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 closed-loop system has the characteristic equation s³ + 6s² − 3s + 11 = 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 + 6s² − 3s + 11

N =

2.Easy

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + 5s² + 7s + 10. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + 5s² + 7s + 10

N =

3.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 3)). 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 + 2)(s + 3))

ω = rad/s

4.Medium

The plant G(s) = (s + 8)/((s − 2)(s + 5)) has an unstable pole at s = 2. A proportional gain K closes a unity-feedback loop around it. Find the smallest gain K_min that makes the closed loop stable.

G(s) = (s + 8)/((s − 2)(s + 5))

K_min =

5.Hard

The root locus of L(s) = K/((s + 2)(s + 3)(s + 4)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 2)(s + 3)(s + 4))

ω = rad/s

6.Hard

For the loop L(s) = K/(s(s + 3)(s + 6)), find the gain K that puts a closed-loop pole at s = −1.5.

L(s) = K/(s(s + 3)(s + 6))s = −1.5

K =

Answer key

1. N = 2
2. N = 2
3. ω = 2.449 rad/s
4. K_min = 1.25
5. ω = 5.099 rad/s
6. K = 10.13

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