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)+K1s + 2C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −4.

G(s) = 1/(s + 2)p = −4
K =
2.Easy
s³1−8s²428s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 4s² − 8s + 28 = 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 + 4s² − 8s + 28
N =
3.Medium
R(s)+K = 121s(s + 1)(s + 5)C(s)−

The loop in the diagram runs with K = 12 and L(s) = K/(s(s + 1)(s + 5)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 1)(s + 5))K = 12
m =
4.Medium
R(s)+501s(s + 4)(s + 10)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 50/(s(s + 4)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 50/(s(s + 4)(s + 10))
N =
5.Hard
ReIm−2−4j2

For L(s) = K/((s + 3)(s² + 2s + 10)), 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 + 3)(s² + 2s + 10))
θd = °
6.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 3)(s + 4)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 3)(s + 4)(s + 6))
ω = rad/s

Answer key

  1. 1. K = 2
  2. 2. N = 2
  3. 3. m = 2.5
  4. 4. N = 1
  5. 5. θd = 33.69 °
  6. 6. ω = 7.348 rad/s

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