Root Locus for 4 Real Poles and 2 Complex Conjugate Zeros

Cuthbert Nyack

Applet shows the root locus of a system with 4 real poles and 2 complex conjugate zeros. This system has the asymptotic behaviour of a second order system with 2 asymptotes at ±90°
For parameters (0.0, 2.0, 1.0, 0.45, 0.05, 0.95, 1.75) the system has 4 real poles at -2.0, -1.0, -0.45, -0.05 and 2 cc zeros at -0.95 ± 1.75 and is stable for K < 2.0 and K > 31.3.

For parameters (0.0, 1.5, 1.0, 0.3, 0.2, 2.0, 1.0) the system has 4 real poles at -1.5, -1.0, -0.3, -0.2 and 2 cc zeros at -2.0 ± 1.0 and is stable for K < 1.8. The asymptotes are to the right of the imaginary axis at +0.5.

For parameters (0.0, 2.0, 1.0, 0.45, 0.05, 0.85, 0.85) the system has 4 real poles at -2.0, -1.0, -0.45, -0.05 and 2 cc zeros at -0.85 ± 0.85 and is stable for all K because none of the loci cross the imaginary axis.

Parameters where the real part of the zero is equal to the asymptote acts as a separation between the 2 cases where the loci to the left ends on the zeros and where the loci on the right ends on the zeros eg (0.0, 2.0, 1.5, 1.0, 0.5, 1.25, 2.0var) when the asymptotes are at -1.25.

Gif image illustrating how applet should appear is shown below.

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