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Negative Gain Root Locus

Consider a negative gain root locus for L (s) = k (s-1)/[s(s^2 + 4s + 8)] with a scale of 20mm = 1 unit.
i. Graphically, find the gain required to give dominant closed loop poles with zeta = 0.5.
ii. For the above value of gain, find natural frequency and hence sketch the expected reference to output step response, assuming that the system is low-passing and dominantly second order.
iii. Add a sketch of the positive gain root locus to the diagram.


Solution Summary

Solutions are shown with full calculations including formulas C(s)/R(s) = L(s)/(1-L(s)) = [k(s-1)]/[s^3 + 4s^2 + (8-k)s + k] and Laplace transform pairs. Sketched graph for (ii) is also included.