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 − 6s + 27 = 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 − 6s + 27

N =

2.Easy

A closed-loop system has the characteristic equation s³ + 3s² + 8s − 8 = 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 + 3s² + 8s − 8

N =

3.Medium

The plant G(s) = (s + 5)/((s − 2)(s + 2)) 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 + 5)/((s − 2)(s + 2))

K_min =

4.Medium

The loop in the diagram uses positive feedback, with L(s) = 10/(s(s + 3)(s + 4)). How many closed-loop poles N lie in the right half-plane?

L(s) = 10/(s(s + 3)(s + 4))

N =

5.Hard

For L(s) = K/((s + 6)(s² + 4s + 5)), 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 + 6)(s² + 4s + 5))

θ_d = °

6.Hard

The root locus of L(s) = K/((s + 4)(s + 6)) 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 + 4)(s + 6))

K =

Answer key

1. N = 2
2. N = 1
3. K_min = 0.8
4. N = 1
5. θ_d = 75.96 °
6. K = 1

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