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 + 4)/(s²(s + 5)). Find the acceleration error constant K_a.

G(s) = 20(s + 4)/(s²(s + 5))

K_a =

2.Easy

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

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

y(∞) =

3.Medium

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

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

e_ss = %

4.Medium

The input r(t) = 10 + 4t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 10(s + 1)/(s(s + 4)). Find the steady-state error e_ss.

G(s) = 10(s + 1)/(s(s + 4))r(t) = 10 + 4t

e_ss =

5.Hard

A proportional controller K = 50 drives the plant G(s) = 5/(s(s + 8)) in a unity-feedback loop. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 8))K = 50

t_s = s

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 1/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 1. Find the integral gain K_i.

G(s) = 1/(s + 5)ωₙ = 8 rad/sζ = 1

K_i =

Answer key

1. K_a = 16
2. y(∞) = 0.9804
3. e_ss = 33.33 %
4. e_ss = 1.6
5. t_s = 1 s
6. K_i = 64

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