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 + 4s³ + 10s² + 8s − 14. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 4s³ + 10s² + 8s − 14

N =

2.Easy

A closed-loop system has the characteristic equation s³ + 6s² + 4s + 34 = 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 + 4s + 34

N =

3.Medium

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

K_min =

4.Medium

The plant G(s) = (s + 10)/((s − 4)(s + 1)) has an unstable pole at s = 4. 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 + 10)/((s − 4)(s + 1))

K_min =

5.Hard

The root locus of L(s) = K(s + 9)/(s(s + 4)) leaves the real axis between s = 0 and s = −4, 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 + 9)/(s(s + 4))

s_i =

6.Hard

The root locus of L(s) = K/(s(s + 1)(s + 8)) leaves the real axis between s = 0 and s = −1. Find the break-away point s_b.

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

s_b =

Answer key

1. N = 1
2. N = 2
3. K_min = 5
4. K_min = 3
5. s_i = −15.71
6. s_b = −0.4834

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