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 + 2)s³(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 2)/(s³(s + 1)). What is the system type N?

G(s) = 5(s + 2)/(s³(s + 1))
N =
2.Easy
r(t) = 2+50(s + 5)(s + 10)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 50/((s + 5)(s + 10)). Find the steady-state error ess.

G(s) = 50/((s + 5)(s + 10))r(t) = 2·u(t)
ess =
3.Medium
R(s)+K = 192.52s + 1C(s)−

A process G(s) = 2.5/(2s + 1) is controlled by a proportional gain K = 19 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/(2s + 1)K = 19
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 25 % for a step input.

G(s) = K/((s + 1)(s + 2))ess = 25 %
K =
5.Hard
r(t) = 2t+1 + 4/s20(s + 1)(s + 2)C(s)−

A PI controller C(s) = 1 + 4/s drives the plant G(s) = 20/((s + 1)(s + 2)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 1 + 4/sG(s) = 20/((s + 1)(s + 2))r(t) = 2t
ess =
6.Hard
R(s)+20 + Kd​s10s(s + 3)C(s)−

A PD controller C(s) = 20 + Kd·s drives the plant G(s) = 10/(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) = 10/(s(s + 3))Kp = 20ζ = 0.5
Kd = s

Answer key

  1. 1. N = 3
  2. 2. ess = 1
  3. 3. ess = 2.062 %
  4. 4. K = 6
  5. 5. ess = 0.05
  6. 6. Kd = 1.114 s

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