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
s³16s²120s¹?s⁰

For the characteristic polynomial Q(s) = s³ + s² + 6s + 20, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + s² + 6s + 20
b1 =
2.Easy
R(s)+K10s + 2C(s)−

A proportional gain K controls the plant G(s) = 10/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −42.

G(s) = 10/(s + 2)p = −42
K =
3.Medium
R(s)+K1(s + 2)(s + 5)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 2)(s + 5)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 2)(s + 5)(s + 6))
Kmax =
4.Medium
R(s)+Ks + 2(s − 4)(s + 1)C(s)−

The plant G(s) = (s + 2)/((s − 4)(s + 1)) 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 + 1))
Kmin =
5.Hard
ReImζ = 0.5−2−4j2j4

The root locus of L(s) = K/(s(s + 4)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.5.

L(s) = K/(s(s + 4))ζ = 0.5
K =
6.Hard
ReIm−2−4−6j2j4

For L(s) = K/((s + 5)(s² + 6s + 25)), find the angle of departure θd of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 5)(s² + 6s + 25))
θd = °

Answer key

  1. 1. b1 = −14
  2. 2. K = 4
  3. 3. Kmax = 616
  4. 4. Kmin = 3
  5. 5. K = 16
  6. 6. θd = 26.57 °

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