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
s³110s²a30s¹(10·a − 30)/as⁰30

The characteristic equation of a closed loop is s³ + a·s² + 10s + 30 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 10s + 30 = 0
a =
2.Easy
s³310s²52s¹?s⁰

For the characteristic polynomial Q(s) = 3s³ + 5s² + 10s + 2, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 3s³ + 5s² + 10s + 2
b1 =
3.Medium
R(s)+Ks + 1(s − 3)(s + 1)C(s)−

The plant G(s) = (s + 1)/((s − 3)(s + 1)) has an unstable pole at s = 3. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain Kmin that makes the closed loop stable.

G(s) = (s + 1)/((s − 3)(s + 1))
Kmin =
4.Medium
R(s)+51s(s + 6)(s + 8)C(s)+

The loop in the diagram uses positive feedback, with L(s) = 5/(s(s + 6)(s + 8)). How many closed-loop poles N lie in the right half-plane?

L(s) = 5/(s(s + 6)(s + 8))
N =
5.Hard
ReIm−2−4−6−8−10−12−14j2j4sb​

The root locus of L(s) = K/((s + 4)(s + 12)) is shown. Find the gain K at the break-away point sb, where the two closed-loop poles meet on the real axis.

L(s) = K/((s + 4)(s + 12))
K =
6.Hard
ReIm−0.5−1−1.5−2−2.5−3−3.5−40.5j0.5j1

A unity-feedback loop has L(s) = K(s + 2)/(s(s + 1)(s + 3)). Find the point σa where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 2)/(s(s + 1)(s + 3))
σa =

Answer key

  1. 1. a = 3
  2. 2. b1 = 8.8
  3. 3. Kmin = 3
  4. 4. N = 1
  5. 5. K = 16
  6. 6. σa = −1

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