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)+K = 51s + 2C(s)−

A proportional controller with gain K = 5 drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 2)K = 5
p =
2.Easy
s³14s²a6s¹(4·a − 6)/as⁰6

The characteristic equation of a closed loop is s³ + a·s² + 4s + 6 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 4s + 6 = 0
a =
3.Medium
R(s)+K = 101s(s + 2)(s + 3)C(s)−

The loop in the diagram runs with K = 10 and L(s) = K/(s(s + 2)(s + 3)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 2)(s + 3))K = 10
m =
4.Medium
R(s)+451s(s + 2)(s + 3)C(s)−

The loop in the diagram has negative feedback and L(s) = 45/(s(s + 2)(s + 3)). How many closed-loop poles N lie in the right half-plane?

L(s) = 45/(s(s + 2)(s + 3))
N =
5.Hard
ReIm−1−2−3−4−5−6−71j1j2

A unity-feedback loop has L(s) = K(s + 2)/(s(s + 6)(s + 4)(s + 5)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 2)/(s(s + 6)(s + 4)(s + 5))
σa =
6.Hard
ReIm−2−4−6j2j4

For L(s) = K/((s + 5)(s² + 4s + 20)), 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 + 5)(s² + 4s + 20))
θd = °

Answer key

  1. 1. p = −7
  2. 2. a = 1.5
  3. 3. m = 3
  4. 4. N = 2
  5. 5. σa = −4.333
  6. 6. θd = 36.87 °

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