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
s³16s²a9s¹(6·a − 9)/as⁰9

The characteristic equation of a closed loop is s³ + a·s² + 6s + 9 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 6s + 9 = 0
a =
2.Easy
R(s)+K1s + 4C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 4) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −9.

G(s) = 1/(s + 4)p = −9
K =
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)+K1s(s + 1)(s + 5)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 5)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 5))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2

A unity-feedback loop has L(s) = K/((s + 8)(s + 10)(s + 6)). Find the point σa where the asymptotes of the root locus cross the real axis.

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

A unity-feedback loop has L(s) = K(s + 8)/(s(s + 2)(s + 10)(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 + 10)(s + 6))
σa =

Answer key

  1. 1. a = 1.5
  2. 2. K = 5
  3. 3. Kmax = 350
  4. 4. Kmax = 30
  5. 5. σa = −8
  6. 6. σa = −3.333

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