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)+521s(s + 8)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 52/(s(s + 8)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 52/(s(s + 8))
ωd = rad/s
2.Easy
s³1−1s²−326s¹?s⁰?

A closed-loop system has the characteristic equation s³ − 3s² − s + 26 = 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 − 3s² − s + 26
N =
3.Medium
R(s)+K1(s + 1)(s + 2)(s + 4)C(s)−

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

L(s) = K/((s + 1)(s + 2)(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
ReImζ = 0.5−1−2j1j2

The root locus of L(s) = K/(s(s + 2)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.5.

L(s) = K/(s(s + 2))ζ = 0.5
K =
6.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 =

Answer key

  1. 1. ωd = 6 rad/s
  2. 2. N = 2
  3. 3. Kmax = 90
  4. 4. Kmax = 6
  5. 5. K = 4
  6. 6. K = 14.06

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