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)+K = 410s + 1C(s)−

A proportional controller with gain K = 4 drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 10/(s + 1)K = 4
p =
2.Easy
s³18s²425s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 4s² + 8s + 25, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 4s² + 8s + 25
b1 =
3.Medium
R(s)+K = 81s(s + 1)(s + 4)C(s)−

The loop in the diagram runs with K = 8 and L(s) = K/(s(s + 1)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 1)(s + 4))K = 8
m =
4.Medium
R(s)+K1(s + 2)(s + 3)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 2)(s + 3)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 2)(s + 3)(s + 6))
Kmax =
5.Hard
ReImζ = 0.8−2−4−6−8−10j2

The root locus of L(s) = K/(s(s + 8)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.8.

L(s) = K/(s(s + 8))ζ = 0.8
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2−0.5

For the loop L(s) = K/(s(s + 2)(s + 6)), find the gain K that puts a closed-loop pole at s = −0.5.

L(s) = K/(s(s + 2)(s + 6))s = −0.5
K =

Answer key

  1. 1. p = −41
  2. 2. b1 = 1.75
  3. 3. m = 2.5
  4. 4. Kmax = 360
  5. 5. K = 25
  6. 6. K = 4.125

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