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)+K = 35s + 1C(s)−

A proportional controller with gain K = 3 drives the plant G(s) = 5/(s + 1) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 1)K = 3
p =
2.Easy
s³37s²529s¹?s⁰

For the characteristic polynomial Q(s) = 3s³ + 5s² + 7s + 29, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 3s³ + 5s² + 7s + 29
b1 =
3.Medium
R(s)+K1s(s + 5)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 5)(s + 6)). 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 + 6))
ω = rad/s
4.Medium
R(s)+K1(s + 2)(s + 4)(s + 5)C(s)−

A unity-feedback loop has L(s) = K/((s + 2)(s + 4)(s + 5)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 2)(s + 4)(s + 5))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K/(s(s + 4)(s + 10)) leaves the real axis between s = 0 and s = −4. Find the break-away point sb.

L(s) = K/(s(s + 4)(s + 10))
sb =
6.Hard
ReImζ = 0.8−1−2−3−4−5−6−71j1j2

The root locus of L(s) = K/(s(s + 6)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.8.

L(s) = K/(s(s + 6))ζ = 0.8
K =

Answer key

  1. 1. p = −16
  2. 2. b1 = −10.4
  3. 3. ω = 5.477 rad/s
  4. 4. Kmax = 378
  5. 5. sb = −1.761
  6. 6. K = 14.06

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