Abstract
The result of J.C. Willems et al. “A note on persistency of excitation”, System & Control Letters, 2005 gives identifiability conditions for system identification as well as data-driven representations for data-driven control. The existing proofs however are proofs by contradiction and do not give insight into the assumptions of controllability and persistency of excitation of the input. Moreover, the existing proofs do not clarify how conservative the assumptions are. We provide an alternative constructive proof for the single-input case. It is shown that persistency of excitation of order more than the time horizon is needed in nongeneric cases, corresponding to special initial conditions. The special initial conditions are explicitly characterized in terms of the solution of a Sylvester equation. Another contribution of the paper is a representation of a scalar persistently exciting input of a finite order as an output of an autonomous linear time-invariant system.
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