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
s³110s²33s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 3s² + 10s + 3, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 3s² + 10s + 3
b1 =
2.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 =
3.Medium
R(s)+Ks + 11s(s + 2)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 11)/(s(s + 2)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 11)/(s(s + 2)(s + 4))
Kmax =
4.Medium
R(s)+K = 82.51s(s + 5)(s + 6)C(s)−

The loop in the diagram runs with K = 82.5 and L(s) = K/(s(s + 5)(s + 6)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 5)(s + 6))K = 82.5
m =
5.Hard
ReIm−1−2−3−4−5−6j1

A unity-feedback loop has L(s) = K/((s + 4)(s + 5)(s + 1)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/((s + 4)(s + 5)(s + 1))
σa =
6.Hard
ReIm−2−4−6−8−10−12−14j2j4sb​

The root locus of L(s) = K/((s + 4)(s + 12)) 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 + 4)(s + 12))
K =

Answer key

  1. 1. b1 = 9
  2. 2. a = 4
  3. 3. Kmax = 9.6
  4. 4. m = 4
  5. 5. σa = −3.333
  6. 6. K = 16

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