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+K = 35s + 4y(t)−

A proportional controller K = 3 drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 4)K = 3r = 5
y(∞) =
2.Easy
R(s)+4(s + 2)(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4/((s + 2)(s + 4)). Find the position error constant Kp.

G(s) = 4/((s + 2)(s + 4))
Kp =
3.Medium
r(t) = 10t+Ks(s + 10)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 10t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 10tess = 0.01
K =
4.Medium
r(t) = 10·t²/2+5(s + 5)s²(s + 5)C(s)−

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 5)/(s²(s + 5)). Find the steady-state error ess.

G(s) = 5(s + 5)/(s²(s + 5))r(t) = 10·t²/2
ess =
5.Hard
R(s)+K = 252s(s + 10)C(s)−

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

G(s) = 2/(s(s + 10))K = 25
ts = s
6.Hard
R(s)+Kp​ + Ki​/s4s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 0.8. Find the proportional gain Kp.

G(s) = 4/(s + 5)ωₙ = 5 rad/sζ = 0.8
Kp =

Answer key

  1. 1. y(∞) = 3.947
  2. 2. Kp = 0.5
  3. 3. K = 10000
  4. 4. ess = 2
  5. 5. ts = 0.8 s
  6. 6. Kp = 0.75

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