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 = 210s + 1C(s)−

A proportional controller with gain K = 2 drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 10/(s + 1)K = 2
p =
2.Easy
s³17s²312s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 3s² + 7s + 12, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 3s² + 7s + 12
b1 =
3.Medium
R(s)+Ks + 5s(s + 1)(s + 3)C(s)−

A unity-feedback loop has L(s) = K(s + 5)/(s(s + 1)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 5)/(s(s + 1)(s + 3))
Kmax =
4.Medium
R(s)+Ks + 1(s − 4)(s + 3)C(s)−

The plant G(s) = (s + 1)/((s − 4)(s + 3)) 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 + 1)/((s − 4)(s + 3))
Kmin =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2

A unity-feedback loop has L(s) = K(s + 8)/(s(s + 2)(s + 5)(s + 6)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 8)/(s(s + 2)(s + 5)(s + 6))
σa =
6.Hard
ReIm−2−4−6j2j4

The root locus of L(s) = K/((s + 1)(s + 2)(s + 5)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 1)(s + 2)(s + 5))
ω = rad/s

Answer key

  1. 1. p = −21
  2. 2. b1 = 3
  3. 3. Kmax = 12
  4. 4. Kmin = 12
  5. 5. σa = −1.667
  6. 6. ω = 4.123 rad/s

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