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(t) = 1+K = 101s + 2y(t)−

A proportional controller K = 10 drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. A step of size 1 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 2)K = 10r = 1
y(∞) =
2.Easy
R(s)+50(s + 1)(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50/((s + 1)(s + 2)). Find the position error constant Kp.

G(s) = 50/((s + 1)(s + 2))
Kp =
3.Medium
r(t) = 2 + 2t+2s(s + 1)(s + 10)C(s)−

The input r(t) = 2 + 2t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 2/(s(s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 2/(s(s + 1)(s + 10))r(t) = 2 + 2t
ess =
4.Medium
r(t) = 2 + 0.5t+5(s + 3)s(s + 4)(s + 5)C(s)−

The input r(t) = 2 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5(s + 3)/(s(s + 4)(s + 5)). Find the steady-state error ess.

G(s) = 5(s + 3)/(s(s + 4)(s + 5))r(t) = 2 + 0.5t
ess =
5.Hard
R(s)+K2s(s + 10)C(s)−

A proportional controller K drives the plant G(s) = 2/(s(s + 10)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 2/(s(s + 10))ζ = 0.5
K =
6.Hard
R(s)+K5s(s + 4)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 5/(s(s + 4))ζ = 0.5
K =

Answer key

  1. 1. y(∞) = 0.8333
  2. 2. Kp = 25
  3. 3. ess = 10
  4. 4. ess = 0.6667
  5. 5. K = 50
  6. 6. K = 3.2

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