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
s³1−1s²13s¹?s⁰?

A closed-loop system has the characteristic equation s³ + s² − s + 3 = 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 + s² − s + 3
N =
2.Easy
s³14s²221s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 2s² + 4s + 21, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 2s² + 4s + 21
b1 =
3.Medium
R(s)+K1s(s + 2)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(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 + 2)(s + 4))
ω = rad/s
4.Medium
R(s)+Ks + 1(s − 4)(s + 5)C(s)−

The plant G(s) = (s + 1)/((s − 4)(s + 5)) has an unstable pole at s = 4. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain Kmin that makes the closed loop stable.

G(s) = (s + 1)/((s − 4)(s + 5))
Kmin =
5.Hard
ReImζ = 0.8−0.5−1−1.5−2−2.50.5j0.5j1

The root locus of L(s) = K/(s(s + 2)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.8.

L(s) = K/(s(s + 2))ζ = 0.8
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2−1.5

For the loop L(s) = K/(s(s + 3)(s + 6)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 3)(s + 6))s = −1.5
K =

Answer key

  1. 1. N = 2
  2. 2. b1 = −6.5
  3. 3. ω = 2.828 rad/s
  4. 4. Kmin = 20
  5. 5. K = 1.562
  6. 6. K = 10.13

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