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+4(s + 2)(s + 4)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 4/((s + 2)(s + 4)). Find the steady-state error ess.

G(s) = 4/((s + 2)(s + 4))r(t) = 1·u(t)
ess =
2.Easy
r(t) = 10+K = 25s + 2y(t)−

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

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

The parabolic input r(t) = 5·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 4)/(s²(s + 4)). Find the steady-state error ess.

G(s) = 5(s + 4)/(s²(s + 4))r(t) = 5·t²/2
ess =
4.Medium
r(t) = 1t+Ks(s + 1)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 1)) must follow the ramp r(t) = 1t with a steady-state error of 0.5. Find the gain K.

G(s) = K/(s(s + 1))r(t) = 1tess = 0.5
K =
5.Hard
R(s)+Kp​ + Ki​/s10s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 1. Find the proportional gain Kp.

G(s) = 10/(s + 2)ωₙ = 10 rad/sζ = 1
Kp =
6.Hard
R(s)+8 + Kd​s2s(s + 1)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 2/(s(s + 1)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.9.

G(s) = 2/(s(s + 1))Kp = 8ζ = 0.9
Kd = s

Answer key

  1. 1. ess = 0.6667
  2. 2. y(∞) = 8.333
  3. 3. ess = 1
  4. 4. K = 2
  5. 5. Kp = 1.8
  6. 6. Kd = 3.1 s

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