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
s³13s²228s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 2s² + 3s + 28, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 2s² + 3s + 28
b1 =
2.Easy
R(s)+K2s + 2C(s)−

A proportional gain K controls the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −6.

G(s) = 2/(s + 2)p = −6
K =
3.Medium
R(s)+K = 1401s(s + 4)(s + 10)C(s)−

The loop in the diagram runs with K = 140 and L(s) = K/(s(s + 4)(s + 10)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 4)(s + 10))K = 140
m =
4.Medium
R(s)+K1s(s + 1)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 8)). Find the largest gain Kmax for which the closed loop is stable.

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

A unity-feedback loop has L(s) = K(s + 6)/(s(s + 5)(s + 10)(s + 8)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 6)/(s(s + 5)(s + 10)(s + 8))
σa =
6.Hard
ReIm−2−4−6−8−10−12j2−1.5

For the loop L(s) = K/(s(s + 5)(s + 10)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 5)(s + 10))s = −1.5
K =

Answer key

  1. 1. b1 = −11
  2. 2. K = 2
  3. 3. m = 4
  4. 4. Kmax = 72
  5. 5. σa = −5.667
  6. 6. K = 44.63

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