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 = 202s + 2y(t)−

A proportional controller K = 20 drives the plant G(s) = 2/(s + 2) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 2)K = 20r = 5
y(∞) =
2.Easy
R(s)+C(s)G(s)C(s)−

The controller C(s) = 5 drives the plant G(s) = 2/(s(s + 10)) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 5G(s) = 2/(s(s + 10))
N =
3.Medium
R(s)+K = 152s + 1C(s)−

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

G(s) = 5/(2s + 1)K = 1
ess = %
4.Medium
R(s)+K = 10510s + 1C(s)−

A process G(s) = 5/(10s + 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) = 5/(10s + 1)K = 10
ess = %
5.Hard
R(s)+10 + Kd​s10s(s + 4)C(s)−

A PD controller C(s) = 10 + 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 = 0.5.

G(s) = 10/(s(s + 4))Kp = 10ζ = 0.5
Kd = s
6.Hard
R(s)+2 + Kd​s5s(s + 2)C(s)−

A PD controller C(s) = 2 + Kd·s drives the plant G(s) = 5/(s(s + 2)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.5.

G(s) = 5/(s(s + 2))Kp = 2ζ = 0.5
Kd = s

Answer key

  1. 1. y(∞) = 4.762
  2. 2. N = 1
  3. 3. ess = 16.67 %
  4. 4. ess = 1.961 %
  5. 5. Kd = 0.6 s
  6. 6. Kd = 0.2325 s

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