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
s³14s²418s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 4s² + 4s + 18, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 4s² + 4s + 18
b1 =
2.Easy
s³15s²435s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 4s² + 5s + 35 = 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 + 5s + 35
N =
3.Medium
R(s)+K1s(s + 6)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 6)(s + 8)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 6)(s + 8))
ω = rad/s
4.Medium
R(s)+K1(s + 1)(s + 2)(s + 5)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 2)(s + 5)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 2)(s + 5))
Kmax =
5.Hard
ReIm−2−4−6−8−10j2

For L(s) = K/((s + 8)(s² + 8s + 25)), 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 + 8)(s² + 8s + 25))
θd = °
6.Hard
ReImζ = 0.6−2−4−6−8−10j2j4j6

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 = 0.6.

L(s) = K/(s(s + 8))ζ = 0.6
K =

Answer key

  1. 1. b1 = −0.5
  2. 2. N = 2
  3. 3. ω = 6.928 rad/s
  4. 4. Kmax = 126
  5. 5. θd = 53.13 °
  6. 6. K = 44.44

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