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)+131s(s + 6)C(s)−

A unity-feedback loop has the open-loop transfer function L(s) = 13/(s(s + 6)). The closed-loop poles are complex. Find their imaginary part ωd.

L(s) = 13/(s(s + 6))
ωd = rad/s
2.Easy
R(s)+K = 31s + 1C(s)−

A proportional controller with gain K = 3 drives the plant G(s) = 1/(s + 1) in a unity-feedback loop. Find the closed-loop pole p.

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

A unity-feedback loop has L(s) = K(s + 6)/(s(s + 1)(s + 3)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 6)/(s(s + 1)(s + 3))
Kmax =
4.Medium
R(s)+K1(s + 2)(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K/((s + 2)(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 2)(s + 3)(s + 4))
Kmax =
5.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 2)(s + 5)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 2)(s + 5)(s + 6))
ω = rad/s
6.Hard
ReIm−2−4−6−8−10−12−14−162j2j4

The root locus of L(s) = K(s + 14)/(s(s + 7)) leaves the real axis between s = 0 and s = −7, circles round and joins the real axis again to the left of the zero. Find the break-away point sb.

L(s) = K(s + 14)/(s(s + 7))
sb =

Answer key

  1. 1. ωd = 2 rad/s
  2. 2. p = −4
  3. 3. Kmax = 6
  4. 4. Kmax = 210
  5. 5. ω = 7.211 rad/s
  6. 6. sb = −4.101

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