It’s reasonably well-established how a single phase-locked loop (PLL) works in terms of its dynamics. You can look this up in many textbooks [1, 2], but in general, one can describe the response of a PLL in the phase-domain (where one thinks of the input not as a sinusoid, but as the argument to that sinusoid – the phase itself). In this case, you can come up with a phase-domain transfer function for the PLL when it is locked onto an input signal. The advantage of modeling a PLL in phase domain is, it allows us to exploit the linear system analysis techniques to understand its dynamics.

Most of the time, PLLs are used as elements that can demodulate the frequency modulation (FM) signals on a carrier frequency as in the case of FM radios. However the case we are interested in is, where the PLL can oscillate a resonator at its resonance frequency in closed loop, by feeding back into the oscillating signal itself. The resonant frequency of the resonator might be changing over time, and the PLL keeps the system oscillating at the varying resonant frequency. So in this closed loop case the linearized model of the system is different than a PLL acting as a demodulator.

We might think of the input of the closed loop system as the resonator’s resonant frequency, and the output as the measured resonant frequency (take the derivative of the input and the output of the phase domain model).

The behavior of the resonator can be approximated by a linear model, where the resonator is a first-order low-pass filter **in the phase-domain **(the blue box below). If you put this model of a resonator in feedback with a PLL, you can simplify the closed loop system as shown below:

Where ki and kp are gains of the proportional-intagral (PI) controller in the PLL, and tau is the resonator time constant (equal to two times the quality factor over the angular resonant frequency). Here the extra system component shown as a box in the PLL side represents extra poles for engineering the closed loop system dynamics. By controlling the PI controller and the location of the (optional) extra poles, we can tailor the frequency response of the overall system. Further details can be found here.

[1] Gardner F.M., Phaselock Techniques, Third edition, 2005

[2] Best, RE, Phase-locked Loops, Sixth edition, 2007