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
s³12s²39s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 3s² + 2s + 9, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 3s² + 2s + 9
b1 =
2.Easy
s⁴16−19s³512s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 5s³ + 6s² + 12s − 19. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 5s³ + 6s² + 12s − 19
N =
3.Medium
R(s)+K1s(s + 5)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 5)(s + 8)). 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 + 5)(s + 8))
ω = rad/s
4.Medium
R(s)+1801s(s + 3)(s + 5)C(s)−

The loop in the diagram has negative feedback and L(s) = 180/(s(s + 3)(s + 5)). How many closed-loop poles N lie in the right half-plane?

L(s) = 180/(s(s + 3)(s + 5))
N =
5.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 =
6.Hard
ReIm−1−2−3−4−5−6−7−81j1j2sb​

The root locus of L(s) = K/((s + 1)(s + 7)) 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 + 1)(s + 7))
K =

Answer key

  1. 1. b1 = −1
  2. 2. N = 1
  3. 3. ω = 6.325 rad/s
  4. 4. N = 2
  5. 5. K = 1
  6. 6. K = 9

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