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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
s³19s²12s¹?s⁰

For the characteristic polynomial Q(s) = s³ + s² + 9s + 2, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + s² + 9s + 2
b1 =
2.Easy
R(s)+K5s + 10C(s)−

A proportional gain K controls the plant G(s) = 5/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −30.

G(s) = 5/(s + 10)p = −30
K =
3.Medium
R(s)+K1s(s + 2)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 4)). 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 + 2)(s + 4))
ω = rad/s
4.Medium
R(s)+Ks + 11s(s + 1)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 11)/(s(s + 1)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 11)/(s(s + 1)(s + 4))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

The root locus of L(s) = K/(s(s + 3)(s + 8)) leaves the real axis between s = 0 and s = −3. Find the break-away point sb.

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

A unity-feedback loop has L(s) = K(s + 2)/((s + 4)(s + 8)(s + 6)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 2)/((s + 4)(s + 8)(s + 6))
σa =

Answer key

  1. 1. b1 = 7
  2. 2. K = 4
  3. 3. ω = 2.828 rad/s
  4. 4. Kmax = 3.333
  5. 5. sb = −1.333
  6. 6. σa = −8

Full step-by-step solutions: practise this topic on examproblems.com.