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
R(s)+K10s + 10C(s)−

A proportional gain K controls the plant G(s) = 10/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −25.

G(s) = 10/(s + 10)p = −25
K =
2.Easy
s³28s²616s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + 6s² + 8s + 16, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + 6s² + 8s + 16
b1 =
3.Medium
R(s)+K1s(s + 4)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 4)(s + 6)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 4)(s + 6))
Kmax =
4.Medium
R(s)+101s(s + 1)(s + 10)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 10/(s(s + 1)(s + 10)). How many closed-loop poles N lie in the right half-plane?

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

The root locus of L(s) = K(s + 11)/(s(s + 3)) leaves the real axis between s = 0 and s = −3, circles round and joins the real axis again to the left of the zero. Find the break-away point sb.

L(s) = K(s + 11)/(s(s + 3))
sb =
6.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2−4

For the loop L(s) = K/(s(s + 6)(s + 8)), find the gain K that puts a closed-loop pole at s = −4.

L(s) = K/(s(s + 6)(s + 8))s = −4
K =

Answer key

  1. 1. K = 1.5
  2. 2. b1 = 2.667
  3. 3. Kmax = 240
  4. 4. N = 1
  5. 5. sb = −1.619
  6. 6. K = 32

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