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) = 1/(s + 4) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −8.

G(s) = 1/(s + 4)p = −8

K =

2.Easy

A closed-loop system has the characteristic equation s³ + s² + 3s − 4 = 0. Use the Routh–Hurwitz criterion to find the number N of closed-loop poles in the right half of the s-plane.

Q(s) = s³ + s² + 3s − 4

N =

3.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 1)(s + 6)). 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 + 6))

ω = rad/s

4.Medium

The loop in the diagram runs with K = 2 and L(s) = K/(s(s + 1)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 1)(s + 4))K = 2

m =

5.Hard

The root locus of L(s) = K(s + 11)/(s(s + 8)) leaves the real axis between s = 0 and s = −8, circles round and joins the real axis again to the left of the zero. Find the break-in point s_i.

L(s) = K(s + 11)/(s(s + 8))

s_i =

6.Hard

The root locus of L(s) = K/((s + 1)(s + 2)(s + 5)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 1)(s + 2)(s + 5))

ω = rad/s

Answer key

1. K = 4
2. N = 1
3. ω = 2.449 rad/s
4. m = 10
5. s_i = −16.74
6. ω = 4.123 rad/s

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