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)+20(s + 2)s³(s + 4)(s + 8)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 2)/(s³(s + 4)(s + 8)). What is the system type N?

G(s) = 20(s + 2)/(s³(s + 4)(s + 8))
N =
2.Easy
R(s)+50(s + 2)s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50(s + 2)/(s(s + 10)). Find the velocity error constant Kv.

G(s) = 50(s + 2)/(s(s + 10))
Kv =
3.Medium
R(s)+K(s + 5)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 5)(s + 10)). Find the gain K that gives a steady-state error of 25 % for a step input.

G(s) = K/((s + 5)(s + 10))ess = 25 %
K =
4.Medium
R(s)+K(s + 5)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 5)(s + 10)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 5)(s + 10))ess = 2 %
K =
5.Hard
R(s)+Kp​ + Ki​/s2s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 2/(s + 2)ωₙ = 5 rad/sζ = 1
Ki =
6.Hard
R(s)+K = 101s(s + 5)C(s)−

A proportional controller K = 10 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 5))K = 10
ts = s

Answer key

  1. 1. N = 3
  2. 2. Kv = 10
  3. 3. K = 150
  4. 4. K = 2450
  5. 5. Ki = 12.5
  6. 6. ts = 1.6 s

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