Quite HOW it all works I don't know, but when the system is stimulated by a PRBS of signals that keep the output UNsaturated, a record is made of the system output. A cross-correlation algorithm is then used to extract the impulse response for the system from the data. With the impulse response, it is possible to apply a control strategy.

We have played with a Ziegler- Nichols method with sampled data systems and it didn't really work well for us - and as Alfredo points out, you can't force a PID solution onto a system with significant third or fourth order poles, or other non-linear systems.

Since Jez seems to have far more maths than the rest of us, I hope he can inject some comments in this thread.
Steve