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
s³15s²123s¹?s⁰?

A closed-loop system has the characteristic equation s³ + s² + 5s + 23 = 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 + s² + 5s + 23
N =
2.Easy
R(s)+K1s + 5C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 5) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −9.

G(s) = 1/(s + 5)p = −9
K =
3.Medium
R(s)+Ks + 1(s − 4)(s + 2)C(s)−

The plant G(s) = (s + 1)/((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 + 1)/((s − 4)(s + 2))
Kmin =
4.Medium
R(s)+K1s(s + 3)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 5)). 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 + 3)(s + 5))
ω = rad/s
5.Hard
ReIm−1−2−3−4−5−6j1

The root locus of L(s) = K/(s(s + 4)(s + 5)) leaves the real axis between s = 0 and s = −4. Find the break-away point sb.

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

The root locus of L(s) = K/(s(s + 1)(s + 6)) leaves the real axis between s = 0 and s = −1. Find the break-away point sb.

L(s) = K/(s(s + 1)(s + 6))
sb =

Answer key

  1. 1. N = 2
  2. 2. K = 4
  3. 3. Kmin = 8
  4. 4. ω = 3.873 rad/s
  5. 5. sb = −1.472
  6. 6. sb = −0.4774

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