A note on input signal generators: A relaxation of Willems' fundamental lemma in the SISO case

TL;DR

This paper relaxes Willems' fundamental lemma for SISO systems by reformulating it through signal generators, enabling more flexible input signals and extending to continuous-time systems.

Contribution

It introduces a practical relaxation of Willems' lemma using signal generators, broadening the class of informative inputs and extending applicability to continuous-time systems.

Findings

Identifies a necessary and sufficient condition on signal generators for informative data.

Allows inputs outside traditional persistency of excitation, such as sinusoidal sequences with fewer frequencies.

Extends the fundamental lemma framework to continuous-time systems.

Abstract

We provide a practical relaxation of Willems' fundamental lemma for discrete-time linear time-invariant (single-input-single-output) systems. Instead of maintaining conventional Willems' persistency of excitation condition in the behavioral theory, we reformulate the problem in terms of signal generators, hence going back to the dynamical systems theory. We discuss the relationship between the persistency of excitation order and the dimension of the signal generator. Furthermore, we identify a necessary and sufficient condition on the signal generator that can generate informative input--output data for almost all systems and initial conditions. This even includes inputs outside the class originally suggested by Willems' fundamental lemma, for example, sinusoidal sequences with fewer frequencies. Finally, the signal generator perspective allows a natural extension to continuous-time systems.