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
s⁴143s³17s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + s³ + 4s² + 7s + 3. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + s³ + 4s² + 7s + 3
N =
2.Easy
s³15s²a25s¹(5·a − 25)/as⁰25

The characteristic equation of a closed loop is s³ + a·s² + 5s + 25 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 5s + 25 = 0
a =
3.Medium
R(s)+Ks + 4(s − 4)(s + 3)C(s)−

The plant G(s) = (s + 4)/((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 + 4)/((s − 4)(s + 3))
Kmin =
4.Medium
R(s)+K = 76.81s(s + 4)(s + 8)C(s)−

The loop in the diagram runs with K = 76.8 and L(s) = K/(s(s + 4)(s + 8)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 4)(s + 8))K = 76.8
m =
5.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 =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.5j0.5j1

The root locus of L(s) = K/(s(s + 2)(s + 3)) leaves the real axis between s = 0 and s = −2. Find the break-away point sb.

L(s) = K/(s(s + 2)(s + 3))
sb =

Answer key

  1. 1. N = 2
  2. 2. a = 5
  3. 3. Kmin = 3
  4. 4. m = 5
  5. 5. sb = −1.333
  6. 6. sb = −0.7847

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