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
s³14s²15s¹?s⁰?

A closed-loop system has the characteristic equation s³ + s² + 4s + 5 = 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 + 4s + 5
N =
2.Easy
s³13s²112s¹?s⁰?

A closed-loop system has the characteristic equation s³ + s² + 3s + 12 = 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 + 3s + 12
N =
3.Medium
R(s)+Ks + 9s(s + 2)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 9)/(s(s + 2)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 9)/(s(s + 2)(s + 4))
Kmax =
4.Medium
R(s)+K1s(s + 1)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 4)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 4))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K(s + 10)/(s(s + 5)) leaves the real axis between s = 0 and s = −5, circles round and joins the real axis again to the left of the zero. Find the break-away point sb.

L(s) = K(s + 10)/(s(s + 5))
sb =
6.Hard
ReImζ = 0.8−1−2−3−4−5j1

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 = 0.8.

L(s) = K/(s(s + 4))ζ = 0.8
K =

Answer key

  1. 1. N = 2
  2. 2. N = 2
  3. 3. Kmax = 16
  4. 4. Kmax = 20
  5. 5. sb = −2.929
  6. 6. K = 6.25

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