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
s³13s²a24s¹(3·a − 24)/as⁰24

The characteristic equation of a closed loop is s³ + a·s² + 3s + 24 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 3s + 24 = 0
a =
2.Easy
s⁴18−20s³4−13s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 4s³ + 8s² − 13s − 20. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 4s³ + 8s² − 13s − 20
N =
3.Medium
R(s)+K1(s + 1)(s + 2)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 2)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 2)(s + 6))
Kmax =
4.Medium
R(s)+51s(s + 1)(s + 6)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 5/(s(s + 1)(s + 6)). How many closed-loop poles N lie in the right half-plane?

L(s) = 5/(s(s + 1)(s + 6))
N =
5.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 4)(s + 5)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 4)(s + 5)(s + 6))
ω = rad/s
6.Hard
ReIm−2−4−6−8−10−12j2

A unity-feedback loop has L(s) = K/((s + 6)(s + 2)(s + 10)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/((s + 6)(s + 2)(s + 10))
σa =

Answer key

  1. 1. a = 8
  2. 2. N = 1
  3. 3. Kmax = 168
  4. 4. N = 1
  5. 5. ω = 8.602 rad/s
  6. 6. σa = −6

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