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 = 12s + 2C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 2/(s + 2)K = 1
p =
2.Easy
s⁴1518s³2−2s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + 5s² − 2s + 18. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + 5s² − 2s + 18
N =
3.Medium
R(s)+K = 61s(s + 2)(s + 3)C(s)−

The loop in the diagram runs with K = 6 and L(s) = K/(s(s + 2)(s + 3)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 2)(s + 3))K = 6
m =
4.Medium
R(s)+K1(s + 1)(s + 2)(s + 4)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 2)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 2)(s + 4))
Kmax =
5.Hard
ReImζ = 0.6−5−10j5

The root locus of L(s) = K/(s(s + 10)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.6.

L(s) = K/(s(s + 10))ζ = 0.6
K =
6.Hard
ReIm−1−2−3−4−5−6j1j2

For L(s) = K/((s + 5)(s² + 6s + 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 + 5)(s² + 6s + 13))
θd = °

Answer key

  1. 1. p = −4
  2. 2. N = 2
  3. 3. m = 5
  4. 4. Kmax = 90
  5. 5. K = 69.44
  6. 6. θd = 45 °

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