Steady-state error & PID control worksheet with answers

Free printable Steady-state error & PID control 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 unity-feedback loop has the open-loop transfer function G(s) = 5/((s + 1)(s + 10)). Find the position error constant K_p.

G(s) = 5/((s + 1)(s + 10))

K_p =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 50/(s(s + 2)(s + 4)). Find the velocity error constant K_v.

G(s) = 50/(s(s + 2)(s + 4))

K_v =

3.Medium

The input r(t) = 5 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4(s + 3)/(s(s + 5)). Find the steady-state error e_ss.

G(s) = 4(s + 3)/(s(s + 5))r(t) = 5 + 3t

e_ss =

4.Medium

A process G(s) = 2.5/(0.5s + 1) is controlled by a proportional gain K = 20 in a unity-feedback loop. Find the steady-state error e_ss for a step, as a percentage of the step size.

G(s) = 2.5/(0.5s + 1)K = 20

e_ss = %

5.Hard

A PI controller C(s) = 5 + 5/s drives the plant G(s) = 10/((s + 2)(s + 5)) in a unity-feedback loop. Find the steady-state error e_ss for the ramp input r(t) = 2t.

C(s) = 5 + 5/sG(s) = 10/((s + 2)(s + 5))r(t) = 2t

e_ss =

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 5/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.7. Find the proportional gain K_p.

G(s) = 5/(s + 5)ωₙ = 5 rad/sζ = 0.7

K_p =

Answer key

1. K_p = 0.5
2. K_v = 6.25
3. e_ss = 1.25
4. e_ss = 1.961 %
5. e_ss = 0.4
6. K_p = 0.4

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