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) = 2(s + 3)/s drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 2(s + 3)/sG(s) = 1/(s(s + 2))
N =
2.Easy
R(s)+5(s + 3)s²(s + 2)C(s)−

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

G(s) = 5(s + 3)/(s²(s + 2))
Ka =
3.Medium
r(t) = 0.5·t²/2+5(s + 5)s²(s + 2)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 5)/(s²(s + 2)). Find the steady-state error ess.

G(s) = 5(s + 5)/(s²(s + 2))r(t) = 0.5·t²/2
ess =
4.Medium
r(t) = 5t+Ks(s + 10)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 5t with a steady-state error of 0.05. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 5tess = 0.05
K =
5.Hard
R(s)+25 + Kd​s5s(s + 3)C(s)−

A PD controller C(s) = 25 + Kd·s drives the plant G(s) = 5/(s(s + 3)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.9.

G(s) = 5/(s(s + 3))Kp = 25ζ = 0.9
Kd = s
6.Hard
R(s)+K5s(s + 8)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 8)) 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 + 8))ζ = 0.6
K =

Answer key

  1. 1. N = 2
  2. 2. Ka = 7.5
  3. 3. ess = 0.04
  4. 4. K = 1000
  5. 5. Kd = 3.425 s
  6. 6. K = 8.889

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