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)+K = 15s + 10C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 10)K = 1
p =
2.Easy
R(s)+K = 51s + 1C(s)−

A proportional controller with gain K = 5 drives the plant G(s) = 1/(s + 1) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 1)K = 5
p =
3.Medium
R(s)+Ks + 13s(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 13)/(s(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 13)/(s(s + 3)(s + 4))
Kmax =
4.Medium
R(s)+Ks + 8s(s + 2)(s + 3)C(s)−

A unity-feedback loop has L(s) = K(s + 8)/(s(s + 2)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 8)/(s(s + 2)(s + 3))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2−1

For the loop L(s) = K/(s(s + 2)(s + 10)), find the gain K that puts a closed-loop pole at s = −1.

L(s) = K/(s(s + 2)(s + 10))s = −1
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2−0.5

For the loop L(s) = K/(s(s + 2)(s + 6)), find the gain K that puts a closed-loop pole at s = −0.5.

L(s) = K/(s(s + 2)(s + 6))s = −0.5
K =

Answer key

  1. 1. p = −15
  2. 2. p = −6
  3. 3. Kmax = 14
  4. 4. Kmax = 10
  5. 5. K = 9
  6. 6. K = 4.125

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