Stability: Routh–Hurwitz & root locus worksheet with answers

A unity-feedback loop has the open-loop transfer function L(s) = 13/(s(s + 6)). The closed-loop poles are complex. Find their imaginary part ω_d.

A proportional gain K controls the plant G(s) = 10/(s + 5) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −35.

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 6)(s + 8)). Find the largest gain K_max for which the closed loop is stable.

The loop in the diagram runs with K = 15 and L(s) = K/(s(s + 2)(s + 3)). By what factor m can the gain be raised before the closed loop becomes unstable?

The root locus of L(s) = K/(s(s + 1)(s + 5)) leaves the real axis between s = 0 and s = −1. Find the break-away point s_b.

For L(s) = K/((s + 6)(s² + 8s + 20)), find the angle of departure θ_d of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.