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 = 45s + 4C(s)−

A proportional controller with gain K = 4 drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 4)K = 4
p =
2.Easy
R(s)+291s(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 29/(s(s + 4)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 29/(s(s + 4))
ωd = rad/s
3.Medium
R(s)+101s(s + 1)(s + 5)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 10/(s(s + 1)(s + 5)). How many closed-loop poles N lie in the right half-plane?

L(s) = 10/(s(s + 1)(s + 5))
N =
4.Medium
R(s)+K1s(s + 1)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 5)). 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 + 1)(s + 5))
ω = rad/s
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−1.5

For the loop L(s) = K/(s(s + 3)(s + 8)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 3)(s + 8))s = −1.5
K =
6.Hard
ReIm−2−4−6−8−10−12j2sb​

The root locus of L(s) = K/((s + 4)(s + 10)) 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 + 4)(s + 10))
K =

Answer key

  1. 1. p = −24
  2. 2. ωd = 5 rad/s
  3. 3. N = 1
  4. 4. ω = 2.236 rad/s
  5. 5. K = 14.63
  6. 6. K = 9

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