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
s⁴1513s³5−9s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 5s³ + 5s² − 9s + 13. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 5s³ + 5s² − 9s + 13
N =
2.Easy
R(s)+321s(s + 8)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 32/(s(s + 8)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 32/(s(s + 8))
ωd = rad/s
3.Medium
R(s)+Ks + 10s(s + 1)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 10)/(s(s + 1)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 10)/(s(s + 1)(s + 4))
Kmax =
4.Medium
R(s)+Ks + 4s(s + 1)(s + 2)C(s)−

A unity-feedback loop has L(s) = K(s + 4)/(s(s + 1)(s + 2)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

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

For the loop L(s) = K/(s(s + 3)(s + 10)), find the gain K that puts a closed-loop pole at s = −2.

L(s) = K/(s(s + 3)(s + 10))s = −2
K =
6.Hard
ReImζ = 0.5−5−10j5j10

The root locus of L(s) = K/(s(s + 10)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.5.

L(s) = K/(s(s + 10))ζ = 0.5
K =

Answer key

  1. 1. N = 2
  2. 2. ωd = 4 rad/s
  3. 3. Kmax = 4
  4. 4. Kmax = 6
  5. 5. K = 16
  6. 6. K = 100

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