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)+2(s + 2)s²(s + 10)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 2)/(s²(s + 10)). Find the acceleration error constant Ka.

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

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 5)/(s(s + 4)(s + 10)). What is the system type N?

G(s) = 2(s + 5)/(s(s + 4)(s + 10))
N =
3.Medium
R(s)+K = 90.51s + 1C(s)−

A process G(s) = 0.5/(1s + 1) is controlled by a proportional gain K = 9 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 0.5/(1s + 1)K = 9
ess = %
4.Medium
r(t) = 1 + 3t+4s(s + 5)C(s)−

The input r(t) = 1 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 4/(s(s + 5)). Find the steady-state error ess.

G(s) = 4/(s(s + 5))r(t) = 1 + 3t
ess =
5.Hard
R(s)+20 + Kd​s10s(s + 4)C(s)−

A PD controller C(s) = 20 + Kd·s drives the plant G(s) = 10/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 10/(s(s + 4))Kp = 20ζ = 0.707
Kd = 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. Ka = 0.4
  2. 2. N = 1
  3. 3. ess = 18.18 %
  4. 4. ess = 3.75
  5. 5. Kd = 1.6 s
  6. 6. ts = 1 s

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