# FW: AW: Re: digital contents

Paolo Predonzani predo at dist.dist.unige.it
Tue Sep 24 15:01:52 CEST 1996

```>
>    It's not so small. The open loop poles almost touch (0,0) in the s-plane.
>    As to the closed-loop poles I never examined them in detail because I
>    would have to find the roots of a 4-th order polinomial. Currently I don't
>    have an algorithm for that.
>
> I'm not sure that I understand you here.  If you mean the positions of
> the (linear) poles with feedback, I can tell you that the poles start
> at -w (real, negative), and, as feedback is applied, they split off in
> an X pattern at 45/135/-45/-135 degrees.  Eventually the two rightmost
> poles hit the axis and it oscillates.

Ok, this is the root locus for constant fcutoff and increasing open-loop
gain but I was considering the case when the poles start
moving as the large input signal makes them move.
The X pattern locus is correct when there are 4 coincident poles in open
loop. As you can see in the plot I posted, the open-loop poles don't
move together. Their trajectories are similar but not identical. In
addition they are slightly shifted in time. So the X shaped root locus
is only an asimptotical approximation (The center of the X being the
average of the open loop pole positions).
So my problem is: let's consider one closed-loop pole (say the one on
the imaginary axis in the self-oscillating case); if the input is null
the pole is exactly on the axis but if the input is large enough the
closed-loop poles move too. How do they move?

--
+-------------------+----------------------------------+
| Paolo Predonzani  |  email: predo at dist.dist.unige.it |
+-------------------+----------------------------------+

```