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
s³11s²613s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 6s² + s + 13 = 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 + 6s² + s + 13
N =
2.Easy
R(s)+K1s + 1C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 1) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −1.5.

G(s) = 1/(s + 1)p = −1.5
K =
3.Medium
R(s)+Ks + 8s(s + 1)(s + 2)C(s)−

A unity-feedback loop has L(s) = K(s + 8)/(s(s + 1)(s + 2)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 8)/(s(s + 1)(s + 2))
Kmax =
4.Medium
R(s)+Ks + 6s(s + 1)(s + 3)C(s)−

A unity-feedback loop has L(s) = K(s + 6)/(s(s + 1)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 6)/(s(s + 1)(s + 3))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12−14−162j2j4sb​

The root locus of L(s) = K/((s + 6)(s + 14)) is shown. Find the gain K at the break-away point sb, where the two closed-loop poles meet on the real axis.

L(s) = K/((s + 6)(s + 14))
K =
6.Hard
ReImζ = 0.3−10j10

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 = 0.3.

L(s) = K/(s(s + 10))ζ = 0.3
K =

Answer key

  1. 1. N = 2
  2. 2. K = 0.5
  3. 3. Kmax = 1.2
  4. 4. Kmax = 6
  5. 5. K = 16
  6. 6. K = 277.8

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