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
s³15s²a40s¹(5·a − 40)/as⁰40

The characteristic equation of a closed loop is s³ + a·s² + 5s + 40 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 5s + 40 = 0
a =
2.Easy
R(s)+611s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 61/(s(s + 10)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 61/(s(s + 10))
ωd = rad/s
3.Medium
R(s)+501s(s + 2)(s + 4)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 50/(s(s + 2)(s + 4)). How many closed-loop poles N lie in the right half-plane?

L(s) = 50/(s(s + 2)(s + 4))
N =
4.Medium
R(s)+21s(s + 3)(s + 4)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 2/(s(s + 3)(s + 4)). How many closed-loop poles N lie in the right half-plane?

L(s) = 2/(s(s + 3)(s + 4))
N =
5.Hard
ReImζ = 0.707−0.5−1−1.5−2−2.50.5j0.5j1

The root locus of L(s) = K/(s(s + 2)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.707.

L(s) = K/(s(s + 2))ζ = 0.707
K =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.5j0.5j1

The root locus of L(s) = K/(s(s + 1)(s + 3)) leaves the real axis between s = 0 and s = −1. Find the break-away point sb.

L(s) = K/(s(s + 1)(s + 3))
sb =

Answer key

  1. 1. a = 8
  2. 2. ωd = 6 rad/s
  3. 3. N = 1
  4. 4. N = 1
  5. 5. K = 2.001
  6. 6. sb = −0.4514

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