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 2nd loop function:
Using Tustin’s method to move HREF[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,