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
s³24s²230s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + 2s² + 4s + 30, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + 2s² + 4s + 30
b1 =
2.Easy
s³1−8s²−5−23s¹?s⁰?

A closed-loop system has the characteristic equation s³ − 5s² − 8s − 23 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ − 5s² − 8s − 23
N =
3.Medium
R(s)+Ks + 1(s − 3)(s + 2)C(s)−

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

The root locus of L(s) = K(s + 8)/(s(s + 3)) leaves the real axis between s = 0 and s = −3, circles round and joins the real axis again to the left of the zero. Find the break-away point sb.

L(s) = K(s + 8)/(s(s + 3))
sb =
6.Hard
ReIm−2−4−6−8−10−12−14j2j4sb​

The root locus of L(s) = K/((s + 6)(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 + 6)(s + 12))
K =

Answer key

  1. 1. b1 = −26
  2. 2. N = 1
  3. 3. Kmin = 6
  4. 4. Kmax = 168
  5. 5. sb = −1.675
  6. 6. K = 9

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