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
s³15s²a20s¹(5·a − 20)/as⁰20

The characteristic equation of a closed loop is s³ + a·s² + 5s + 20 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 5s + 20 = 0
a =
2.Easy
s³14s²428s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 4s² + 4s + 28 = 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 + 4s² + 4s + 28
N =
3.Medium
R(s)+8401s(s + 4)(s + 10)C(s)−

The loop in the diagram has negative feedback and L(s) = 840/(s(s + 4)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 840/(s(s + 4)(s + 10))
N =
4.Medium
R(s)+K1s(s + 4)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 5)). Find the largest gain Kmax for which the closed loop is stable.

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

A unity-feedback loop has L(s) = K/((s + 5)(s + 3)(s + 4)(s + 6)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/((s + 5)(s + 3)(s + 4)(s + 6))
σa =
6.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−4

For the loop L(s) = K/(s(s + 6)(s + 8)), find the gain K that puts a closed-loop pole at s = −4.

L(s) = K/(s(s + 6)(s + 8))s = −4
K =

Answer key

  1. 1. a = 4
  2. 2. N = 2
  3. 3. N = 2
  4. 4. Kmax = 180
  5. 5. σa = −4.5
  6. 6. K = 32

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