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 + 4)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 20/(s(s + 4)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 20/(s(s + 4))
ωd = rad/s
2.Easy
R(s)+K = 11s + 10C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 1/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 10)K = 1
p =
3.Medium
R(s)+K1s(s + 1)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 8)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 8))
Kmax =
4.Medium
R(s)+Ks + 12s(s + 2)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 12)/(s(s + 2)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

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

For L(s) = K/((s + 7)(s² + 4s + 13)), find the angle of departure θd of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 7)(s² + 4s + 13))
θd = °
6.Hard
ReIm−1−2−3−4−5−6−71j1j2

A unity-feedback loop has L(s) = K(s + 3)/(s(s + 2)(s + 6)(s + 5)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 3)/(s(s + 2)(s + 6)(s + 5))
σa =

Answer key

  1. 1. ωd = 4 rad/s
  2. 2. p = −11
  3. 3. Kmax = 72
  4. 4. Kmax = 8
  5. 5. θd = 59.04 °
  6. 6. σa = −3.333

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