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 = 310s + 1y(t)−

A proportional controller K = 3 drives the plant G(s) = 10/(s + 1) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 10/(s + 1)K = 3r = 5
y(∞) =
2.Easy
R(s)+5(s + 4)(s + 5)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 5/((s + 4)(s + 5)). Find the position error constant Kp.

G(s) = 5/((s + 4)(s + 5))
Kp =
3.Medium
R(s)+K = 1920.5s + 1C(s)−

A process G(s) = 2/(0.5s + 1) is controlled by a proportional gain K = 19 in a unity-feedback loop. Find the steady-state error ess for a step, as a percentage of the step size.

G(s) = 2/(0.5s + 1)K = 19
ess = %
4.Medium
R(s)+K(s + 1)(s + 5)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 5)). Find the gain K that gives a steady-state error of 5 % for a step input.

G(s) = K/((s + 1)(s + 5))ess = 5 %
K =
5.Hard
R(s)+8 + Kd​s2s(s + 4)C(s)−

A PD controller C(s) = 8 + Kd·s drives the plant G(s) = 2/(s(s + 4)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.8.

G(s) = 2/(s(s + 4))Kp = 8ζ = 0.8
Kd = s
6.Hard
R(s)+Kp​ + Ki​/s4s + 4C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 4/(s + 4) in a unity-feedback loop. The closed loop must have ωₙ = 6 rad/s and ζ = 0.7. Find the proportional gain Kp.

G(s) = 4/(s + 4)ωₙ = 6 rad/sζ = 0.7
Kp =

Answer key

  1. 1. y(∞) = 4.839
  2. 2. Kp = 0.25
  3. 3. ess = 2.564 %
  4. 4. K = 95
  5. 5. Kd = 1.2 s
  6. 6. Kp = 1.1

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