I wish to calculate the Final Value of systems in which a high pass filter of the output feeds back into the input.
A simple example would be:
where is a 1st order high pass filter with transfer function:
I was expecting the
y in the above example to have an infinite final value to a step in
x, because keeps feeding
However, the workings below give a different answer:
hp1(z)in terms of its inputs only:
Add to both sides of the system's equation:
Write the system's transfer function:
Re-write the infinite sum in the denominator:
Apply the Final Value Theorem to the response of this system to a step in x:
Taking the limit:
The above suggests that the system has a well defined terminal value to a step in x. However I don't think that can be the case.
Where am I going wrong? Help much appreciated