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)+10(s + 2)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 2)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 10(s + 2)/(s²(s + 1))
Ka =
2.Easy
R(s)+20(s + 2)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/((s + 2)(s + 10)). Find the position error constant Kp.

G(s) = 20/((s + 2)(s + 10))
Kp =
3.Medium
r(t) = 1t+50(s + 2)s(s + 1)C(s)−

A ramp r(t) = 1t is applied to a unity-feedback loop with G(s) = 50(s + 2)/(s(s + 1)). Find the steady-state error ess.

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

A unity-feedback loop has G(s) = K/((s + 1)(s + 2)). Find the gain K that gives a steady-state error of 5 % for a step input.

G(s) = K/((s + 1)(s + 2))ess = 5 %
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 4/(s + 4)ωₙ = 8 rad/sζ = 0.6
Kp =
6.Hard
R(s)+10 + Kd​s2s(s + 3)C(s)−

A PD controller C(s) = 10 + Kd·s drives the plant G(s) = 2/(s(s + 3)) 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 + 3))Kp = 10ζ = 0.5
Kd = s

Answer key

  1. 1. Ka = 20
  2. 2. Kp = 1
  3. 3. ess = 0.01
  4. 4. K = 38
  5. 5. Kp = 1.4
  6. 6. Kd = 0.7361 s

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