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
R(s)+20s³(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s³(s + 4)). What is the system type N?

G(s) = 20/(s³(s + 4))
N =
2.Easy
r(t) = 1+K = 42s + 1y(t)−

A proportional controller K = 4 drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. A step of size 1 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 1)K = 4r = 1
y(∞) =
3.Medium
r(t) = 2t+4(s + 3)s(s + 2)(s + 4)C(s)−

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

G(s) = 4(s + 3)/(s(s + 2)(s + 4))r(t) = 2t
ess =
4.Medium
R(s)+K = 200.510s + 1C(s)−

A process G(s) = 0.5/(10s + 1) is controlled by a proportional gain K = 20 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 0.5/(10s + 1)K = 20
ess = %
5.Hard
R(s)+K5s(s + 8)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 8)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.4.

G(s) = 5/(s(s + 8))ζ = 0.4
K =
6.Hard
R(s)+Kp​ + Ki​/s10s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 10/(s + 4)ωₙ = 10 rad/sζ = 0.6
Ki =

Answer key

  1. 1. N = 3
  2. 2. y(∞) = 0.8889
  3. 3. ess = 1.333
  4. 4. ess = 9.091 %
  5. 5. K = 20
  6. 6. Ki = 10

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