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 + 4)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 20/(s(s + 4)). Find the velocity error constant Kv.

G(s) = 20/(s(s + 4))
Kv =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 4(s + 2)/s drives the plant G(s) = 5/(s(s + 3)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 4(s + 2)/sG(s) = 5/(s(s + 3))
N =
3.Medium
r(t) = 10 + 2t+2(s + 4)s(s + 1)(s + 5)C(s)−

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

G(s) = 2(s + 4)/(s(s + 1)(s + 5))r(t) = 10 + 2t
ess =
4.Medium
R(s)+K = 1025s + 1C(s)−

A process G(s) = 2/(5s + 1) is controlled by a proportional gain K = 10 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 2/(5s + 1)K = 10
ess = %
5.Hard
R(s)+Kp​ + Ki​/s10s + 1C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.5. Find the integral gain Ki.

G(s) = 10/(s + 1)ωₙ = 10 rad/sζ = 0.5
Ki =
6.Hard
R(s)+K5s(s + 10)C(s)−

A proportional controller K drives the plant G(s) = 5/(s(s + 10)) in a unity-feedback loop. Find the gain K that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 5/(s(s + 10))ζ = 0.6
K =

Answer key

  1. 1. Kv = 5
  2. 2. N = 2
  3. 3. ess = 1.25
  4. 4. ess = 4.762 %
  5. 5. Ki = 10
  6. 6. K = 13.89

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