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(s)+4(s + 4)s²(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 4)/(s²(s + 2)). Find the acceleration error constant Ka.

G(s) = 4(s + 4)/(s²(s + 2))
Ka =
2.Easy
r(t) = 5+K = 25s + 10y(t)−

A proportional controller K = 2 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 10)K = 2r = 5
y(∞) =
3.Medium
r(t) = 1 + 5t+5s(s + 4)(s + 5)C(s)−

The input r(t) = 1 + 5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5/(s(s + 4)(s + 5)). Find the steady-state error ess.

G(s) = 5/(s(s + 4)(s + 5))r(t) = 1 + 5t
ess =
4.Medium
r(t) = 5t+5(s + 4)s(s + 1)(s + 4)C(s)−

A ramp r(t) = 5t is applied to a unity-feedback loop with G(s) = 5(s + 4)/(s(s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 5(s + 4)/(s(s + 1)(s + 4))r(t) = 5t
ess =
5.Hard
R(s)+K = 251s(s + 6)C(s)−

A proportional controller K = 25 drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 1/(s(s + 6))K = 25
tp = s
6.Hard
R(s)+4 + Kd​s4s(s + 3)C(s)−

A PD controller C(s) = 4 + Kd·s drives the plant G(s) = 4/(s(s + 3)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 4/(s(s + 3))Kp = 4ζ = 0.707
Kd = s

Answer key

  1. 1. Ka = 8
  2. 2. y(∞) = 2.5
  3. 3. ess = 20
  4. 4. ess = 1
  5. 5. tp = 0.7854 s
  6. 6. Kd = 0.664 s

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