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)+5(s + 2)s²(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5(s + 2)/(s²(s + 5)). Find the acceleration error constant Ka.

G(s) = 5(s + 2)/(s²(s + 5))
Ka =
2.Easy
r(t) = 1+K = 22s + 1y(t)−

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

G(s) = 2/(s + 1)K = 2r = 1
y(∞) =
3.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 4 % for a step input.

G(s) = K/((s + 1)(s + 2))ess = 4 %
K =
4.Medium
R(s)+K = 192.50.5s + 1C(s)−

A process G(s) = 2.5/(0.5s + 1) is controlled by a proportional gain K = 19 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 2.5/(0.5s + 1)K = 19
ess = %
5.Hard
R(s)+K = 101s(s + 5)C(s)−

A proportional controller K = 10 drives the plant G(s) = 1/(s(s + 5)) in a unity-feedback loop. Find the percent overshoot %OS of the step response.

G(s) = 1/(s(s + 5))K = 10
%OS = %
6.Hard
R(s)+K = 210s(s + 4)C(s)−

A proportional controller K = 2 drives the plant G(s) = 10/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 10/(s(s + 4))K = 2
ts = s

Answer key

  1. 1. Ka = 2
  2. 2. y(∞) = 0.8
  3. 3. K = 48
  4. 4. ess = 2.062 %
  5. 5. %OS = 1.732 %
  6. 6. ts = 2 s

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