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) = 2+40(s + 1)(s + 4)C(s)−

A step of size 2 is applied to a unity-feedback loop with G(s) = 40/((s + 1)(s + 4)). Find the steady-state error ess.

G(s) = 40/((s + 1)(s + 4))r(t) = 2·u(t)
ess =
2.Easy
r(t) = 1+50(s + 1)(s + 2)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 50/((s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 50/((s + 1)(s + 2))r(t) = 1·u(t)
ess =
3.Medium
R(s)+K(s + 2)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 2)(s + 10)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 2)(s + 10))ess = 2 %
K =
4.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 0.1. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 10tess = 0.1
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 4/(s + 2)ωₙ = 3 rad/sζ = 0.7
Kp =
6.Hard
R(s)+8 + Kd​s10s(s + 1)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 10/(s(s + 1)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 10/(s(s + 1))Kp = 8ζ = 0.8
Kd = s

Answer key

  1. 1. ess = 0.1818
  2. 2. ess = 0.03846
  3. 3. K = 980
  4. 4. K = 1000
  5. 5. Kp = 0.55
  6. 6. Kd = 1.331 s

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