varying degrees of persistently exciting data.