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
s³22s²415s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + 4s² + 2s + 15, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + 4s² + 2s + 15
b1 =
2.Easy
R(s)+K = 0.52s + 2C(s)−

A proportional controller with gain K = 0.5 drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 2/(s + 2)K = 0.5
p =
3.Medium
R(s)+K1s(s + 2)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 10)). 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 + 2)(s + 10))
ω = rad/s
4.Medium
R(s)+K = 16.21s(s + 3)(s + 6)C(s)−

The loop in the diagram runs with K = 16.2 and L(s) = K/(s(s + 3)(s + 6)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 3)(s + 6))K = 16.2
m =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

The root locus of L(s) = K/(s(s + 5)(s + 8)) leaves the real axis between s = 0 and s = −5. Find the break-away point sb.

L(s) = K/(s(s + 5)(s + 8))
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. b1 = −5.5
  2. 2. p = −3
  3. 3. ω = 4.472 rad/s
  4. 4. m = 10
  5. 5. sb = −2
  6. 6. sb = −0.4774

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