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)+2(s + 3)s²(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 3)/(s²(s + 5)). Find the acceleration error constant Ka.

G(s) = 2(s + 3)/(s²(s + 5))
Ka =
2.Easy
r(t) = 2+K = 410s + 10y(t)−

A proportional controller K = 4 drives the plant G(s) = 10/(s + 10) in a unity-feedback loop. A step of size 2 is applied. Find the final value of the output y(∞).

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

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

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

A ramp r(t) = 3t is applied to a unity-feedback loop with G(s) = 4(s + 2)/(s(s + 1)). Find the steady-state error ess.

G(s) = 4(s + 2)/(s(s + 1))r(t) = 3t
ess =
5.Hard
R(s)+K5s(s + 5)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 5)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 5/(s(s + 5))ζ = 0.6
K =
6.Hard
R(s)+Kp​ + Ki​/s1s + 1C(s)−

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

G(s) = 1/(s + 1)ωₙ = 5 rad/sζ = 0.6
Ki =

Answer key

  1. 1. Ka = 1.2
  2. 2. y(∞) = 1.6
  3. 3. ess = 0.625
  4. 4. ess = 0.375
  5. 5. K = 3.472
  6. 6. Ki = 25

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