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)+50(s + 2)s(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50(s + 2)/(s(s + 4)). Find the velocity error constant Kv.

G(s) = 50(s + 2)/(s(s + 4))
Kv =
2.Easy
R(s)+20s²(s + 4)(s + 6)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s²(s + 4)(s + 6)). What is the system type N?

G(s) = 20/(s²(s + 4)(s + 6))
N =
3.Medium
r(t) = 2t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 2t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 2tess = 0.01
K =
4.Medium
R(s)+K = 10110s + 1C(s)−

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

G(s) = 1/(10s + 1)K = 10
ess = %
5.Hard
R(s)+K = 202s(s + 6)C(s)−

A proportional controller K = 20 drives the plant G(s) = 2/(s(s + 6)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 2/(s(s + 6))K = 20
%OS = %
6.Hard
R(s)+8 + Kd​s4s(s + 4)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 4/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.9.

G(s) = 4/(s(s + 4))Kp = 8ζ = 0.9
Kd = s

Answer key

  1. 1. Kv = 25
  2. 2. N = 2
  3. 3. K = 200
  4. 4. ess = 9.091 %
  5. 5. %OS = 18.4 %
  6. 6. Kd = 1.546 s

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