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 = 22s + 5C(s)−

A proportional controller with gain K = 2 drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 2/(s + 5)K = 2
p =
2.Easy
R(s)+K = 32s + 2C(s)−

A proportional controller with gain K = 3 drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 2/(s + 2)K = 3
p =
3.Medium
R(s)+K1s(s + 4)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 5)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 4)(s + 5))
Kmax =
4.Medium
R(s)+K1s(s + 1)(s + 2)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 2)). 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 + 1)(s + 2))
ω = rad/s
5.Hard
ReImζ = 0.4−2−4j2j4

The root locus of L(s) = K/(s(s + 4)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.4.

L(s) = K/(s(s + 4))ζ = 0.4
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2−1.5

For the loop L(s) = K/(s(s + 5)(s + 6)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 5)(s + 6))s = −1.5
K =

Answer key

  1. 1. p = −9
  2. 2. p = −8
  3. 3. Kmax = 180
  4. 4. ω = 1.414 rad/s
  5. 5. K = 25
  6. 6. K = 23.63

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