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
s⁴17−12s³416s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 4s³ + 7s² + 16s − 12. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 4s³ + 7s² + 16s − 12
N =
2.Easy
R(s)+K1s + 2C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −3.5.

G(s) = 1/(s + 2)p = −3.5
K =
3.Medium
R(s)+Ks + 13s(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 13)/(s(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 13)/(s(s + 3)(s + 4))
Kmax =
4.Medium
R(s)+K1s(s + 3)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 8)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 3)(s + 8))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2−5

For the loop L(s) = K/(s(s + 6)(s + 10)), find the gain K that puts a closed-loop pole at s = −5.

L(s) = K/(s(s + 6)(s + 10))s = −5
K =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.5−50.5j0.5j1sb​

The root locus of L(s) = K/((s + 2)(s + 4)) is shown. Find the gain K at the break-away point sb, where the two closed-loop poles meet on the real axis.

L(s) = K/((s + 2)(s + 4))
K =

Answer key

  1. 1. N = 1
  2. 2. K = 1.5
  3. 3. Kmax = 14
  4. 4. Kmax = 264
  5. 5. K = 25
  6. 6. K = 1

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