Oscillator circuit using comparator and opamps
At the moment, all I can say is the following:

The circuit is an electronic model of a pendulum clock - that means: Each period it receives (from the comparator) an additional &quot;kick&quot; which can compensate the losses of the analog loop. In the literature, this principle is called &quot;restoration of initial conditions&quot;.

The transfer function of the closed-analog loop is a second-order lowpass with a pole-frequency of app wp=6.28*160 rad/s and a pole quality factor Qp=10.

Hence, the step response of this closed-loop has a decaying function exp[-(wp/2Q)t]

Periodically, the comparator output (+- 1V) injects energy into the loop thereby compensating the losses (that means: working against the decaying property of the lossy loop).

This state of compensation (a kind of equilibrium) defines the final steady-state amplitude of the circuit. This amplitude strongly depends on the comparator output voltage. This &quot;kick&quot; signal may be as low as 1µVolt - the circuit still oscillates, but with a very ow amplitude.


EDIT: Are you still interested in a formula for the amplitude?
Comment:I must admit that I am rather surprised that - up to now - there is no further comment or answer. For my opinion, the shown circuit is very interesting in its function and opens the question: Is it really true that all oscillators are non-linear circuits?
EDIT2: The considerations for finding the oscillation amplitude are as follows:

The high-Q lowpass oscillations will decay (without any &quot;kick&quot;) resulting in an amplitude difference

[Vmax- Vmax*exp(-t/tau)] = Vmax[1-exp(-t/tau)].

This loss must be compensated by the external &quot;kick&quot; (comparator voltage Vc) each half period. For this reason, we set t=1/2fo ; tau=wo/2Q (filter theory). In the design example Q=10 (Bode diagram for the closed.loop magnitude).

Equating the above expression with the amplified comparator voltage Vc*Ac, we can solve for Vmax and the result is Vmax=12.5 V. This is in full agreement with corresponding simulation.
The full swing comparator voltage is Vc=2V and the gain Ac for the comparator voltage is Ac=Zf/R with feedback impedance Zf=10R||(1/sC). (In our case: Ac=0.91).