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)+451s(s + 6)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 45/(s(s + 6)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 45/(s(s + 6))
ωd = rad/s
2.Easy
R(s)+K = 55s + 10C(s)−

A proportional controller with gain K = 5 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 10)K = 5
p =
3.Medium
R(s)+K1s(s + 2)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 6)). 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 + 2)(s + 6))
ω = rad/s
4.Medium
R(s)+14401s(s + 6)(s + 10)C(s)−

The loop in the diagram has negative feedback and L(s) = 1440/(s(s + 6)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 1440/(s(s + 6)(s + 10))
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−2−4−6−8−10−12−14−162j2j4sb​

The root locus of L(s) = K/((s + 6)(s + 14)) 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 + 6)(s + 14))
K =

Answer key

  1. 1. ωd = 6 rad/s
  2. 2. p = −35
  3. 3. ω = 3.464 rad/s
  4. 4. N = 2
  5. 5. K = 14.06
  6. 6. K = 16

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