A New Perspective on Willems’ Fundamental Lemma: Universality of Persistently Exciting Inputs

In this letter, we provide new insight into Willems et al.’s fundamental lemma by studying the concept of universal inputs. An input is called universal if, when applied to any controllable system, it leads to input-output data that parameterizes all finite trajectories of the system. By the fundamental lemma, inputs that are persistently exciting of sufficiently high order are universal. The main contribution of this letter is to prove the converse. Therefore, universality and persistency of excitation are equivalent.