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)+K = 11s + 10C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 1/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 10)K = 1
p =
2.Easy
R(s)+K = 210s + 4C(s)−

A proportional controller with gain K = 2 drives the plant G(s) = 10/(s + 4) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 10/(s + 4)K = 2
p =
3.Medium
R(s)+2881s(s + 2)(s + 6)C(s)−

The loop in the diagram has negative feedback and L(s) = 288/(s(s + 2)(s + 6)). How many closed-loop poles N lie in the right half-plane?

L(s) = 288/(s(s + 2)(s + 6))
N =
4.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)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 2)(s + 4))
Kmax =
5.Hard
ReImζ = 0.4−5−10j5j10

The root locus of L(s) = K/(s(s + 8)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.4.

L(s) = K/(s(s + 8))ζ = 0.4
K =
6.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

A unity-feedback loop has L(s) = K/(s(s + 3)(s + 8)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/(s(s + 3)(s + 8))
σa =

Answer key

  1. 1. p = −11
  2. 2. p = −24
  3. 3. N = 2
  4. 4. Kmax = 48
  5. 5. K = 100
  6. 6. σa = −3.667

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