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

C(s) = 5G(s) = 5/(s(s + 5))
N =
2.Easy
R(s)+10(s + 4)s²(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 4)/(s²(s + 5)). Find the acceleration error constant Ka.

G(s) = 10(s + 4)/(s²(s + 5))
Ka =
3.Medium
r(t) = 2t+Ks(s + 10)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 10)) must follow the ramp r(t) = 2t with a steady-state error of 0.01. Find the gain K.

G(s) = K/(s(s + 10))r(t) = 2tess = 0.01
K =
4.Medium
r(t) = 10·t²/2+2(s + 5)s²(s + 4)C(s)−

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

G(s) = 2(s + 5)/(s²(s + 4))r(t) = 10·t²/2
ess =
5.Hard
R(s)+5 + Kd​s4s(s + 2)C(s)−

A PD controller C(s) = 5 + Kd·s drives the plant G(s) = 4/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 4/(s(s + 2))Kp = 5ζ = 0.6
Kd = s
6.Hard
R(s)+Kp​ + Ki​/s1s + 2C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 1/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 3 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 1/(s + 2)ωₙ = 3 rad/sζ = 0.6
Ki =

Answer key

  1. 1. N = 1
  2. 2. Ka = 8
  3. 3. K = 2000
  4. 4. ess = 4
  5. 5. Kd = 0.8416 s
  6. 6. Ki = 9

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