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)+K2s + 2C(s)−

A proportional gain K controls the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −4.

G(s) = 2/(s + 2)p = −4
K =
2.Easy
R(s)+K = 55s + 4C(s)−

A proportional controller with gain K = 5 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 = 5
p =
3.Medium
R(s)+K1s(s + 3)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 10)). 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 + 10))
ω = rad/s
4.Medium
R(s)+Ks + 10s(s + 1)(s + 3)C(s)−

A unity-feedback loop has L(s) = K(s + 10)/(s(s + 1)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 10)/(s(s + 1)(s + 3))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6−71j1j2

The root locus of L(s) = K/(s(s + 3)(s + 6)) leaves the real axis between s = 0 and s = −3. Find the break-away point sb.

L(s) = K/(s(s + 3)(s + 6))
sb =
6.Hard
ReIm−1−2−3−4−5−6j1j2

For L(s) = K/((s + 5)(s² + 8s + 20)), 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 + 5)(s² + 8s + 20))
θd = °

Answer key

  1. 1. K = 1
  2. 2. p = −29
  3. 3. ω = 5.477 rad/s
  4. 4. Kmax = 2
  5. 5. sb = −1.268
  6. 6. θd = 26.57 °

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