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
s³12s²a4s¹(2·a − 4)/as⁰4

The characteristic equation of a closed loop is s³ + a·s² + 2s + 4 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 2s + 4 = 0
a =
2.Easy
s³15s²a15s¹(5·a − 15)/as⁰15

The characteristic equation of a closed loop is s³ + a·s² + 5s + 15 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 5s + 15 = 0
a =
3.Medium
R(s)+Ks + 5(s − 4)(s + 3)C(s)−

The plant G(s) = (s + 5)/((s − 4)(s + 3)) 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 + 3))
Kmin =
4.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)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 2)(s + 4))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6j1

The root locus of L(s) = K(s + 5)/(s(s + 1)) leaves the real axis between s = 0 and s = −1, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

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

The root locus of L(s) = K/(s(s + 4)(s + 8)) leaves the real axis between s = 0 and s = −4. Find the break-away point sb.

L(s) = K/(s(s + 4)(s + 8))
sb =

Answer key

  1. 1. a = 2
  2. 2. a = 3
  3. 3. Kmin = 2.4
  4. 4. Kmax = 48
  5. 5. si = −9.472
  6. 6. sb = −1.691

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