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)+C(s)G(s)C(s)−

The controller C(s) = 4(s + 2)/s drives the plant G(s) = 2/(s + 10) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 4(s + 2)/sG(s) = 2/(s + 10)
N =
2.Easy
r(t) = 5+K = 35s + 10y(t)−

A proportional controller K = 3 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 10)K = 3r = 5
y(∞) =
3.Medium
R(s)+K = 20210s + 1C(s)−

A process G(s) = 2/(10s + 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) = 2/(10s + 1)K = 20
ess = %
4.Medium
r(t) = 4·t²/2+10(s + 5)s²(s + 5)C(s)−

The parabolic input r(t) = 4·t²/2 is applied to a unity-feedback loop with G(s) = 10(s + 5)/(s²(s + 5)). Find the steady-state error ess.

G(s) = 10(s + 5)/(s²(s + 5))r(t) = 4·t²/2
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 = 5 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 1/(s + 5)ωₙ = 5 rad/sζ = 0.6
Ki =
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 = 8 rad/s and ζ = 0.5. Find the proportional gain Kp.

G(s) = 2/(s + 2)ωₙ = 8 rad/sζ = 0.5
Kp =

Answer key

  1. 1. N = 1
  2. 2. y(∞) = 3
  3. 3. ess = 2.439 %
  4. 4. ess = 0.4
  5. 5. Ki = 25
  6. 6. Kp = 3

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