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
s³34s²616s¹?s⁰

For the characteristic polynomial Q(s) = 3s³ + 6s² + 4s + 16, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 3s³ + 6s² + 4s + 16
b1 =
2.Easy
s⁴18−17s³59s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 5s³ + 8s² + 9s − 17. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 5s³ + 8s² + 9s − 17
N =
3.Medium
R(s)+Ks + 13s(s + 3)(s + 4)C(s)−

A unity-feedback loop has L(s) = K(s + 13)/(s(s + 3)(s + 4)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

L(s) = K(s + 13)/(s(s + 3)(s + 4))
Kmax =
4.Medium
R(s)+601s(s + 1)(s + 4)C(s)−

The loop in the diagram has negative feedback and L(s) = 60/(s(s + 1)(s + 4)). How many closed-loop poles N lie in the right half-plane?

L(s) = 60/(s(s + 1)(s + 4))
N =
5.Hard
ReIm−2−4j2

For L(s) = K/((s + 3)(s² + 4s + 13)), find the angle of departure θd of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 3)(s² + 4s + 13))
θd = °
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.5j0.5j1

The root locus of L(s) = K(s + 3)/(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 + 3)/(s(s + 1))
sb =

Answer key

  1. 1. b1 = −4
  2. 2. N = 1
  3. 3. Kmax = 14
  4. 4. N = 2
  5. 5. θd = 18.43 °
  6. 6. sb = −0.5505

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