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
s³1−5s²3−5s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 3s² − 5s − 5 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ + 3s² − 5s − 5
N =
2.Easy
R(s)+181s(s + 6)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 18/(s(s + 6)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 18/(s(s + 6))
ωd = rad/s
3.Medium
R(s)+K1s(s + 3)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 10)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 3)(s + 10))
Kmax =
4.Medium
R(s)+K1s(s + 3)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 4)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 3)(s + 4))
Kmax =
5.Hard
ReImζ = 0.5−5−10j5j10

The root locus of L(s) = K/(s(s + 10)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.5.

L(s) = K/(s(s + 10))ζ = 0.5
K =
6.Hard
ReImζ = 0.707−1−2−3−4−5−6j1j2

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 = 0.707.

L(s) = K/(s(s + 5))ζ = 0.707
K =

Answer key

  1. 1. N = 1
  2. 2. ωd = 3 rad/s
  3. 3. Kmax = 390
  4. 4. Kmax = 84
  5. 5. K = 100
  6. 6. K = 12.5

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