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
s³1−7s²119s¹?s⁰?

A closed-loop system has the characteristic equation s³ + s² − 7s + 19 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ + s² − 7s + 19
N =
2.Easy
R(s)+K5s + 2C(s)−

A proportional gain K controls the plant G(s) = 5/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −22.

G(s) = 5/(s + 2)p = −22
K =
3.Medium
R(s)+Ks + 8(s − 2)(s + 1)C(s)−

The plant G(s) = (s + 8)/((s − 2)(s + 1)) has an unstable pole at s = 2. 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 + 8)/((s − 2)(s + 1))
Kmin =
4.Medium
R(s)+K1(s + 1)(s + 2)(s + 3)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 2)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 2)(s + 3))
Kmax =
5.Hard
ReImζ = 0.6−5−10j5

The root locus of L(s) = K/(s(s + 10)) is shown. Find the gain K for which the closed-loop poles have the damping ratio ζ = 0.6.

L(s) = K/(s(s + 10))ζ = 0.6
K =
6.Hard
ReIm−1−2−3−4−5−6j1j2

For L(s) = K/((s + 5)(s² + 8s + 20)), 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 + 8s + 20))
θd = °

Answer key

  1. 1. N = 2
  2. 2. K = 4
  3. 3. Kmin = 1
  4. 4. Kmax = 60
  5. 5. K = 69.44
  6. 6. θd = 26.57 °

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