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) = 2(s + 2)/s drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 2(s + 2)/sG(s) = 5/(s(s + 4))
N =
2.Easy
r(t) = 5+K = 52s + 10y(t)−

A proportional controller K = 5 drives the plant G(s) = 2/(s + 10) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 10)K = 5r = 5
y(∞) =
3.Medium
r(t) = 10·t²/2+20(s + 5)s²(s + 2)C(s)−

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 20(s + 5)/(s²(s + 2)). Find the steady-state error ess.

G(s) = 20(s + 5)/(s²(s + 2))r(t) = 10·t²/2
ess =
4.Medium
R(s)+K = 141s + 1C(s)−

A process G(s) = 4/(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) = 4/(1s + 1)K = 1
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 = 0.5. Find the proportional gain Kp.

G(s) = 1/(s + 5)ωₙ = 6 rad/sζ = 0.5
Kp =
6.Hard
R(s)+K1s(s + 10)C(s)−

A proportional controller K drives the plant G(s) = 1/(s(s + 10)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 1/(s(s + 10))ζ = 0.8
K =

Answer key

  1. 1. N = 2
  2. 2. y(∞) = 2.5
  3. 3. ess = 0.2
  4. 4. ess = 20 %
  5. 5. Kp = 1
  6. 6. K = 39.06

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