I am trying to reproduce the transfer functions of the PID controllers. I am running into difficulties in some parameter regimes, for example when the integrator saturation frequency is close to the the proportional gain frequency. To model a controller with proportional gain, integral gain, and integral saturation I guess the following transfer function

G(s) = prop_gain*(s+int_crossover)/(s+prop_gain*int_crossover/int_saturation)

By taking the limit s->0 we see that the DC gain of this transfer function is int_saturation. For a clear example, when we take the integrator saturation to be equal to the proportional gain, this transfer function simplifies to a controller with only proportional gain.

When I set the integrator saturation equal to the proportional gain and look at the bode plot on the Moku GUI, it is clear that this transfer function is not the one that I guessed. The DC gain shown by the Moku appears to be about 5dB.

What is the transfer function of the PID controller?