# Root Locus for 4 Real Poles and 2 Real Zeros

Cuthbert Nyack
Applet shows root locus of a system with 4 real poles and 2 real zeros.
This system has the asymptotic behaviour of a 2nd order system with 2 asymptotes at ±90°.
With parameters (0.0, 1.0, 1.0, 1.0, 1.0, 2.5, 2.5) There are 4 poles at -1 and 2 zeros at -2.5. This system is stable for K < 3.2.

With parameters (0.0, 1.0, 1.0, 1.0, 1.0, 2.0, 2.0) The center of gravity of the pole zero configuration is 0 and the asymptotes lie on the imaginary axis. This system is therefore stable for all K.

The root locus behaviour for different pole zero locations can be illustrated by the following parameters:-
(0.0, 3.5, 2.5, 1.5, 0.5, 4.0, 3.75) z-z-p-p-p-p
(0.0, 3.5, 2.5, 1.5, 0.5, 3.85, 3.0) z-p-z-p-p-p
(0.0, 3.5, 2.5, 1.5, 0.5, 2.85, 3.0) p-z-z-p-p-p
(0.0, 3.5, 2.5, 1.5, 0.5, 3.85, 2.0) z-p-p-z-p-p
(0.0, 3.5, 2.5, 1.5, 0.5, 2.25, 1.85) p-p-z-z-p-p
(0.0, 3.5, 2.6, 1.5, 0.5, 2.25, 1.85) p-p-z-z-p-p
(0.0, 3.5, 2.5, 1.5, 0.5, 2.25, 1.0) p-p-z-p-z-p
(0.0, 3.5, 2.6, 1.65, 0.5, 1.25, 1.0) p-p-p-z-z-p
(0.0, 3.5, 3.0, 2.5, 2.0, 0.2, 0.8) p-p-p-p-z-z

Gif image illustrating how applet should appear is shown below.