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
s³12s²a16s¹(2·a − 16)/as⁰16

The characteristic equation of a closed loop is s³ + a·s² + 2s + 16 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 2s + 16 = 0
a =
2.Easy
s³37s²38s¹?s⁰

For the characteristic polynomial Q(s) = 3s³ + 3s² + 7s + 8, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 3s³ + 3s² + 7s + 8
b1 =
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)). 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 + 3)(s + 4))
ω = rad/s
5.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.5j0.5j1

A unity-feedback loop has L(s) = K/(s(s + 3)(s + 2)(s + 1)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/(s(s + 3)(s + 2)(s + 1))
σa =
6.Hard
ReIm−1−2−3−4−5−6j1sb​

The root locus of L(s) = K/((s + 3)(s + 5)) is shown. Find the gain K at the break-away point sb, where the two closed-loop poles meet on the real axis.

L(s) = K/((s + 3)(s + 5))
K =

Answer key

  1. 1. a = 8
  2. 2. b1 = −1
  3. 3. Kmax = 390
  4. 4. ω = 3.464 rad/s
  5. 5. σa = −1.5
  6. 6. K = 1

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