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
s³21s²220s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + 2s² + s + 20, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + 2s² + s + 20
b1 =
2.Easy
R(s)+K10s + 1C(s)−

A proportional gain K controls the plant G(s) = 10/(s + 1) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −6.

G(s) = 10/(s + 1)p = −6
K =
3.Medium
R(s)+Ks + 2(s − 4)(s + 5)C(s)−

The plant G(s) = (s + 2)/((s − 4)(s + 5)) has an unstable pole at s = 4. 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 + 2)/((s − 4)(s + 5))
Kmin =
4.Medium
R(s)+K1(s + 2)(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K/((s + 2)(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 2)(s + 3)(s + 4))
Kmax =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

A unity-feedback loop has L(s) = K/((s + 8)(s + 4)(s + 3)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/((s + 8)(s + 4)(s + 3))
σa =
6.Hard
ReImζ = 0.5−2−4−6j2j4j6

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 = 0.5.

L(s) = K/(s(s + 6))ζ = 0.5
K =

Answer key

  1. 1. b1 = −19
  2. 2. K = 0.5
  3. 3. Kmin = 10
  4. 4. Kmax = 210
  5. 5. σa = −5
  6. 6. K = 36

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