The Free Response of an Identified Model of a Linear System with a Completely Unknown Harmonic Disturbance Exactly Forcasts the Free-Plus-Forced Response of the True System Thereby Enabling Adaptive MPC for Harmonic Disturbance Rejection

This paper considers system identification in the presence of an unmeasured, unknown, and unmatched multitone harmonic disturbance with completely unknown spectrum. It is shown that the identified model possesses spurious poles at the disturbance frequencies that are cancelled by coincident, spurious zeros. Although the presence of the spurious poles is expected, this paper shows that the free response of the identified model is identical—in frequencies, amplitudes, and phases—to the free-plus-forced response of the true system. Consequently, by retaining—rather than cancelling—the coincident, spurious poles and zeros, the identified model has the ability to forecast the future response to an unknown harmonic input over a prediction horizon during which the harmonic disturbance persists. A numerical example illustrates the usefulness of this property to model predictive control with concurrent system identification for rejecting unmeasured, unknown, and unmatched harmonic disturbances with completely unknown spectrum.