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)+20s(s + 4)C(s)−

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

G(s) = 20/(s(s + 4))
N =
2.Easy
R(s)+5(s + 5)s²(s + 4)C(s)−

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

G(s) = 5(s + 5)/(s²(s + 4))
Ka =
3.Medium
r(t) = 1t+50s(s + 1)C(s)−

A ramp r(t) = 1t is applied to a unity-feedback loop with G(s) = 50/(s(s + 1)). Find the steady-state error ess.

G(s) = 50/(s(s + 1))r(t) = 1t
ess =
4.Medium
r(t) = 10 + 1t+2s(s + 2)C(s)−

The input r(t) = 10 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 2/(s(s + 2)). Find the steady-state error ess.

G(s) = 2/(s(s + 2))r(t) = 10 + 1t
ess =
5.Hard
R(s)+K = 252s(s + 2)C(s)−

A proportional controller K = 25 drives the plant G(s) = 2/(s(s + 2)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 2/(s(s + 2))K = 25
ts = s
6.Hard
R(s)+Kp​ + Ki​/s4s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 4/(s + 1)ωₙ = 8 rad/sζ = 0.7
Kp =

Answer key

  1. 1. N = 1
  2. 2. Ka = 6.25
  3. 3. ess = 0.02
  4. 4. ess = 1
  5. 5. ts = 4 s
  6. 6. Kp = 2.55

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