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

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

y(∞) =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 4)/(s²(s + 4)). Find the acceleration error constant K_a.

G(s) = 5(s + 4)/(s²(s + 4))

K_a =

3.Medium

The input r(t) = 2 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 10(s + 2)/(s(s + 10)). Find the steady-state error e_ss.

G(s) = 10(s + 2)/(s(s + 10))r(t) = 2 + 3t

e_ss =

4.Medium

A process G(s) = 4/(5s + 1) is controlled by a proportional gain K = 2 in a unity-feedback loop. Find the steady-state error e_ss for a step, as a percentage of the step size.

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

e_ss = %

5.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 10/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.7. Find the integral gain K_i.

G(s) = 10/(s + 2)ωₙ = 3 rad/sζ = 0.7

K_i =

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.8. Find the integral gain K_i.

G(s) = 1/(s + 2)ωₙ = 5 rad/sζ = 0.8

K_i =

Answer key

1. y(∞) = 1.667
2. K_a = 5
3. e_ss = 1.5
4. e_ss = 11.11 %
5. K_i = 0.9
6. K_i = 25

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