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
s³110s²521s¹?s⁰

For the characteristic polynomial Q(s) = s³ + 5s² + 10s + 21, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = s³ + 5s² + 10s + 21
b1 =
2.Easy
s³310s²611s¹?s⁰

For the characteristic polynomial Q(s) = 3s³ + 6s² + 10s + 11, find the first-column entry b1 of the s¹ row of the Routh array.

Q(s) = 3s³ + 6s² + 10s + 11
b1 =
3.Medium
R(s)+38401s(s + 6)(s + 10)C(s)−

The loop in the diagram has negative feedback and L(s) = 3840/(s(s + 6)(s + 10)). How many closed-loop poles N lie in the right half-plane?

L(s) = 3840/(s(s + 6)(s + 10))
N =
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−2−4−6−8−10−12j2

The root locus of L(s) = K/(s(s + 3)(s + 10)) leaves the real axis between s = 0 and s = −3. Find the break-away point sb.

L(s) = K/(s(s + 3)(s + 10))
sb =
6.Hard
ReIm−2−4−6−8−10−12j2

The root locus of L(s) = K(s + 10)/(s(s + 3)) leaves the real axis between s = 0 and s = −3, circles round and joins the real axis again to the left of the zero. Find the break-in point si.

L(s) = K(s + 10)/(s(s + 3))
si =

Answer key

  1. 1. b1 = 5.8
  2. 2. b1 = 4.5
  3. 3. N = 2
  4. 4. Kmax = 630
  5. 5. sb = −1.371
  6. 6. si = −18.37

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