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)+171s(s + 8)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 17/(s(s + 8)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 17/(s(s + 8))
ωd = rad/s
2.Easy
R(s)+K1s + 1C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 1) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −2.

G(s) = 1/(s + 1)p = −2
K =
3.Medium
R(s)+K = 2.51s(s + 1)(s + 4)C(s)−

The loop in the diagram runs with K = 2.5 and L(s) = K/(s(s + 1)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 1)(s + 4))K = 2.5
m =
4.Medium
R(s)+K1s(s + 5)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 5)(s + 6)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 5)(s + 6))
Kmax =
5.Hard
ReImζ = 0.8−1−2−3−4−5−6j1j2

The root locus of L(s) = K/(s(s + 5)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.8.

L(s) = K/(s(s + 5))ζ = 0.8
K =
6.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 2)(s + 4)(s + 5)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 2)(s + 4)(s + 5))
ω = rad/s

Answer key

  1. 1. ωd = 1 rad/s
  2. 2. K = 1
  3. 3. m = 8
  4. 4. Kmax = 330
  5. 5. K = 9.766
  6. 6. ω = 6.164 rad/s

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