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
R(s)+261s(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 26/(s(s + 2)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 26/(s(s + 2))
ωd = rad/s
2.Easy
s³39s²325s¹?s⁰

For the characteristic polynomial Q(s) = 3s³ + 3s² + 9s + 25, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 3s³ + 3s² + 9s + 25
b1 =
3.Medium
R(s)+K = 1.21s(s + 1)(s + 3)C(s)−

The loop in the diagram runs with K = 1.2 and L(s) = K/(s(s + 1)(s + 3)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 1)(s + 3))K = 1.2
m =
4.Medium
R(s)+Ks + 1(s − 3)(s + 1)C(s)−

The plant G(s) = (s + 1)/((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 + 1)/((s − 3)(s + 1))
Kmin =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

The root locus of L(s) = K/(s(s + 2)(s + 8)) leaves the real axis between s = 0 and s = −2. Find the break-away point sb.

L(s) = K/(s(s + 2)(s + 8))
sb =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−4−4.5−50.5j0.5j1−0.5

For the loop L(s) = K/(s(s + 3)(s + 4)), find the gain K that puts a closed-loop pole at s = −0.5.

L(s) = K/(s(s + 3)(s + 4))s = −0.5
K =

Answer key

  1. 1. ωd = 5 rad/s
  2. 2. b1 = −16
  3. 3. m = 10
  4. 4. Kmin = 3
  5. 5. sb = −0.9296
  6. 6. K = 4.375

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