  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.
