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 + 4)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 4)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 2(s + 4)/(s²(s + 1))
Ka =
2.Easy
r(t) = 2+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 2 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 5)K = 4r = 2
y(∞) =
3.Medium
r(t) = 2 + 0.5t+5s(s + 1)(s + 2)C(s)−

The input r(t) = 2 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5/(s(s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 5/(s(s + 1)(s + 2))r(t) = 2 + 0.5t
ess =
4.Medium
R(s)+K = 542s + 1C(s)−

A process G(s) = 4/(2s + 1) is controlled by a proportional gain K = 5 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 4/(2s + 1)K = 5
ess = %
5.Hard
r(t) = 2t+2 + 0.5/s20(s + 1)(s + 4)C(s)−

A PI controller C(s) = 2 + 0.5/s drives the plant G(s) = 20/((s + 1)(s + 4)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 2t.

C(s) = 2 + 0.5/sG(s) = 20/((s + 1)(s + 4))r(t) = 2t
ess =
6.Hard
R(s)+Kp​ + Ki​/s2s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 2/(s + 2)ωₙ = 3 rad/sζ = 1
Ki =

Answer key

  1. 1. Ka = 8
  2. 2. y(∞) = 1.231
  3. 3. ess = 0.2
  4. 4. ess = 4.762 %
  5. 5. ess = 0.8
  6. 6. Ki = 4.5

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