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)+40(s + 1)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40/((s + 1)(s + 5)). Find the position error constant Kp.

G(s) = 40/((s + 1)(s + 5))
Kp =
2.Easy
r(t) = 2+K = 25s + 4y(t)−

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

G(s) = 5/(s + 4)K = 2r = 2
y(∞) =
3.Medium
r(t) = 2·t²/2+20(s + 2)s²(s + 10)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 + 10)). Find the steady-state error ess.

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

A unity-feedback loop has G(s) = K/((s + 2)(s + 4)). Find the gain K that gives a steady-state error of 4 % for a step input.

G(s) = K/((s + 2)(s + 4))ess = 4 %
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 4/(s + 2)ωₙ = 3 rad/sζ = 0.7
Ki =
6.Hard
R(s)+K = 5010s(s + 8)C(s)−

A proportional controller K = 50 drives the plant G(s) = 10/(s(s + 8)) in a unity-feedback loop. Find the peak time tp of the step response.

G(s) = 10/(s(s + 8))K = 50
tp = s

Answer key

  1. 1. Kp = 8
  2. 2. y(∞) = 1.429
  3. 3. ess = 0.5
  4. 4. K = 192
  5. 5. Ki = 2.25
  6. 6. tp = 0.1428 s

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