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+K = 42s + 5y(t)−

A proportional controller K = 4 drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 5)K = 4r = 5
y(∞) =
2.Easy
R(s)+2s(s + 4)C(s)−

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

G(s) = 2/(s(s + 4))
Kv =
3.Medium
R(s)+K = 10.52s + 1C(s)−

A process G(s) = 0.5/(2s + 1) is controlled by a proportional gain K = 1 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 0.5/(2s + 1)K = 1
ess = %
4.Medium
r(t) = 2t+Ks(s + 5)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 5)) must follow the ramp r(t) = 2t with a steady-state error of 0.02. Find the gain K.

G(s) = K/(s(s + 5))r(t) = 2tess = 0.02
K =
5.Hard
R(s)+5 + Kd​s2s(s + 3)C(s)−

A PD controller C(s) = 5 + Kd·s drives the plant G(s) = 2/(s(s + 3)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 2/(s(s + 3))Kp = 5ζ = 0.5
Kd = s
6.Hard
r(t) = 5t+5 + 4/s10(s + 4)(s + 10)C(s)−

A PI controller C(s) = 5 + 4/s drives the plant G(s) = 10/((s + 4)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 5 + 4/sG(s) = 10/((s + 4)(s + 10))r(t) = 5t
ess =

Answer key

  1. 1. y(∞) = 3.077
  2. 2. Kv = 0.5
  3. 3. ess = 66.67 %
  4. 4. K = 500
  5. 5. Kd = 0.08114 s
  6. 6. ess = 5

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