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
R(s)+201s(s + 8)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 20/(s(s + 8)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 20/(s(s + 8))
ωd = rad/s
2.Easy
s³110s²a15s¹(10·a − 15)/as⁰15

The characteristic equation of a closed loop is s³ + a·s² + 10s + 15 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 10s + 15 = 0
a =
3.Medium
R(s)+K1s(s + 4)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 8)). 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 + 4)(s + 8))
ω = rad/s
4.Medium
R(s)+K1s(s + 5)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 5)(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 + 5)(s + 10))
ω = rad/s
5.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 2)(s + 4)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

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

The root locus of L(s) = K/(s(s + 6)(s + 10)) leaves the real axis between s = 0 and s = −6. Find the break-away point sb.

L(s) = K/(s(s + 6)(s + 10))
sb =

Answer key

  1. 1. ωd = 2 rad/s
  2. 2. a = 1.5
  3. 3. ω = 5.657 rad/s
  4. 4. ω = 7.071 rad/s
  5. 5. ω = 6.633 rad/s
  6. 6. sb = −2.427

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