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)+5(s + 1)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/((s + 1)(s + 10)). Find the position error constant Kp.

G(s) = 5/((s + 1)(s + 10))
Kp =
2.Easy
r(t) = 2+20(s + 1)(s + 10)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 20/((s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 20/((s + 1)(s + 10))r(t) = 2·u(t)
ess =
3.Medium
r(t) = 1t+Ks(s + 4)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 4)) must follow the ramp r(t) = 1t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 4))r(t) = 1tess = 0.05
K =
4.Medium
R(s)+K = 22.50.5s + 1C(s)−

A process G(s) = 2.5/(0.5s + 1) is controlled by a proportional gain K = 2 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 2.5/(0.5s + 1)K = 2
ess = %
5.Hard
R(s)+Kp​ + Ki​/s2s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 2/(s + 1)ωₙ = 8 rad/sζ = 0.7
Ki =
6.Hard
R(s)+2 + Kd​s2s(s + 1)C(s)−

A PD controller C(s) = 2 + Kd·s drives the plant G(s) = 2/(s(s + 1)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 2/(s(s + 1))Kp = 2ζ = 0.707
Kd = s

Answer key

  1. 1. Kp = 0.5
  2. 2. ess = 0.6667
  3. 3. K = 80
  4. 4. ess = 16.67 %
  5. 5. Ki = 32
  6. 6. Kd = 0.914 s

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