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
s⁴1111s³24s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + s² + 4s + 11. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + s² + 4s + 11
N =
2.Easy
s³13s²a12s¹(3·a − 12)/as⁰12

The characteristic equation of a closed loop is s³ + a·s² + 3s + 12 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 3s + 12 = 0
a =
3.Medium
R(s)+K1s(s + 2)(s + 6)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 6)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 2)(s + 6))
Kmax =
4.Medium
R(s)+K1s(s + 2)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 10)). Find the largest gain Kmax for which the closed loop is stable.

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

The root locus of L(s) = K(s + 10)/(s(s + 4)) leaves the real axis between s = 0 and s = −4, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

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

A unity-feedback loop has L(s) = K/((s + 6)(s + 10)(s + 4)(s + 8)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/((s + 6)(s + 10)(s + 4)(s + 8))
σa =

Answer key

  1. 1. N = 2
  2. 2. a = 4
  3. 3. Kmax = 96
  4. 4. Kmax = 240
  5. 5. si = −17.75
  6. 6. σa = −7

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