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

The characteristic polynomial of a feedback system is Q(s) = s⁴ + 2s³ + 3s² + 8s + 4. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + 3s² + 8s + 4

N =

2.Easy

A proportional gain K controls the plant G(s) = 5/(s + 4) in a unity-feedback loop. Find the gain K that places the closed-loop pole at s = −6.5.

G(s) = 5/(s + 4)p = −6.5

K =

3.Medium

The loop in the diagram has the open-loop transfer function L(s) = K/(s(s + 2)(s + 4)). 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 + 4))

ω = rad/s

4.Medium

A unity-feedback loop has L(s) = K/((s + 2)(s + 5)(s + 6)), with K > 0. Find the largest gain K_max for which the closed loop is stable.

L(s) = K/((s + 2)(s + 5)(s + 6))

K_max =

5.Hard

The root locus of L(s) = K/((s + 3)(s + 4)(s + 6)) is sketched. Find the frequency ω at which the locus crosses the imaginary axis.

L(s) = K/((s + 3)(s + 4)(s + 6))

ω = rad/s

6.Hard

For L(s) = K/((s + 8)(s² + 6s + 13)), 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 + 8)(s² + 6s + 13))

θ_d = °

Answer key

1. N = 2
2. K = 0.5
3. ω = 2.828 rad/s
4. K_max = 616
5. ω = 7.348 rad/s
6. θ_d = 68.2 °

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