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)+20s(s + 1)(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s(s + 1)(s + 10)). Find the velocity error constant Kv.

G(s) = 20/(s(s + 1)(s + 10))
Kv =
2.Easy
R(s)+5s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/(s²(s + 1)). What is the system type N?

G(s) = 5/(s²(s + 1))
N =
3.Medium
r(t) = 10t+Ks(s + 4)C(s)−

A unity-feedback loop with G(s) = K/(s(s + 4)) must follow the ramp r(t) = 10t with a steady-state error of 0.2. Find the gain K.

G(s) = K/(s(s + 4))r(t) = 10tess = 0.2
K =
4.Medium
r(t) = 2 + 3t+5(s + 3)s(s + 5)C(s)−

The input r(t) = 2 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 5(s + 3)/(s(s + 5)). Find the steady-state error ess.

G(s) = 5(s + 3)/(s(s + 5))r(t) = 2 + 3t
ess =
5.Hard
R(s)+5 + Kd​s10s(s + 1)C(s)−

A PD controller C(s) = 5 + Kd·s drives the plant G(s) = 10/(s(s + 1)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 10/(s(s + 1))Kp = 5ζ = 0.6
Kd = s
6.Hard
R(s)+Kp​ + Ki​/s10s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.7. Find the integral gain Ki.

G(s) = 10/(s + 5)ωₙ = 8 rad/sζ = 0.7
Ki =

Answer key

  1. 1. Kv = 2
  2. 2. N = 2
  3. 3. K = 200
  4. 4. ess = 1
  5. 5. Kd = 0.7485 s
  6. 6. Ki = 6.4

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