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 + 1)/s drives the plant G(s) = 1/(s + 10) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 2(s + 1)/sG(s) = 1/(s + 10)
N =
2.Easy
r(t) = 10+K = 52s + 2y(t)−

A proportional controller K = 5 drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 2)K = 5r = 10
y(∞) =
3.Medium
r(t) = 0.5·t²/2+20(s + 4)s²(s + 2)C(s)−

The parabolic input r(t) = 0.5·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 4)/(s²(s + 2)). Find the steady-state error ess.

G(s) = 20(s + 4)/(s²(s + 2))r(t) = 0.5·t²/2
ess =
4.Medium
r(t) = 3t+50s(s + 2)(s + 5)C(s)−

A ramp r(t) = 3t is applied to a unity-feedback loop with G(s) = 50/(s(s + 2)(s + 5)). Find the steady-state error ess.

G(s) = 50/(s(s + 2)(s + 5))r(t) = 3t
ess =
5.Hard
R(s)+K = 201s(s + 6)C(s)−

A proportional controller K = 20 drives the plant G(s) = 1/(s(s + 6)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 6))K = 20
ts = s
6.Hard
R(s)+20 + Kd​s1s(s + 2)C(s)−

A PD controller C(s) = 20 + Kd·s drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 1/(s(s + 2))Kp = 20ζ = 0.6
Kd = s

Answer key

  1. 1. N = 1
  2. 2. y(∞) = 8.333
  3. 3. ess = 0.0125
  4. 4. ess = 0.6
  5. 5. ts = 1.333 s
  6. 6. Kd = 3.367 s

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