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
s³12s²−1−26s¹?s⁰?

A closed-loop system has the characteristic equation s³ − s² + 2s − 26 = 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 + 2s − 26
N =
2.Easy
R(s)+K5s + 10C(s)−

A proportional gain K controls the plant G(s) = 5/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −25.

G(s) = 5/(s + 10)p = −25
K =
3.Medium
R(s)+K = 331s(s + 3)(s + 8)C(s)−

The loop in the diagram runs with K = 33 and L(s) = K/(s(s + 3)(s + 8)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 3)(s + 8))K = 33
m =
4.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 =
5.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 =
6.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−1

For the loop L(s) = K/(s(s + 4)(s + 8)), find the gain K that puts a closed-loop pole at s = −1.

L(s) = K/(s(s + 4)(s + 8))s = −1
K =

Answer key

  1. 1. N = 1
  2. 2. K = 3
  3. 3. m = 8
  4. 4. Kmax = 390
  5. 5. K = 12.5
  6. 6. K = 21

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