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 = 105s + 4C(s)−

A proportional controller with gain K = 10 drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 4)K = 10
p =
2.Easy
s³110s²a15s¹(10·a − 15)/as⁰15

The characteristic equation of a closed loop is s³ + a·s² + 10s + 15 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 10s + 15 = 0
a =
3.Medium
R(s)+K = 421s(s + 3)(s + 4)C(s)−

The loop in the diagram runs with K = 42 and L(s) = K/(s(s + 3)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 3)(s + 4))K = 42
m =
4.Medium
R(s)+Ks + 5(s − 4)(s + 2)C(s)−

The plant G(s) = (s + 5)/((s − 4)(s + 2)) 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 + 5)/((s − 4)(s + 2))
Kmin =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−3

For the loop L(s) = K/(s(s + 4)(s + 8)), find the gain K that puts a closed-loop pole at s = −3.

L(s) = K/(s(s + 4)(s + 8))s = −3
K =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.5−50.5j0.5j1

For L(s) = K/((s + 4)(s² + 2s + 2)), find the angle of departure θd of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 4)(s² + 2s + 2))
θd = °

Answer key

  1. 1. p = −54
  2. 2. a = 1.5
  3. 3. m = 2
  4. 4. Kmin = 2
  5. 5. K = 15
  6. 6. θd = 71.57 °

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