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(t) = 2+20(s + 1)(s + 4)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 20/((s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 20/((s + 1)(s + 4))r(t) = 2·u(t)
ess =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 5(s + 3)/s drives the plant G(s) = 1/(s(s + 3)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 5(s + 3)/sG(s) = 1/(s(s + 3))
N =
3.Medium
r(t) = 10 + 1t+50(s + 2)s(s + 4)(s + 5)C(s)−

The input r(t) = 10 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50(s + 2)/(s(s + 4)(s + 5)). Find the steady-state error ess.

G(s) = 50(s + 2)/(s(s + 4)(s + 5))r(t) = 10 + 1t
ess =
4.Medium
r(t) = 0.5·t²/2+4(s + 3)s²(s + 5)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 4(s + 3)/(s²(s + 5)). Find the steady-state error ess.

G(s) = 4(s + 3)/(s²(s + 5))r(t) = 0.5·t²/2
ess =
5.Hard
R(s)+Kp​ + Ki​/s5s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.5. Find the proportional gain Kp.

G(s) = 5/(s + 5)ωₙ = 8 rad/sζ = 0.5
Kp =
6.Hard
R(s)+20 + Kd​s2s(s + 4)C(s)−

A PD controller C(s) = 20 + Kd·s drives the plant G(s) = 2/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 2/(s(s + 4))Kp = 20ζ = 0.5
Kd = s

Answer key

  1. 1. ess = 0.3333
  2. 2. N = 2
  3. 3. ess = 0.2
  4. 4. ess = 0.2083
  5. 5. Kp = 0.6
  6. 6. Kd = 1.162 s

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