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
s⁴1620s³514s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 5s³ + 6s² + 14s + 20. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 5s³ + 6s² + 14s + 20
N =
2.Easy
s³14s²a16s¹(4·a − 16)/as⁰16

The characteristic equation of a closed loop is s³ + a·s² + 4s + 16 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 4s + 16 = 0
a =
3.Medium
R(s)+101s(s + 1)(s + 3)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 10/(s(s + 1)(s + 3)). How many closed-loop poles N lie in the right half-plane?

L(s) = 10/(s(s + 1)(s + 3))
N =
4.Medium
R(s)+K1(s + 1)(s + 3)(s + 5)C(s)−

A unity-feedback loop has L(s) = K/((s + 1)(s + 3)(s + 5)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/((s + 1)(s + 3)(s + 5))
Kmax =
5.Hard
ReIm−0.5−1−1.5−2−2.50.5j0.5j1

The root locus of L(s) = K(s + 2)/(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 + 2)/(s(s + 1))
sb =
6.Hard
ReIm−5j5

The root locus of L(s) = K/((s + 3)(s + 4)(s + 5)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 3)(s + 4)(s + 5))
ω = rad/s

Answer key

  1. 1. N = 2
  2. 2. a = 4
  3. 3. N = 1
  4. 4. Kmax = 192
  5. 5. sb = −0.5858
  6. 6. ω = 6.856 rad/s

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