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) = 10+20(s + 4)(s + 5)C(s)−

A step of size 10 is applied to a unity-feedback loop with G(s) = 20/((s + 4)(s + 5)). Find the steady-state error ess.

G(s) = 20/((s + 4)(s + 5))r(t) = 10·u(t)
ess =
2.Easy
R(s)+5(s + 2)s²(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 2)/(s²(s + 5)). Find the acceleration error constant Ka.

G(s) = 5(s + 2)/(s²(s + 5))
Ka =
3.Medium
R(s)+K = 2050.5s + 1C(s)−

A process G(s) = 5/(0.5s + 1) is controlled by a proportional gain K = 20 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 5/(0.5s + 1)K = 20
ess = %
4.Medium
r(t) = 10 + 1t+4s(s + 10)C(s)−

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

G(s) = 4/(s(s + 10))r(t) = 10 + 1t
ess =
5.Hard
R(s)+Kp​ + Ki​/s1s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 5) 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) = 1/(s + 5)ωₙ = 6 rad/sζ = 0.6
Kp =
6.Hard
R(s)+K = 202s(s + 6)C(s)−

A proportional controller K = 20 drives the plant G(s) = 2/(s(s + 6)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 2/(s(s + 6))K = 20
%OS = %

Answer key

  1. 1. ess = 5
  2. 2. Ka = 2
  3. 3. ess = 0.9901 %
  4. 4. ess = 2.5
  5. 5. Kp = 2.2
  6. 6. %OS = 18.4 %

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