Willems’ Lemma Reformulations: Which Operators Preserve LTI System Behavior?

In the behavioral approach, dynamical systems are abstracted as sets of trajectories. This approach gave birth to the celebrated Willems’ Fundamental Lemma, for which various reformulations have been proposed in the literature, for instance, using frequency-domain data. In this note, we show that all reformulations are necessarily based on linear shift-invariant transformations, which have the fundamental property that they preserve the trajectory space of all LTI systems. Our results unify the existing reformulations and provide guidelines for designing novel ones.