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
s³16s²327s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 3s² + 6s + 27, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 3s² + 6s + 27
b1 =
2.Easy
s⁴111s³27s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + s² + 7s + 1. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + s² + 7s + 1
N =
3.Medium
R(s)+K1(s + 2)(s + 3)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 2)(s + 3)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 2)(s + 3)(s + 6))
Kmax =
4.Medium
R(s)+Ks + 5(s − 4)(s + 2)C(s)−

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

For L(s) = K/((s + 1)(s² + 6s + 13)), 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 + 1)(s² + 6s + 13))
θd = °
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. b1 = −3
  2. 2. N = 2
  3. 3. Kmax = 360
  4. 4. Kmin = 2
  5. 5. θd = −45 °
  6. 6. ω = 4.123 rad/s

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