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
s³19s²514s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 5s² + 9s + 14, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 5s² + 9s + 14
b1 =
2.Easy
s³12s²a10s¹(2·a − 10)/as⁰10

The characteristic equation of a closed loop is s³ + a·s² + 2s + 10 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 2s + 10 = 0
a =
3.Medium
R(s)+K1s(s + 4)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 8)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 4)(s + 8))
Kmax =
4.Medium
R(s)+K1(s + 1)(s + 3)(s + 5)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 3)(s + 5)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 3)(s + 5))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−1.5

For the loop L(s) = K/(s(s + 6)(s + 8)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 6)(s + 8))s = −1.5
K =
6.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 4)(s + 5)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 4)(s + 5)(s + 6))
ω = rad/s

Answer key

  1. 1. b1 = 6.2
  2. 2. a = 5
  3. 3. Kmax = 384
  4. 4. Kmax = 192
  5. 5. K = 43.88
  6. 6. ω = 8.602 rad/s

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