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) = 5+K = 42s + 5y(t)−

A proportional controller K = 4 drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 5)K = 4r = 5
y(∞) =
2.Easy
R(s)+4(s + 5)s³(s + 3)(s + 6)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 5)/(s³(s + 3)(s + 6)). What is the system type N?

G(s) = 4(s + 5)/(s³(s + 3)(s + 6))
N =
3.Medium
r(t) = 10t+Ks(s + 2)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 2)) must follow the ramp r(t) = 10t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 2))r(t) = 10tess = 0.2
K =
4.Medium
r(t) = 2·t²/2+20(s + 2)s²(s + 4)C(s)−

The parabolic input r(t) = 2·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 2)/(s²(s + 4)). Find the steady-state error ess.

G(s) = 20(s + 2)/(s²(s + 4))r(t) = 2·t²/2
ess =
5.Hard
R(s)+Kp​ + Ki​/s5s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 4 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 5/(s + 1)ωₙ = 4 rad/sζ = 0.6
Ki =
6.Hard
r(t) = 1t+2 + 4/s10(s + 2)(s + 4)C(s)−

A PI controller C(s) = 2 + 4/s drives the plant G(s) = 10/((s + 2)(s + 4)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 1t.

C(s) = 2 + 4/sG(s) = 10/((s + 2)(s + 4))r(t) = 1t
ess =

Answer key

  1. 1. y(∞) = 3.077
  2. 2. N = 3
  3. 3. K = 100
  4. 4. ess = 0.2
  5. 5. Ki = 3.2
  6. 6. ess = 0.2

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