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 + 10C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −10.5.

G(s) = 1/(s + 10)p = −10.5
K =
2.Easy
R(s)+K = 25s + 5C(s)−

A proportional controller with gain K = 2 drives the plant G(s) = 5/(s + 5) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 5/(s + 5)K = 2
p =
3.Medium
R(s)+K1(s + 3)(s + 4)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 3)(s + 4)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 3)(s + 4)(s + 6))
Kmax =
4.Medium
R(s)+38401s(s + 6)(s + 10)C(s)−

The loop in the diagram has negative feedback and L(s) = 3840/(s(s + 6)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 3840/(s(s + 6)(s + 10))
N =
5.Hard
ReIm−1−2−3−4−5−6−7−8−9−101j1j2sb​

The root locus of L(s) = K/((s + 2)(s + 8)) 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 + 2)(s + 8))
K =
6.Hard
ReIm−1−2−3−4−5−6−71j1j2−0.5

For the loop L(s) = K/(s(s + 2)(s + 6)), find the gain K that puts a closed-loop pole at s = −0.5.

L(s) = K/(s(s + 2)(s + 6))s = −0.5
K =

Answer key

  1. 1. K = 0.5
  2. 2. p = −15
  3. 3. Kmax = 630
  4. 4. N = 2
  5. 5. K = 9
  6. 6. K = 4.125

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