In the figure of the control loop above, we want to find the transfer function, so we solve for:

In the time domain representation, we have:

The last integral is a first-order filter, which is easily shown to be represented in the z-domain as:

So using the bilinear z-transform, we convert the discrete time series to:

Rearranging, weget:

Multiplying both sides by   gives us:

Therefore,

We want to reformat this equation into the classical 2^nd loop function:

Using Tustin’s method to move H_REF[s] to the z-domain:

If we work through the algebra, and substitute , where  is the undamped natural frequency, we get:

This looks messy now, but we can find substitutions for this equation in terms of our loop gains, α and β, where α is known as the proportional gain and β is known as the integral gain. Specifically,

Similarly,

Which leaves,