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 = 2 drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 4)K = 2r = 10

y(∞) =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s(s + 5)(s + 10)). Find the velocity error constant K_v.

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

K_v =

3.Medium

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 3)/(s²(s + 1)). Find the steady-state error e_ss.

G(s) = 5(s + 3)/(s²(s + 1))r(t) = 0.5·t²/2

e_ss =

4.Medium

A ramp r(t) = 3t is applied to a unity-feedback loop with G(s) = 20(s + 2)/(s(s + 4)(s + 5)). Find the steady-state error e_ss.

G(s) = 20(s + 2)/(s(s + 4)(s + 5))r(t) = 3t

e_ss =

5.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 1/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 1. Find the proportional gain K_p.

G(s) = 1/(s + 4)ωₙ = 6 rad/sζ = 1

K_p =

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.6. Find the integral gain K_i.

G(s) = 2/(s + 1)ωₙ = 5 rad/sζ = 0.6

K_i =

Answer key

1. y(∞) = 7.143
2. K_v = 0.4
3. e_ss = 0.03333
4. e_ss = 1.5
5. K_p = 8
6. K_i = 12.5

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