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.

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1.Easy
s³14s²19s¹?s⁰?

A closed-loop system has the characteristic equation s³ + s² + 4s + 9 = 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 + s² + 4s + 9
N =
2.Easy
s³13s²628s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 6s² + 3s + 28 = 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 + 3s + 28
N =
3.Medium
R(s)+K = 841s(s + 6)(s + 8)C(s)−

The loop in the diagram runs with K = 84 and L(s) = K/(s(s + 6)(s + 8)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 6)(s + 8))K = 84
m =
4.Medium
R(s)+Ks + 8(s − 2)(s + 5)C(s)−

The plant G(s) = (s + 8)/((s − 2)(s + 5)) has an unstable pole at s = 2. 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 + 8)/((s − 2)(s + 5))
Kmin =
5.Hard
ReIm−1−2−3−4−5−6−71j1j2−1

For the loop L(s) = K/(s(s + 2)(s + 6)), find the gain K that puts a closed-loop pole at s = −1.

L(s) = K/(s(s + 2)(s + 6))s = −1
K =
6.Hard
ReIm−2−4−6−8−10−12−14−16j2j4

The root locus of L(s) = K(s + 13)/(s(s + 7)) leaves the real axis between s = 0 and s = −7, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

L(s) = K(s + 13)/(s(s + 7))
si =

Answer key

  1. 1. N = 2
  2. 2. N = 2
  3. 3. m = 8
  4. 4. Kmin = 1.25
  5. 5. K = 5
  6. 6. si = −21.83

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