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 = 11s + 4C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 1/(s + 4) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 4)K = 1
p =
2.Easy
R(s)+K5s + 2C(s)−

A proportional gain K controls the plant G(s) = 5/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −17.

G(s) = 5/(s + 2)p = −17
K =
3.Medium
R(s)+Ks + 12s(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 12)/(s(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 12)/(s(s + 3)(s + 4))
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−1−2−3−4−5j1j2

For L(s) = K/((s + 2)(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 + 2)(s² + 8s + 20))
θd = °
6.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−3

For the loop L(s) = K/(s(s + 6)(s + 8)), find the gain K that puts a closed-loop pole at s = −3.

L(s) = K/(s(s + 6)(s + 8))s = −3
K =

Answer key

  1. 1. p = −5
  2. 2. K = 3
  3. 3. Kmax = 16.8
  4. 4. Kmax = 8
  5. 5. θd = −45 °
  6. 6. K = 45

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