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)+K = 110s + 5C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 10/(s + 5) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 10/(s + 5)K = 1
p =
2.Easy
R(s)+K5s + 1C(s)−

A proportional gain K controls the plant G(s) = 5/(s + 1) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −26.

G(s) = 5/(s + 1)p = −26
K =
3.Medium
R(s)+K = 40.51s(s + 3)(s + 6)C(s)−

The loop in the diagram runs with K = 40.5 and L(s) = K/(s(s + 3)(s + 6)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 3)(s + 6))K = 40.5
m =
4.Medium
R(s)+K1s(s + 1)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 10)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 10))
Kmax =
5.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 2)(s + 4)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

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

The root locus of L(s) = K/(s(s + 6)(s + 10)) leaves the real axis between s = 0 and s = −6. Find the break-away point sb.

L(s) = K/(s(s + 6)(s + 10))
sb =

Answer key

  1. 1. p = −15
  2. 2. K = 5
  3. 3. m = 4
  4. 4. Kmax = 110
  5. 5. ω = 6.633 rad/s
  6. 6. sb = −2.427

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