Finding transfer function of a compensator circuit similar to a type 2 compensator topology
A contribution: Using the Thevenin's Theorem it is possible to transform the circuit to a simpler one:



Using the superposition principle, the output \$V_{out}\$ is:

$$ V_{out}=\left [ 1 + \frac{Z_f(R_1+R_0)}{R_0R_1+R_0R_3+R_1R_3}\right ]V_{ref}  - \frac{Z_fR_0}{R_0R_1+R_0R_3+R_1R_3}V_{in}$$

You can't get a transfer function in closed form relating \$V_{out}\$ and \$V_{in}\$. This situation is similar to what happens in control systems, when the plant has two inputs: Reference and disturbance. In that case, the principle of superposition can be used, resulting in two separate transfer functions: one relating the output to the reference input and the other relating the output to the disturbance input.

However, assuming that \$V_{ref}\$ is constant, a transfer function related to input with the output can be obtained, considering the ratio between the variations of these quantities (linear relationship):

$$ \frac{\Delta V_{out}}{\Delta V_{in}} = G_1$$

where:

$$ G_1 = - \frac{Z_fR_0}{R_0R_1+R_0R_3+R_1R_3}$$