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
R(s)+K2s + 2C(s)−

A proportional gain K controls the plant G(s) = 2/(s + 2) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −4.

G(s) = 2/(s + 2)p = −4
K =
2.Easy
s³12s²a10s¹(2·a − 10)/as⁰10

The characteristic equation of a closed loop is s³ + a·s² + 2s + 10 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 2s + 10 = 0
a =
3.Medium
R(s)+K1s(s + 1)(s + 8)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 8)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 8))
Kmax =
4.Medium
R(s)+K1(s + 3)(s + 4)(s + 6)C(s)−

A unity-feedback loop has L(s) = K/((s + 3)(s + 4)(s + 6)), with K > 0. Find the largest gain Kmax for which the closed loop is stable.

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

For L(s) = K/((s + 7)(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 + 7)(s² + 4s + 13))
θd = °
6.Hard
ReIm−1−2−3−4−5−6−71j1j2

The root locus of L(s) = K/(s(s + 4)(s + 6)) leaves the real axis between s = 0 and s = −4. Find the break-away point sb.

L(s) = K/(s(s + 4)(s + 6))
sb =

Answer key

  1. 1. K = 1
  2. 2. a = 5
  3. 3. Kmax = 72
  4. 4. Kmax = 630
  5. 5. θd = 59.04 °
  6. 6. sb = −1.569

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