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)+4(s + 1)s²(s + 1)C(s)−

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

G(s) = 4(s + 1)/(s²(s + 1))
Ka =
2.Easy
r(t) = 5+K = 21s + 1y(t)−

A proportional controller K = 2 drives the plant G(s) = 1/(s + 1) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 1)K = 2r = 5
y(∞) =
3.Medium
R(s)+K = 2041s + 1C(s)−

A process G(s) = 4/(1s + 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) = 4/(1s + 1)K = 20
ess = %
4.Medium
R(s)+K(s + 1)(s + 2)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 2)). Find the gain K that gives a steady-state error of 20 % for a step input.

G(s) = K/((s + 1)(s + 2))ess = 20 %
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(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) = 4/(s + 2)ωₙ = 8 rad/sζ = 0.5
Kp =
6.Hard
R(s)+Kp​ + Ki​/s1s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.8. Find the integral gain Ki.

G(s) = 1/(s + 4)ωₙ = 10 rad/sζ = 0.8
Ki =

Answer key

  1. 1. Ka = 4
  2. 2. y(∞) = 3.333
  3. 3. ess = 1.235 %
  4. 4. K = 8
  5. 5. Kp = 1.5
  6. 6. Ki = 100

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