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
s³13s²a9s¹(3·a − 9)/as⁰9

The characteristic equation of a closed loop is s³ + a·s² + 3s + 9 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 3s + 9 = 0
a =
2.Easy
R(s)+K5s + 10C(s)−

A proportional gain K controls the plant G(s) = 5/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −30.

G(s) = 5/(s + 10)p = −30
K =
3.Medium
R(s)+Ks + 12s(s + 2)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 12)/(s(s + 2)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 12)/(s(s + 2)(s + 4))
Kmax =
4.Medium
R(s)+Ks + 7s(s + 2)(s + 3)C(s)−

A unity-feedback loop has L(s) = K(s + 7)/(s(s + 2)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 7)/(s(s + 2)(s + 3))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2−2

For the loop L(s) = K/(s(s + 6)(s + 10)), find the gain K that puts a closed-loop pole at s = −2.

L(s) = K/(s(s + 6)(s + 10))s = −2
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2−1.5

For the loop L(s) = K/(s(s + 2)(s + 6)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 2)(s + 6))s = −1.5
K =

Answer key

  1. 1. a = 3
  2. 2. K = 4
  3. 3. Kmax = 8
  4. 4. Kmax = 15
  5. 5. K = 64
  6. 6. K = 3.375

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