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 step of size 1 is applied to a unity-feedback loop with G(s) = 2(s + 2)/((s + 5)(s + 10)). Find the steady-state error e_ss.

G(s) = 2(s + 2)/((s + 5)(s + 10))r(t) = 1·u(t)

e_ss =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 10/((s + 5)(s + 10)). Find the position error constant K_p.

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

K_p =

3.Medium

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 2)/(s²(s + 10)). Find the steady-state error e_ss.

G(s) = 10(s + 2)/(s²(s + 10))r(t) = 4·t²/2

e_ss =

4.Medium

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 4)/(s²(s + 4)). Find the steady-state error e_ss.

G(s) = 10(s + 4)/(s²(s + 4))r(t) = 5·t²/2

e_ss =

5.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 0.5. Find the proportional gain K_p.

G(s) = 2/(s + 2)ωₙ = 6 rad/sζ = 0.5

K_p =

6.Hard

A PD controller C(s) = 8 + K_d·s drives the plant G(s) = 5/(s(s + 2)) in a unity-feedback loop. Find the derivative gain K_d that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 5/(s(s + 2))K_p = 8ζ = 0.6

K_d = s

Answer key

1. e_ss = 0.9259
2. K_p = 0.2
3. e_ss = 2
4. e_ss = 0.5
5. K_p = 2
6. K_d = 1.118 s

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