Stability: Routh–Hurwitz & root locus worksheet with answers

A proportional gain K controls the plant G(s) = 1/(s + 5) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −9.

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 4s³ + 7s² − 9s + 15. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

The plant G(s) = (s + 5)/((s − 3)(s + 3)) has an unstable pole at s = 3. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain K_min that makes the closed loop stable.

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 6)(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 root locus of L(s) = K/(s(s + 4)(s + 10)) leaves the real axis between s = 0 and s = −4. Find the break-away point s_b.

For L(s) = K/((s + 7)(s² + 6s + 18)), 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.