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
s³16s²327s¹?s⁰?

A closed-loop system has the characteristic equation s³ + 3s² + 6s + 27 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ + 3s² + 6s + 27
N =
2.Easy
s³210s²127s¹?s⁰

For the characteristic polynomial Q(s) = 2s³ + s² + 10s + 27, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 2s³ + s² + 10s + 27
b1 =
3.Medium
R(s)+K1(s + 1)(s + 4)(s + 5)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 4)(s + 5)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 4)(s + 5))
Kmax =
4.Medium
R(s)+K = 651s(s + 5)(s + 8)C(s)−

The loop in the diagram runs with K = 65 and L(s) = K/(s(s + 5)(s + 8)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 5)(s + 8))K = 65
m =
5.Hard
ReIm−1−2−3−4−5−6−71j1j2

A unity-feedback loop has L(s) = K/(s(s + 6)(s + 3)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K/(s(s + 6)(s + 3))
σa =
6.Hard
ReIm−2−4−6−8−10j2

The root locus of L(s) = K(s + 9)/(s(s + 1)) leaves the real axis between s = 0 and s = −1, circles round and joins the real axis again to the left of the zero. Find the break-away point sb.

L(s) = K(s + 9)/(s(s + 1))
sb =

Answer key

  1. 1. N = 2
  2. 2. b1 = −44
  3. 3. Kmax = 270
  4. 4. m = 8
  5. 5. σa = −3
  6. 6. sb = −0.5147

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