Stability: Routh–Hurwitz & root locus worksheet with answers

Free printable Stability: Routh–Hurwitz & root locus worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
R(s)+K = 35s + 1C(s)−

A proportional controller with gain K = 3 drives the plant G(s) = 5/(s + 1) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 1)K = 3
p =
2.Easy
s⁴110−14s³215s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + 10s² + 15s − 14. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + 10s² + 15s − 14
N =
3.Medium
R(s)+K1s(s + 2)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 6)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 2)(s + 6))
ω = rad/s
4.Medium
R(s)+K = 9.61s(s + 2)(s + 4)C(s)−

The loop in the diagram runs with K = 9.6 and L(s) = K/(s(s + 2)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 2)(s + 4))K = 9.6
m =
5.Hard
ReIm−2j2

For L(s) = K/((s + 2)(s² + 2s + 10)), find the angle of departure θd of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 2)(s² + 2s + 10))
θd = °
6.Hard
ReImζ = 0.8−1−2−3−4−5−6j1j2

The root locus of L(s) = K/(s(s + 5)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.8.

L(s) = K/(s(s + 5))ζ = 0.8
K =

Answer key

  1. 1. p = −16
  2. 2. N = 1
  3. 3. ω = 3.464 rad/s
  4. 4. m = 5
  5. 5. θd = 18.43 °
  6. 6. K = 9.766

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