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)+50(s + 1)s(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 50(s + 1)/(s(s + 10)). Find the velocity error constant Kv.

G(s) = 50(s + 1)/(s(s + 10))
Kv =
2.Easy
R(s)+10(s + 1)s²(s + 2)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s²(s + 2)). What is the system type N?

G(s) = 10(s + 1)/(s²(s + 2))
N =
3.Medium
R(s)+K = 111s + 1C(s)−

A process G(s) = 1/(1s + 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) = 1/(1s + 1)K = 1
ess = %
4.Medium
R(s)+K(s + 4)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 4)(s + 10)). Find the gain K that gives a steady-state error of 2 % for a step input.

G(s) = K/((s + 4)(s + 10))ess = 2 %
K =
5.Hard
r(t) = 2t+1 + 2/s2(s + 4)(s + 10)C(s)−

A PI controller C(s) = 1 + 2/s drives the plant G(s) = 2/((s + 4)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 1 + 2/sG(s) = 2/((s + 4)(s + 10))r(t) = 2t
ess =
6.Hard
R(s)+Kp​ + Ki​/s10s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.5. Find the proportional gain Kp.

G(s) = 10/(s + 4)ωₙ = 10 rad/sζ = 0.5
Kp =

Answer key

  1. 1. Kv = 5
  2. 2. N = 2
  3. 3. ess = 50 %
  4. 4. K = 1960
  5. 5. ess = 20
  6. 6. Kp = 0.6

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