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
s⁴110−11s³24s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + 10s² + 4s − 11. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + 10s² + 4s − 11
N =
2.Easy
R(s)+K = 42s + 10C(s)−

A proportional controller with gain K = 4 drives the plant G(s) = 2/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 2/(s + 10)K = 4
p =
3.Medium
R(s)+101s(s + 8)(s + 10)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 10/(s(s + 8)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 10/(s(s + 8)(s + 10))
N =
4.Medium
R(s)+K1s(s + 2)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 8)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 2)(s + 8))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6−71j1j2

The root locus of L(s) = K/(s(s + 2)(s + 6)) leaves the real axis between s = 0 and s = −2. Find the break-away point sb.

L(s) = K/(s(s + 2)(s + 6))
sb =
6.Hard
ReImζ = 0.3−5−10j5j10j15

The root locus of L(s) = K/(s(s + 8)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.3.

L(s) = K/(s(s + 8))ζ = 0.3
K =

Answer key

  1. 1. N = 1
  2. 2. p = −18
  3. 3. N = 1
  4. 4. Kmax = 160
  5. 5. sb = −0.9028
  6. 6. K = 177.8

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