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 + 4C(s)−

A proportional gain K controls the plant G(s) = 1/(s + 4) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −4.5.

G(s) = 1/(s + 4)p = −4.5
K =
2.Easy
R(s)+K = 41s + 2C(s)−

A proportional controller with gain K = 4 drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 2)K = 4
p =
3.Medium
R(s)+601s(s + 1)(s + 5)C(s)−

The loop in the diagram has negative feedback and L(s) = 60/(s(s + 1)(s + 5)). How many closed-loop poles N lie in the right half-plane?

L(s) = 60/(s(s + 1)(s + 5))
N =
4.Medium
R(s)+K = 71s(s + 2)(s + 5)C(s)−

The loop in the diagram runs with K = 7 and L(s) = K/(s(s + 2)(s + 5)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 2)(s + 5))K = 7
m =
5.Hard
ReIm−2−4−6j2j4

The root locus of L(s) = K/((s + 1)(s + 3)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 1)(s + 3)(s + 6))
ω = rad/s
6.Hard
ReIm−2−4−6−8−10−12−14j2j4

The root locus of L(s) = K(s + 12)/(s(s + 5)) leaves the real axis between s = 0 and s = −5, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

L(s) = K(s + 12)/(s(s + 5))
si =

Answer key

  1. 1. K = 0.5
  2. 2. p = −6
  3. 3. N = 2
  4. 4. m = 10
  5. 5. ω = 5.196 rad/s
  6. 6. si = −21.17

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