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 + 1)/s drives the plant G(s) = 5/(s + 4) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 4(s + 1)/sG(s) = 5/(s + 4)
N =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 5 drives the plant G(s) = 1/(s(s + 2)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 5G(s) = 1/(s(s + 2))
N =
3.Medium
r(t) = 2 + 0.5t+2(s + 2)s(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) = 2(s + 2)/(s(s + 2)). Find the steady-state error ess.

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

A unity-feedback loop has G(s) = K/((s + 2)(s + 4)). Find the gain K that gives a steady-state error of 5 % for a step input.

G(s) = K/((s + 2)(s + 4))ess = 5 %
K =
5.Hard
R(s)+Kp​ + Ki​/s5s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 5/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 5/(s + 1)ωₙ = 10 rad/sζ = 0.7
Kp =
6.Hard
R(s)+K = 104s(s + 2)C(s)−

A proportional controller K = 10 drives the plant G(s) = 4/(s(s + 2)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 4/(s(s + 2))K = 10
%OS = %

Answer key

  1. 1. N = 1
  2. 2. N = 1
  3. 3. ess = 0.25
  4. 4. K = 152
  5. 5. Kp = 2.6
  6. 6. %OS = 60.47 %

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