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 + s² + 13s + 17. Build the Routh array and find how many roots N of Q(s) lie in the right half-plane.

Q(s) = s⁴ + 2s³ + s² + 13s + 17

N =

2.Easy

The characteristic equation of a closed loop is s³ + a·s² + 3s + 18 = 0. Find the value of a above which the system is stable.

s³ + a·s² + 3s + 18 = 0

a =

3.Medium

The plant G(s) = (s + 10)/((s − 3)(s + 5)) has an unstable pole at s = 3. 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 − 3)(s + 5))

K_min =

4.Medium

A unity-feedback loop has L(s) = K/((s + 1)(s + 4)(s + 5)), with K > 0. Find the largest gain K_max for which the closed loop is stable.

L(s) = K/((s + 1)(s + 4)(s + 5))

K_max =

5.Hard

The root locus of L(s) = K(s + 8)/(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-away point s_b.

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

s_b =

6.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

Answer key

1. N = 2
2. a = 6
3. K_min = 1.5
4. K_max = 270
5. s_b = −2.343
6. ω = 5.099 rad/s

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