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
s³28s²67s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + 6s² + 8s + 7, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + 6s² + 8s + 7
b1 =
2.Easy
s³18s²a32s¹(8·a − 32)/as⁰32

The characteristic equation of a closed loop is s³ + a·s² + 8s + 32 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 8s + 32 = 0
a =
3.Medium
R(s)+K1(s + 1)(s + 4)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 4)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 4)(s + 6))
Kmax =
4.Medium
R(s)+Ks + 10s(s + 2)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 10)/(s(s + 2)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 10)/(s(s + 2)(s + 4))
Kmax =
5.Hard
ReImζ = 0.3−5−10j5j10j15

The root locus of L(s) = K/(s(s + 8)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.3.

L(s) = K/(s(s + 8))ζ = 0.3
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2

The root locus of L(s) = K/(s(s + 5)(s + 6)) leaves the real axis between s = 0 and s = −5. Find the break-away point sb.

L(s) = K/(s(s + 5)(s + 6))
sb =

Answer key

  1. 1. b1 = 5.667
  2. 2. a = 4
  3. 3. Kmax = 350
  4. 4. Kmax = 12
  5. 5. K = 177.8
  6. 6. sb = −1.811

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