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
s⁴1916s³47s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 4s³ + 9s² + 7s + 16. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 4s³ + 9s² + 7s + 16
N =
2.Easy
R(s)+K2s + 1C(s)−

A proportional gain K controls the plant G(s) = 2/(s + 1) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −4.

G(s) = 2/(s + 1)p = −4
K =
3.Medium
R(s)+201s(s + 4)(s + 10)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 20/(s(s + 4)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 20/(s(s + 4)(s + 10))
N =
4.Medium
R(s)+K1s(s + 2)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 5)). 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 + 5))
ω = rad/s
5.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 =
6.Hard
ReImζ = 0.3−5j5

The root locus of L(s) = K/(s(s + 4)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.3.

L(s) = K/(s(s + 4))ζ = 0.3
K =

Answer key

  1. 1. N = 2
  2. 2. K = 1.5
  3. 3. N = 1
  4. 4. ω = 3.162 rad/s
  5. 5. K = 16
  6. 6. K = 44.44

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