Stability: Routh–Hurwitz & root locus worksheet with answers

The characteristic polynomial of a feedback system is Q(s) = s⁴ + s³ + s² + 3s + 7. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

A proportional gain K controls the plant G(s) = 5/(s + 4) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −14.

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 10)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

The loop in the diagram runs with K = 20 and L(s) = K/(s(s + 3)(s + 5)). By what factor m can the gain be raised before the closed loop becomes unstable?

The root locus of L(s) = K/((s + 1)(s + 5)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

The root locus of L(s) = K(s + 11)/(s(s + 7)) leaves the real axis between s = 0 and s = −7, circles round and joins the real axis again to the left of the zero. Find the break-away point s_b.