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 controller with gain K = 10 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. Find the closed-loop pole p.

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

p =

2.Easy

The characteristic equation of a closed loop is s³ + a·s² + 6s + 36 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 6s + 36 = 0

a =

3.Medium

The loop in the diagram runs with K = 10.5 and L(s) = K/(s(s + 3)(s + 4)). By what factor m can the gain be raised before the closed loop becomes unstable?

L(s) = K/(s(s + 3)(s + 4))K = 10.5

m =

4.Medium

The plant G(s) = (s + 5)/((s − 1)(s + 1)) has an unstable pole at s = 1. 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 + 5)/((s − 1)(s + 1))

K_min =

5.Hard

The root locus of L(s) = K(s + 4)/(s(s + 2)) leaves the real axis between s = 0 and s = −2, 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 + 4)/(s(s + 2))

s_i =

6.Hard

A unity-feedback loop has L(s) = K/(s(s + 1)(s + 4)(s + 8)). Find the point σ_a where the asymptotes of the root locus cross the real axis.

L(s) = K/(s(s + 1)(s + 4)(s + 8))

σ_a =

Answer key

1. p = −60
2. a = 6
3. m = 8
4. K_min = 0.2
5. s_i = −6.828
6. σ_a = −3.25

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