Finding Root Locus for 3 Real Poles

Cuthbert Nyack

Applet shows the determination of the root locus using the angle and magnitude condition.
For parameters (0.0, 0.5, 1.0, 1.5, sr, si) the root locus for different K is shown starting from the open loop poles(sr = 0.5, 1.0, 1.5), breaking away from the real axis at sr = ~-0.71 and tending towards the asymptotes. To find the closed loop poles for any K sr and si are adjusted along the locus until the desired K is obtained.
To find poles for K = 1.0, sr = -0.46 and si = 0.8 give K = 1.014 and f = -180.68. Also sr = -2.09, si = 0.0 give K = 1.02 f = -540°. Therefore for K = 1, the poles are ~-2.09, ~-0.46 ± j0.8.
These parameters (1.0, 0.5, 1.0, 1.5, -0.46, 0.8) show that the locus starts from the closed loop poles. The step response is shown in white and the frequency in light magenta.
For K = 2, poles are at ~2.33, ~ -0.34 ± j1.04.

This system may have an open loop pole to the right of the imaginary axis and still be controllable.
eg parameters (0.0, 2.0, 2.5, -0.5, sr, si) has a pole at +0.5.
Setting sr = 0.0, si = 0.0, shows K = 2.5 when the locus crosses the axis moving to the left.
Setting sr = 0.0, si = 1.66 show K = ~13.5 when the locus crosses the axis to the right.
Increasing K from 0 shows the system is stable for 13.5 > K > 2.5 and unstable for other K's.

The calculated values of the closed loop poles are also shown on the applet.

Gif image below show how applet should appear.

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