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)+C(s)G(s)C(s)−

The controller C(s) = 5 drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 5G(s) = 5/(s(s + 5))
N =
2.Easy
R(s)+10(s + 1)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 10(s + 1)/(s²(s + 1))
Ka =
3.Medium
r(t) = 1 + 2t+4s(s + 5)(s + 10)C(s)−

The input r(t) = 1 + 2t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 5)(s + 10)). Find the steady-state error ess.

G(s) = 4/(s(s + 5)(s + 10))r(t) = 1 + 2t
ess =
4.Medium
r(t) = 5 + 5t+5(s + 4)s(s + 1)(s + 2)C(s)−

The input r(t) = 5 + 5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5(s + 4)/(s(s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 5(s + 4)/(s(s + 1)(s + 2))r(t) = 5 + 5t
ess =
5.Hard
R(s)+K5s(s + 4)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 5/(s(s + 4))ζ = 0.5
K =
6.Hard
R(s)+K = 255s(s + 5)C(s)−

A proportional controller K = 25 drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 5))K = 25
ts = s

Answer key

  1. 1. N = 1
  2. 2. Ka = 10
  3. 3. ess = 25
  4. 4. ess = 0.5
  5. 5. K = 3.2
  6. 6. ts = 1.6 s

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