A Note on Persistency of Excitation

Authors

Jan C. Willems, Paolo Rapisarda, Ivan Markovsky, Bart L.M. De Moor

Year

2005

Journal Paper

Algorithm

Willems' Fundamental Lemma

Source

Systems and Control Letters

Identifiers

DOI 10.1016/j.sysconle.2004.09.003

Abstract¶

We prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This is then applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The related question of when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts is of central importance in subspace system identification.

Summary¶

Proves the trajectory-spanning result now known as Willems' fundamental lemma: sufficiently exciting measured trajectories of a controllable LTI system span its finite-horizon behavior. The result is a key bridge from behavioral systems theory to modern data-enabled predictive control and subspace identification.