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) = 20(s + 2)/(s²(s + 1)). Find the acceleration error constant K_a.

G(s) = 20(s + 2)/(s²(s + 1))

K_a =

2.Easy

A proportional controller K = 4 drives the plant G(s) = 1/(s + 5) in a unity-feedback loop. A step of size 2 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 5)K = 4r = 2

y(∞) =

3.Medium

A process G(s) = 1/(1s + 1) is controlled by a proportional gain K = 9 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/(1s + 1)K = 9

e_ss = %

4.Medium

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 5)/(s²(s + 1)). Find the steady-state error e_ss.

G(s) = 20(s + 5)/(s²(s + 1))r(t) = 10·t²/2

e_ss =

5.Hard

A PI controller C(s) = 5 + 4/s drives the plant G(s) = 20/((s + 2)(s + 10)) in a unity-feedback loop. Find the steady-state error e_ss for the ramp input r(t) = 2t.

C(s) = 5 + 4/sG(s) = 20/((s + 2)(s + 10))r(t) = 2t

e_ss =

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 1. Find the proportional gain K_p.

G(s) = 5/(s + 4)ωₙ = 3 rad/sζ = 1

K_p =

Answer key

1. K_a = 40
2. y(∞) = 0.8889
3. e_ss = 10 %
4. e_ss = 0.1
5. e_ss = 0.5
6. K_p = 0.4

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