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

A proportional gain K controls the plant G(s) = 10/(s + 4) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −19.

G(s) = 10/(s + 4)p = −19

K =

2.Easy

A proportional gain K controls the plant G(s) = 10/(s + 10) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −60.

G(s) = 10/(s + 10)p = −60

K =

3.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 3)(s + 8)). 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 + 3)(s + 8))

ω = rad/s

4.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 6)(s + 8)). 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 + 6)(s + 8))

ω = rad/s

5.Hard

A unity-feedback loop has L(s) = K(s + 6)/((s + 10)(s + 2)(s + 3)(s + 1)). Find the point σ_a where the asymptotes of the root locus cross the real axis.

L(s) = K(s + 6)/((s + 10)(s + 2)(s + 3)(s + 1))

σ_a =

6.Hard

The root locus of L(s) = K/((s + 6)(s + 10)) 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 + 6)(s + 10))

K =

Answer key

1. K = 1.5
2. K = 5
3. ω = 4.899 rad/s
4. ω = 6.928 rad/s
5. σ_a = −3.333
6. K = 4

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