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

For the characteristic polynomial Q(s) = s³ + 4s² + 7s + 21, find the first-column entry b₁ of the s¹ row of the Routh array.

Q(s) = s³ + 4s² + 7s + 21

b₁ =

2.Easy

A unity-feedback loop has the open-loop transfer function L(s) = 13/(s(s + 6)). The closed-loop poles are complex. Find their imaginary part ω_d.

L(s) = 13/(s(s + 6))

ω_d = rad/s

3.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 5)(s + 10)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 5)(s + 10))

ω = rad/s

4.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 10)). The gain is raised until the closed loop is on the edge of stability. Find the frequency ω of the sustained oscillation.

L(s) = K/(s(s + 1)(s + 10))

ω = rad/s

5.Hard

For L(s) = K/((s + 4)(s² + 2s + 10)), find the angle of departure θ_d of the root locus from the complex pole in the upper half-plane, measured from the positive real axis.

L(s) = K/((s + 4)(s² + 2s + 10))

θ_d = °

6.Hard

The root locus of L(s) = K/((s + 4)(s + 6)) is shown. Find the gain K at the break-away point s_b, where the two closed-loop poles meet on the real axis.

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

K =

Answer key

1. b₁ = 1.75
2. ω_d = 2 rad/s
3. ω = 7.071 rad/s
4. ω = 3.162 rad/s
5. θ_d = 45 °
6. K = 1

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