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
s³1−4s²6−26s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 6s² − 4s − 26 = 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 + 6s² − 4s − 26
N =
2.Easy
R(s)+261s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 26/(s(s + 10)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 26/(s(s + 10))
ωd = rad/s
3.Medium
R(s)+K = 961s(s + 6)(s + 10)C(s)−

The loop in the diagram runs with K = 96 and L(s) = K/(s(s + 6)(s + 10)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 6)(s + 10))K = 96
m =
4.Medium
R(s)+K1s(s + 4)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 5)). Find the largest gain Kmax for which the closed loop is stable.

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

The root locus of L(s) = K(s + 9)/(s(s + 6)) leaves the real axis between s = 0 and s = −6, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

L(s) = K(s + 9)/(s(s + 6))
si =
6.Hard
ReIm−2−4−6−8−10−12j2sb​

The root locus of L(s) = K/((s + 5)(s + 11)) 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 + 5)(s + 11))
K =

Answer key

  1. 1. N = 1
  2. 2. ωd = 1 rad/s
  3. 3. m = 10
  4. 4. Kmax = 180
  5. 5. si = −14.2
  6. 6. K = 9

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