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
s⁴14−5s³410s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 4s³ + 4s² + 10s − 5. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 4s³ + 4s² + 10s − 5
N =
2.Easy
R(s)+K10s + 5C(s)−

A proportional gain K controls the plant G(s) = 10/(s + 5) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −45.

G(s) = 10/(s + 5)p = −45
K =
3.Medium
R(s)+451s(s + 1)(s + 5)C(s)−

The loop in the diagram has negative feedback and L(s) = 45/(s(s + 1)(s + 5)). How many closed-loop poles N lie in the right half-plane?

L(s) = 45/(s(s + 1)(s + 5))
N =
4.Medium
R(s)+K1s(s + 6)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 6)(s + 10)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 6)(s + 10))
ω = rad/s
5.Hard
ReIm−2−4−6−8−10−12j2−0.5

For the loop L(s) = K/(s(s + 5)(s + 10)), find the gain K that puts a closed-loop pole at s = −0.5.

L(s) = K/(s(s + 5)(s + 10))s = −0.5
K =
6.Hard
ReIm−1−2−3−4−5−6−7−81j1j2

For L(s) = K/((s + 7)(s² + 6s + 13)), 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 + 7)(s² + 6s + 13))
θd = °

Answer key

  1. 1. N = 1
  2. 2. K = 4
  3. 3. N = 2
  4. 4. ω = 7.746 rad/s
  5. 5. K = 21.38
  6. 6. θd = 63.43 °

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