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)+40(s + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40/((s + 1)(s + 2)). Find the position error constant Kp.

G(s) = 40/((s + 1)(s + 2))
Kp =
2.Easy
r(t) = 1+5(s + 1)(s + 4)(s + 10)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 5(s + 1)/((s + 4)(s + 10)). Find the steady-state error ess.

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

A unity-feedback loop has G(s) = K/((s + 4)(s + 5)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 4)(s + 5))ess = 20 %
K =
4.Medium
R(s)+K = 225s + 1C(s)−

A process G(s) = 2/(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/(5s + 1)K = 2
ess = %
5.Hard
R(s)+25 + Kd​s2s(s + 4)C(s)−

A PD controller C(s) = 25 + Kd·s drives the plant G(s) = 2/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 2/(s(s + 4))Kp = 25ζ = 0.8
Kd = s
6.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 1/(s + 2)ωₙ = 6 rad/sζ = 0.6
Kp =

Answer key

  1. 1. Kp = 20
  2. 2. ess = 0.8889
  3. 3. K = 80
  4. 4. ess = 20 %
  5. 5. Kd = 3.657 s
  6. 6. Kp = 5.2

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