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)+4(s + 1)s(s + 4)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 1)/(s(s + 4)(s + 10)). Find the velocity error constant Kv.

G(s) = 4(s + 1)/(s(s + 4)(s + 10))
Kv =
3.Medium
r(t) = 4t+4(s + 3)s(s + 1)(s + 5)C(s)−

A ramp r(t) = 4t is applied to a unity-feedback loop with G(s) = 4(s + 3)/(s(s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 4(s + 3)/(s(s + 1)(s + 5))r(t) = 4t
ess =
4.Medium
R(s)+K(s + 2)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 10)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 2)(s + 10))ess = 20 %
K =
5.Hard
R(s)+K = 810s(s + 6)C(s)−

A proportional controller K = 8 drives the plant G(s) = 10/(s(s + 6)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 10/(s(s + 6))K = 8
tp = s
6.Hard
R(s)+10 + Kd​s2s(s + 2)C(s)−

A PD controller C(s) = 10 + Kd·s drives the plant G(s) = 2/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 2/(s(s + 2))Kp = 10ζ = 0.6
Kd = s

Answer key

  1. 1. Ka = 20
  2. 2. Kv = 0.1
  3. 3. ess = 1.667
  4. 4. K = 80
  5. 5. tp = 0.3728 s
  6. 6. Kd = 1.683 s

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