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.

New worksheetPractise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________   Date: ____________

1.Easy
r(t) = 10+K = 210s + 10y(t)−

A proportional controller K = 2 drives the plant G(s) = 10/(s + 10) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 10)K = 2r = 10
y(∞) =
2.Easy
R(s)+20(s + 1)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/((s + 1)(s + 4)). Find the position error constant Kp.

G(s) = 20/((s + 1)(s + 4))
Kp =
3.Medium
R(s)+K = 9410s + 1C(s)−

A process G(s) = 4/(10s + 1) is controlled by a proportional gain K = 9 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 4/(10s + 1)K = 9
ess = %
4.Medium
r(t) = 10t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 10t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 10tess = 0.05
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 4/(s + 1)ωₙ = 8 rad/sζ = 0.8
Kp =
6.Hard
R(s)+Kp​ + Ki​/s10s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 10/(s + 5)ωₙ = 5 rad/sζ = 0.8
Ki =

Answer key

  1. 1. y(∞) = 6.667
  2. 2. Kp = 5
  3. 3. ess = 2.703 %
  4. 4. K = 200
  5. 5. Kp = 2.95
  6. 6. Ki = 2.5

Full step-by-step solutions: practise this topic on examproblems.com.