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)+201s(s + 8)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 20/(s(s + 8)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 20/(s(s + 8))
ωd = rad/s
2.Easy
s³17s²−424s¹?s⁰?

A closed-loop system has the characteristic equation s³ − 4s² + 7s + 24 = 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 − 4s² + 7s + 24
N =
3.Medium
R(s)+K1s(s + 3)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 4)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 3)(s + 4))
ω = rad/s
4.Medium
R(s)+K = 1.21s(s + 1)(s + 3)C(s)−

The loop in the diagram runs with K = 1.2 and L(s) = K/(s(s + 1)(s + 3)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 1)(s + 3))K = 1.2
m =
5.Hard
ReIm−2−4−6−8−10−12−14−16j2j4sb​

The root locus of L(s) = K/((s + 5)(s + 13)) is shown. Find the gain K at the break-away point sb, where the two closed-loop poles meet on the real axis.

L(s) = K/((s + 5)(s + 13))
K =
6.Hard
ReIm−2−4−6−8−10−12j2−2

For the loop L(s) = K/(s(s + 5)(s + 10)), find the gain K that puts a closed-loop pole at s = −2.

L(s) = K/(s(s + 5)(s + 10))s = −2
K =

Answer key

  1. 1. ωd = 2 rad/s
  2. 2. N = 2
  3. 3. ω = 3.464 rad/s
  4. 4. m = 10
  5. 5. K = 16
  6. 6. K = 48

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