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
R(s)+K1s + 10C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −13.

G(s) = 1/(s + 10)p = −13
K =
2.Easy
s³110s²424s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 4s² + 10s + 24, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 4s² + 10s + 24
b1 =
3.Medium
R(s)+K1s(s + 1)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 4)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 4))
Kmax =
4.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)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 6)(s + 8))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6j1

The root locus of L(s) = K(s + 5)/(s(s + 2)) leaves the real axis between s = 0 and s = −2, circles round and joins the real axis again to the left of the zero. Find the break-away point sb.

L(s) = K(s + 5)/(s(s + 2))
sb =
6.Hard
ReIm−2−4−6−8−10−12j2−1

For the loop L(s) = K/(s(s + 8)(s + 10)), find the gain K that puts a closed-loop pole at s = −1.

L(s) = K/(s(s + 8)(s + 10))s = −1
K =

Answer key

  1. 1. K = 3
  2. 2. b1 = 4
  3. 3. Kmax = 20
  4. 4. Kmax = 672
  5. 5. sb = −1.127
  6. 6. K = 63

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