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)+4(s + 3)s(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s(s + 5)). Find the velocity error constant Kv.

G(s) = 4(s + 3)/(s(s + 5))
Kv =
2.Easy
R(s)+4(s + 3)s²(s + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 4(s + 3)/(s²(s + 4)). Find the acceleration error constant Ka.

G(s) = 4(s + 3)/(s²(s + 4))
Ka =
3.Medium
r(t) = 5t+2s(s + 4)(s + 5)C(s)−

A ramp r(t) = 5t is applied to a unity-feedback loop with G(s) = 2/(s(s + 4)(s + 5)). Find the steady-state error ess.

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

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

G(s) = 4/(s(s + 1)(s + 4))r(t) = 4t
ess =
5.Hard
R(s)+4 + Kd​s4s(s + 4)C(s)−

A PD controller C(s) = 4 + 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 = 0.8.

G(s) = 4/(s(s + 4))Kp = 4ζ = 0.8
Kd = s
6.Hard
r(t) = 5t+1 + 1/s5(s + 5)(s + 10)C(s)−

A PI controller C(s) = 1 + 1/s drives the plant G(s) = 5/((s + 5)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 1 + 1/sG(s) = 5/((s + 5)(s + 10))r(t) = 5t
ess =

Answer key

  1. 1. Kv = 2.4
  2. 2. Ka = 3
  3. 3. ess = 50
  4. 4. ess = 4
  5. 5. Kd = 0.6 s
  6. 6. ess = 50

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