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)+2(s + 1)s(s + 4)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 1)/(s(s + 4)(s + 10)). Find the velocity error constant Kv.

G(s) = 2(s + 1)/(s(s + 4)(s + 10))
Kv =
2.Easy
R(s)+20(s + 5)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20(s + 5)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 20(s + 5)/(s²(s + 4))
Ka =
3.Medium
r(t) = 3t+10(s + 1)s(s + 4)C(s)−

A ramp r(t) = 3t is applied to a unity-feedback loop with G(s) = 10(s + 1)/(s(s + 4)). Find the steady-state error ess.

G(s) = 10(s + 1)/(s(s + 4))r(t) = 3t
ess =
4.Medium
R(s)+K(s + 1)(s + 4)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 4)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 1)(s + 4))ess = 2 %
K =
5.Hard
r(t) = 1t+2 + 5/s2(s + 4)(s + 5)C(s)−

A PI controller C(s) = 2 + 5/s drives the plant G(s) = 2/((s + 4)(s + 5)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 1t.

C(s) = 2 + 5/sG(s) = 2/((s + 4)(s + 5))r(t) = 1t
ess =
6.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 = 6 rad/s and ζ = 0.6. Find the proportional gain Kp.

G(s) = 4/(s + 2)ωₙ = 6 rad/sζ = 0.6
Kp =

Answer key

  1. 1. Kv = 0.05
  2. 2. Ka = 25
  3. 3. ess = 1.2
  4. 4. K = 196
  5. 5. ess = 2
  6. 6. Kp = 1.3

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