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
s⁴16−20s³1−12s²??s¹?s⁰?

The characteristic polynomial of a feedback system is Q(s) = s⁴ + s³ + 6s² − 12s − 20. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + s³ + 6s² − 12s − 20
N =
2.Easy
R(s)+K = 11s + 10C(s)−

A proportional controller with gain K = 1 drives the plant G(s) = 1/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

G(s) = 1/(s + 10)K = 1
p =
3.Medium
R(s)+4861s(s + 3)(s + 6)C(s)−

The loop in the diagram has negative feedback and L(s) = 486/(s(s + 3)(s + 6)). How many closed-loop poles N lie in the right half-plane?

L(s) = 486/(s(s + 3)(s + 6))
N =
4.Medium
R(s)+K1s(s + 1)(s + 10)C(s)−

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 10)). Find the largest gain Kmax for which the closed loop is stable.

L(s) = K/(s(s + 1)(s + 10))
Kmax =
5.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K/(s(s + 6)(s + 10)) leaves the real axis between s = 0 and s = −6. Find the break-away point sb.

L(s) = K/(s(s + 6)(s + 10))
sb =
6.Hard
ReIm−1−2−3−4−5−6−7−81j1j2

For L(s) = K/((s + 7)(s² + 4s + 8)), 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 + 8))
θd = °

Answer key

  1. 1. N = 1
  2. 2. p = −11
  3. 3. N = 2
  4. 4. Kmax = 110
  5. 5. sb = −2.427
  6. 6. θd = 68.2 °

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