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
s⁴16−19s³18s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + s³ + 6s² + 8s − 19. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + s³ + 6s² + 8s − 19
N =
2.Easy
s³14s²a20s¹(4·a − 20)/as⁰20

The characteristic equation of a closed loop is s³ + a·s² + 4s + 20 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 4s + 20 = 0
a =
3.Medium
R(s)+21s(s + 1)(s + 4)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 2/(s(s + 1)(s + 4)). How many closed-loop poles N lie in the right half-plane?

L(s) = 2/(s(s + 1)(s + 4))
N =
4.Medium
R(s)+K1s(s + 1)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 4)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 4))
Kmax =
5.Hard
ReIm−0.5−1−1.5−2−2.50.5j0.5j1

The root locus of L(s) = K/(s(s + 1)(s + 2)) leaves the real axis between s = 0 and s = −1. Find the break-away point sb.

L(s) = K/(s(s + 1)(s + 2))
sb =
6.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K/(s(s + 2)(s + 10)) leaves the real axis between s = 0 and s = −2. Find the break-away point sb.

L(s) = K/(s(s + 2)(s + 10))
sb =

Answer key

  1. 1. N = 1
  2. 2. a = 5
  3. 3. N = 1
  4. 4. Kmax = 20
  5. 5. sb = −0.4226
  6. 6. sb = −0.9449

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