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)+131s(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 13/(s(s + 4)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 13/(s(s + 4))
ωd = rad/s
2.Easy
R(s)+291s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 29/(s(s + 10)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 29/(s(s + 10))
ωd = rad/s
3.Medium
R(s)+K1s(s + 1)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 6)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 6))
Kmax =
4.Medium
R(s)+Ks + 8s(s + 2)(s + 3)C(s)−

A unity-feedback loop has L(s) = K(s + 8)/(s(s + 2)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 8)/(s(s + 2)(s + 3))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2−1.5

For the loop L(s) = K/(s(s + 5)(s + 10)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 5)(s + 10))s = −1.5
K =
6.Hard
ReIm−2−4−6−8−10−12j2−1

For the loop L(s) = K/(s(s + 3)(s + 10)), find the gain K that puts a closed-loop pole at s = −1.

L(s) = K/(s(s + 3)(s + 10))s = −1
K =

Answer key

  1. 1. ωd = 3 rad/s
  2. 2. ωd = 2 rad/s
  3. 3. Kmax = 42
  4. 4. Kmax = 10
  5. 5. K = 44.63
  6. 6. K = 18

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