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)+40(s + 5)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 40/((s + 5)(s + 10)). Find the position error constant Kp.

G(s) = 40/((s + 5)(s + 10))
Kp =
2.Easy
r(t) = 2+K = 15s + 1y(t)−

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

G(s) = 5/(s + 1)K = 1r = 2
y(∞) =
3.Medium
R(s)+K(s + 4)(s + 10)C(s)−

A unity-feedback loop has G(s) = K/((s + 4)(s + 10)). Find the gain K that gives a steady-state error of 5 % for a step input.

G(s) = K/((s + 4)(s + 10))ess = 5 %
K =
4.Medium
r(t) = 10 + 0.5t+10s(s + 5)C(s)−

The input r(t) = 10 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 10/(s(s + 5)). Find the steady-state error ess.

G(s) = 10/(s(s + 5))r(t) = 10 + 0.5t
ess =
5.Hard
R(s)+K = 45s(s + 4)C(s)−

A proportional controller K = 4 drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 4))K = 4
ts = s
6.Hard
R(s)+K = 251s(s + 8)C(s)−

A proportional controller K = 25 drives the plant G(s) = 1/(s(s + 8)) in a unity-feedback loop. Find the 2 % settling time ts of the step response, using ts ≈ 4/(ζωₙ).

G(s) = 1/(s(s + 8))K = 25
ts = s

Answer key

  1. 1. Kp = 0.8
  2. 2. y(∞) = 1.667
  3. 3. K = 760
  4. 4. ess = 0.25
  5. 5. ts = 2 s
  6. 6. ts = 1 s

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