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 proportional controller K = 1 drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. A step of size 1 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 2)K = 1r = 1

y(∞) =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 4/((s + 1)(s + 5)). Find the position error constant K_p.

G(s) = 4/((s + 1)(s + 5))

K_p =

3.Medium

The input r(t) = 2 + 4t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 4)/(s(s + 1)(s + 2)). Find the steady-state error e_ss.

G(s) = 50(s + 4)/(s(s + 1)(s + 2))r(t) = 2 + 4t

e_ss =

4.Medium

A process G(s) = 1/(2s + 1) is controlled by a proportional gain K = 5 in a unity-feedback loop. Find the steady-state error e_ss for a step, as a percentage of the step size.

G(s) = 1/(2s + 1)K = 5

e_ss = %

5.Hard

A PI controller C(s) = 2 + 1/s drives the plant G(s) = 2/((s + 1)(s + 10)) in a unity-feedback loop. Find the steady-state error e_ss for the ramp input r(t) = 1t.

C(s) = 2 + 1/sG(s) = 2/((s + 1)(s + 10))r(t) = 1t

e_ss =

6.Hard

A proportional controller K = 20 drives the plant G(s) = 10/(s(s + 5)) in a unity-feedback loop. Find the peak time t_p of the step response.

G(s) = 10/(s(s + 5))K = 20

t_p = s

Answer key

1. y(∞) = 0.3333
2. K_p = 0.8
3. e_ss = 0.04
4. e_ss = 16.67 %
5. e_ss = 5
6. t_p = 0.2257 s

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