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
s³18s²a16s¹(8·a − 16)/as⁰16

The characteristic equation of a closed loop is s³ + a·s² + 8s + 16 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 8s + 16 = 0
a =
2.Easy
R(s)+K2s + 2C(s)−

A proportional gain K controls the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −3.

G(s) = 2/(s + 2)p = −3
K =
3.Medium
R(s)+K1s(s + 2)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 10)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 2)(s + 10))
Kmax =
4.Medium
R(s)+K1s(s + 1)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 10)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 1)(s + 10))
ω = rad/s
5.Hard
ReImζ = 0.707−1−2−3−4−5j1j2

The root locus of L(s) = K/(s(s + 4)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.707.

L(s) = K/(s(s + 4))ζ = 0.707
K =
6.Hard
ReIm−2−4−6−8−10−12−14−16j2j4

The root locus of L(s) = K(s + 13)/(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-away point sb.

L(s) = K(s + 13)/(s(s + 6))
sb =

Answer key

  1. 1. a = 2
  2. 2. K = 0.5
  3. 3. Kmax = 240
  4. 4. ω = 3.162 rad/s
  5. 5. K = 8.002
  6. 6. sb = −3.461

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