  The present paper develops recursive algorithms to track shifts in the
resonance frequency of linear systems in real time. To date, automatic
resonance tracking has been limited to non-model-based approaches, which rely
solely on the phase difference between a specific input and output of the
system. Instead, we propose a transformation of the system into a
complex-valued representation, which allows us to abstract the resonance shifts
as an exogenous disturbance acting on the excitation frequency, perturbing the
excitation frequency from the natural frequency of the plant. We then discuss
the resonance tracking task in two parts: recursively identifying the frequency
disturbance and incorporating an update of the excitation frequency in the
algorithm. The complex representation of the system simplifies the design of
resonance tracking algorithms due to the applicability of well-established
techniques. We discuss the stability of the proposed scheme, even in cases that
seriously challenge current phase-based approaches, such as nonmonotonic phase
differences and multiple-input multiple-output systems. Numerical simulations
further demonstrate the performance of the proposed resonance tracking scheme.
