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) = 4(s + 4)/(s²(s + 2)). Find the acceleration error constant K_a.

G(s) = 4(s + 4)/(s²(s + 2))

K_a =

2.Easy

A proportional controller K = 2 drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 1)K = 2r = 10

y(∞) =

3.Medium

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 1)/(s²(s + 2)). Find the steady-state error e_ss.

G(s) = 20(s + 1)/(s²(s + 2))r(t) = 4·t²/2

e_ss =

4.Medium

The parabolic input r(t) = 1·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 2)/(s²(s + 2)). Find the steady-state error e_ss.

G(s) = 4(s + 2)/(s²(s + 2))r(t) = 1·t²/2

e_ss =

5.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 4/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 4 rad/s and ζ = 0.7. Find the proportional gain K_p.

G(s) = 4/(s + 1)ωₙ = 4 rad/sζ = 0.7

K_p =

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 4/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.8. Find the proportional gain K_p.

G(s) = 4/(s + 1)ωₙ = 8 rad/sζ = 0.8

K_p =

Answer key

1. K_a = 8
2. y(∞) = 9.524
3. e_ss = 0.4
4. e_ss = 0.25
5. K_p = 1.15
6. K_p = 2.95

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