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
s³21s²510s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + 5s² + s + 10, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + 5s² + s + 10
b1 =
2.Easy
s³110s²a20s¹(10·a − 20)/as⁰20

The characteristic equation of a closed loop is s³ + a·s² + 10s + 20 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 10s + 20 = 0
a =
3.Medium
R(s)+K1s(s + 2)(s + 4)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 4)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 2)(s + 4))
Kmax =
4.Medium
R(s)+K = 51s(s + 2)(s + 3)C(s)−

The loop in the diagram runs with K = 5 and L(s) = K/(s(s + 2)(s + 3)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 2)(s + 3))K = 5
m =
5.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K(s + 10)/(s(s + 7)) leaves the real axis between s = 0 and s = −7, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

L(s) = K(s + 10)/(s(s + 7))
si =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2sb​

The root locus of L(s) = K/((s + 4)(s + 6)) 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 + 4)(s + 6))
K =

Answer key

  1. 1. b1 = −3
  2. 2. a = 2
  3. 3. Kmax = 48
  4. 4. m = 6
  5. 5. si = −15.48
  6. 6. K = 1

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