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
s⁴14−19s³17s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + s³ + 4s² + 7s − 19. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + s³ + 4s² + 7s − 19
N =
2.Easy
s³16s²a48s¹(6·a − 48)/as⁰48

The characteristic equation of a closed loop is s³ + a·s² + 6s + 48 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 6s + 48 = 0
a =
3.Medium
R(s)+Ks + 4s(s + 1)(s + 2)C(s)−

A unity-feedback loop has L(s) = K(s + 4)/(s(s + 1)(s + 2)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

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

The plant G(s) = (s + 4)/((s − 3)(s + 2)) has an unstable pole at s = 3. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain Kmin that makes the closed loop stable.

G(s) = (s + 4)/((s − 3)(s + 2))
Kmin =
5.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K(s + 11)/(s(s + 8)) leaves the real axis between s = 0 and s = −8, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

L(s) = K(s + 11)/(s(s + 8))
si =
6.Hard
ReIm−2−4−6−8−10−12−14−16j2j4

The root locus of L(s) = K(s + 13)/(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-away point sb.

L(s) = K(s + 13)/(s(s + 5))
sb =

Answer key

  1. 1. N = 3
  2. 2. a = 8
  3. 3. Kmax = 6
  4. 4. Kmin = 1.5
  5. 5. si = −16.74
  6. 6. sb = −2.802

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