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)+10(s + 1)s(s + 5)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s(s + 5)(s + 10)). Find the velocity error constant Kv.

G(s) = 10(s + 1)/(s(s + 5)(s + 10))
Kv =
2.Easy
R(s)+10(s + 5)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 5)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 10(s + 5)/(s²(s + 4))
Ka =
3.Medium
r(t) = 4·t²/2+2(s + 5)s²(s + 1)C(s)−

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 2(s + 5)/(s²(s + 1)). Find the steady-state error ess.

G(s) = 2(s + 5)/(s²(s + 1))r(t) = 4·t²/2
ess =
4.Medium
R(s)+K = 52.510s + 1C(s)−

A process G(s) = 2.5/(10s + 1) is controlled by a proportional gain K = 5 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 2.5/(10s + 1)K = 5
ess = %
5.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 1/(s + 2)ωₙ = 10 rad/sζ = 0.7
Kp =
6.Hard
R(s)+K4s(s + 10)C(s)−

A proportional controller K drives the plant G(s) = 4/(s(s + 10)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 4/(s(s + 10))ζ = 0.707
K =

Answer key

  1. 1. Kv = 0.2
  2. 2. Ka = 12.5
  3. 3. ess = 0.4
  4. 4. ess = 7.407 %
  5. 5. Kp = 12
  6. 6. K = 12.5

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