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
s³11s²511s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 5s² + s + 11 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ + 5s² + s + 11
N =
2.Easy
s³16s²a9s¹(6·a − 9)/as⁰9

The characteristic equation of a closed loop is s³ + a·s² + 6s + 9 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 6s + 9 = 0
a =
3.Medium
R(s)+K1s(s + 5)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 5)(s + 8)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 5)(s + 8))
ω = rad/s
4.Medium
R(s)+K = 301s(s + 4)(s + 6)C(s)−

The loop in the diagram runs with K = 30 and L(s) = K/(s(s + 4)(s + 6)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 4)(s + 6))K = 30
m =
5.Hard
ReIm−1−2−3−4−5−6j1−1.5

For the loop L(s) = K/(s(s + 2)(s + 5)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 2)(s + 5))s = −1.5
K =
6.Hard
ReIm−2−4−6−8−10−12j2

A unity-feedback loop has L(s) = K(s + 6)/(s(s + 8)(s + 10)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 6)/(s(s + 8)(s + 10))
σa =

Answer key

  1. 1. N = 2
  2. 2. a = 1.5
  3. 3. ω = 6.325 rad/s
  4. 4. m = 8
  5. 5. K = 2.625
  6. 6. σa = −6

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