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) = 5+2(s + 1)(s + 10)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 2/((s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 2/((s + 1)(s + 10))r(t) = 5·u(t)
ess =
2.Easy
r(t) = 5+K = 15s + 1y(t)−

A proportional controller K = 1 drives the plant G(s) = 5/(s + 1) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 1)K = 1r = 5
y(∞) =
3.Medium
r(t) = 5t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 5t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 5tess = 0.2
K =
4.Medium
R(s)+K(s + 1)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 10)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 1)(s + 10))ess = 2 %
K =
5.Hard
R(s)+K = 2510s(s + 5)C(s)−

A proportional controller K = 25 drives the plant G(s) = 10/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 10/(s(s + 5))K = 25
ts = s
6.Hard
R(s)+K = 401s(s + 5)C(s)−

A proportional controller K = 40 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 5))K = 40
ts = s

Answer key

  1. 1. ess = 4.167
  2. 2. y(∞) = 4.167
  3. 3. K = 125
  4. 4. K = 490
  5. 5. ts = 1.6 s
  6. 6. ts = 1.6 s

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