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
s³18s²−114s¹?s⁰?

A closed-loop system has the characteristic equation s³ − s² + 8s + 14 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ − s² + 8s + 14
N =
2.Easy
R(s)+131s(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 13/(s(s + 4)). The closed-loop poles are complex. Find their imaginary part ωd.

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

The plant G(s) = (s + 8)/((s − 4)(s + 2)) has an unstable pole at s = 4. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain Kmin that makes the closed loop stable.

G(s) = (s + 8)/((s − 4)(s + 2))
Kmin =
4.Medium
R(s)+601s(s + 2)(s + 3)C(s)−

The loop in the diagram has negative feedback and L(s) = 60/(s(s + 2)(s + 3)). How many closed-loop poles N lie in the right half-plane?

L(s) = 60/(s(s + 2)(s + 3))
N =
5.Hard
ReImζ = 0.8−1−2−3−4−5−6−71j1j2

The root locus of L(s) = K/(s(s + 6)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.8.

L(s) = K/(s(s + 6))ζ = 0.8
K =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.5j0.5j1−1

For the loop L(s) = K/(s(s + 2)(s + 3)), find the gain K that puts a closed-loop pole at s = −1.

L(s) = K/(s(s + 2)(s + 3))s = −1
K =

Answer key

  1. 1. N = 2
  2. 2. ωd = 3 rad/s
  3. 3. Kmin = 2
  4. 4. N = 2
  5. 5. K = 14.06
  6. 6. K = 2

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