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
s³12s²a8s¹(2·a − 8)/as⁰8

The characteristic equation of a closed loop is s³ + a·s² + 2s + 8 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 2s + 8 = 0
a =
2.Easy
s³15s²−118s¹?s⁰?

A closed-loop system has the characteristic equation s³ − s² + 5s + 18 = 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 − s² + 5s + 18
N =
3.Medium
R(s)+Ks + 10(s − 3)(s + 1)C(s)−

The plant G(s) = (s + 10)/((s − 3)(s + 1)) has an unstable pole at s = 3. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain Kmin that makes the closed loop stable.

G(s) = (s + 10)/((s − 3)(s + 1))
Kmin =
4.Medium
R(s)+Ks + 7s(s + 1)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 7)/(s(s + 1)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

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

A unity-feedback loop has L(s) = K(s + 4)/(s(s + 6)(s + 8)(s + 1)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 4)/(s(s + 6)(s + 8)(s + 1))
σa =
6.Hard
ReIm−1−2−3−4−5−6−7−81j1j2

For L(s) = K/((s + 7)(s² + 6s + 10)), find the angle of departure θd of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 7)(s² + 6s + 10))
θd = °

Answer key

  1. 1. a = 4
  2. 2. N = 2
  3. 3. Kmin = 2
  4. 4. Kmax = 10
  5. 5. σa = −3.667
  6. 6. θd = 75.96 °

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