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(t) = 5+K = 110s + 1y(t)−

A proportional controller K = 1 drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 1)K = 1r = 5
y(∞) =
2.Easy
r(t) = 1+4(s + 1)(s + 2)C(s)−

A step of size 1 is applied to a unity-feedback loop with G(s) = 4/((s + 1)(s + 2)). Find the steady-state error ess.

G(s) = 4/((s + 1)(s + 2))r(t) = 1·u(t)
ess =
3.Medium
r(t) = 10·t²/2+5(s + 1)s²(s + 4)C(s)−

The parabolic input r(t) = 10·t²/2 is applied to a unity-feedback loop with G(s) = 5(s + 1)/(s²(s + 4)). Find the steady-state error ess.

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

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

G(s) = K/((s + 4)(s + 5))ess = 5 %
K =
5.Hard
R(s)+Kp​ + Ki​/s4s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 1. Find the integral gain Ki.

G(s) = 4/(s + 5)ωₙ = 3 rad/sζ = 1
Ki =
6.Hard
R(s)+10 + Kd​s2s(s + 3)C(s)−

A PD controller C(s) = 10 + Kd·s drives the plant G(s) = 2/(s(s + 3)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 2/(s(s + 3))Kp = 10ζ = 0.5
Kd = s

Answer key

  1. 1. y(∞) = 4.545
  2. 2. ess = 0.3333
  3. 3. ess = 8
  4. 4. K = 380
  5. 5. Ki = 2.25
  6. 6. Kd = 0.7361 s

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