Root Locus for Minimum and Non Minimum Phase

Cuthbert Nyack

The first Applet below show the Root Locus of (2) as well as the step response of (1) and (2)
(1) a second order type 0 system with a zero to the left of the imaginary axis.
G(s) = K(1 + Tns)/((T1s + 1)(T2s + 1))), C(s)/R(s) = K(1 + Tns)/(((T1s + 1)(T2s + 1)) + K(1 + Tns))

(2) a second order type 0 system with a zero to the right of the imaginary axis(nonminimum phase).
G(s) = K(1 - Tns)/(((T1s + 1)(T2s + 1))), C(s)/R(s) = K(1 - Tns)/(((T1s + 1)(T2s + 1)) + K(1 - Tns))

The eg parameters below show the nonminimum phase system becoming unstable, in each case the closed loop poles can be located by the cyan x's and the period and oscillation frequency can be determined from the step response.
(20.0, 1.0, 2.0, -0.15, 1.0) closed loop pole ~±j3.2, period=~1.94, w=~3.24
(10.0, 1.0, 2.0, -0.3, 1.0) closed loop pole ~±j2.38, period=~2.66, w=~2.35
(3.3, 1.0, 2.0, -0.9, 1.0) closed loop pole ~±j1.4, period=~4.3, w=~1.46

The applet below show the root locus for both cases where the zero is to the left and to the right of the imaginary axis.:- eg parameters (0.0, 1.0, 2.0, 0.4, 1.0) show the locus when the zero is at -2.5. There is one asymptote at -180°. Changing K from 0 to 40 show that the system cannot become unstable.

eg parameters (0.0, 1.0, 2.0, 0.0, 1.0) show the locus when there is no zero. There are 2 asymptotes at ±90°. Changing K from 0 to 40 show that the system can develop transient oscillations but cannot become unstable.

eg parameters (0.0, 1.0, 2.0, -0.5, 1.0) show the locus when the zero is to the right at +2. There is 1 asymptote at +180°. Changing K show that the system becomes oscillatory at K = 6.0. At this value of K the closed loop poles are at ~±j1.8, the period is ~3.36s and the angular frequency is ~1.86 which is close enough to the closed loop pole.

Gif images below shows how applet should appear.

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