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 + 5)(s + 10)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 4/((s + 5)(s + 10)). Find the steady-state error ess.

G(s) = 4/((s + 5)(s + 10))r(t) = 1·u(t)
ess =
2.Easy
R(s)+4s²(s + 4)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4/(s²(s + 4)(s + 10)). What is the system type N?

G(s) = 4/(s²(s + 4)(s + 10))
N =
3.Medium
r(t) = 5t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 5t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 5tess = 0.01
K =
4.Medium
R(s)+K(s + 1)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 5)). Find the gain K that gives a steady-state error of 5 % for a step input.

G(s) = K/((s + 1)(s + 5))ess = 5 %
K =
5.Hard
R(s)+K1s(s + 2)C(s)−

A proportional controller K drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 1/(s(s + 2))ζ = 0.707
K =
6.Hard
R(s)+Kp​ + Ki​/s2s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 4 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 2/(s + 1)ωₙ = 4 rad/sζ = 0.6
Kp =

Answer key

  1. 1. ess = 0.9259
  2. 2. N = 2
  3. 3. K = 2500
  4. 4. K = 95
  5. 5. K = 2.001
  6. 6. Kp = 1.9

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