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
s³16s²−619s¹?s⁰?

A closed-loop system has the characteristic equation s³ − 6s² + 6s + 19 = 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 − 6s² + 6s + 19
N =
2.Easy
s³14s²428s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 4s² + 4s + 28 = 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 + 4s² + 4s + 28
N =
3.Medium
R(s)+K1s(s + 1)(s + 2)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 2)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 2))
Kmax =
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 = 0.4−2−4j2j4

The root locus of L(s) = K/(s(s + 4)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.4.

L(s) = K/(s(s + 4))ζ = 0.4
K =
6.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

The root locus of L(s) = K/(s(s + 3)(s + 8)) leaves the real axis between s = 0 and s = −3. Find the break-away point sb.

L(s) = K/(s(s + 3)(s + 8))
sb =

Answer key

  1. 1. N = 2
  2. 2. N = 2
  3. 3. Kmax = 6
  4. 4. Kmax = 390
  5. 5. K = 25
  6. 6. sb = −1.333

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