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
s⁴1416s³57s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 5s³ + 4s² + 7s + 16. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 5s³ + 4s² + 7s + 16
N =
2.Easy
s³14s²a20s¹(4·a − 20)/as⁰20

The characteristic equation of a closed loop is s³ + a·s² + 4s + 20 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 4s + 20 = 0
a =
3.Medium
R(s)+K = 16.81s(s + 3)(s + 4)C(s)−

The loop in the diagram runs with K = 16.8 and L(s) = K/(s(s + 3)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 3)(s + 4))K = 16.8
m =
4.Medium
R(s)+Ks + 12s(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 12)/(s(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 12)/(s(s + 3)(s + 4))
Kmax =
5.Hard
ReIm−2−4j2

For L(s) = K/((s + 2)(s² + 8s + 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 + 2)(s² + 8s + 25))
θd = °
6.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K/(s(s + 2)(s + 10)) leaves the real axis between s = 0 and s = −2. Find the break-away point sb.

L(s) = K/(s(s + 2)(s + 10))
sb =

Answer key

  1. 1. N = 2
  2. 2. a = 5
  3. 3. m = 5
  4. 4. Kmax = 16.8
  5. 5. θd = −33.69 °
  6. 6. sb = −0.9449

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