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+20(s + 1)(s + 10)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 20/((s + 1)(s + 10)). Find the steady-state error ess.

G(s) = 20/((s + 1)(s + 10))r(t) = 5·u(t)
ess =
2.Easy
r(t) = 1+K = 11s + 4y(t)−

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

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

The input r(t) = 5 + 2t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5/(s(s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 5/(s(s + 1)(s + 5))r(t) = 5 + 2t
ess =
4.Medium
r(t) = 5t+Ks(s + 10)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 5t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 5tess = 0.2
K =
5.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 5 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 1/(s + 2)ωₙ = 5 rad/sζ = 1
Ki =
6.Hard
R(s)+8 + Kd​s4s(s + 4)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 4/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 4/(s(s + 4))Kp = 8ζ = 0.5
Kd = s

Answer key

  1. 1. ess = 1.667
  2. 2. y(∞) = 0.2
  3. 3. ess = 2
  4. 4. K = 250
  5. 5. Ki = 25
  6. 6. Kd = 0.4142 s

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