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) = 2+K = 110s + 1y(t)−

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

G(s) = 10/(s + 1)K = 1r = 2
y(∞) =
2.Easy
r(t) = 5+K = 21s + 4y(t)−

A proportional controller K = 2 drives the plant G(s) = 1/(s + 4) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

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

The input r(t) = 10 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5/(s(s + 4)(s + 5)). Find the steady-state error ess.

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

The input r(t) = 5 + 5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 4/(s(s + 1)(s + 10))r(t) = 5 + 5t
ess =
5.Hard
R(s)+Kp​ + Ki​/s10s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 10/(s + 5)ωₙ = 6 rad/sζ = 0.7
Kp =
6.Hard
R(s)+K = 1010s(s + 8)C(s)−

A proportional controller K = 10 drives the plant G(s) = 10/(s(s + 8)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 10/(s(s + 8))K = 10
ts = s

Answer key

  1. 1. y(∞) = 1.818
  2. 2. y(∞) = 1.667
  3. 3. ess = 12
  4. 4. ess = 12.5
  5. 5. Kp = 0.34
  6. 6. ts = 1 s

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