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)+C(s)G(s)C(s)−

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

C(s) = 4G(s) = 2/(s(s + 3))
N =
2.Easy
R(s)+2(s + 4)s²(s + 1)C(s)−

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 4)/(s²(s + 1)). Find the acceleration error constant Ka.

G(s) = 2(s + 4)/(s²(s + 1))
Ka =
3.Medium
R(s)+K(s + 1)(s + 2)C(s)−

A unity-feedback loop has G(s) = K/((s + 1)(s + 2)). Find the gain K that gives a steady-state error of 25 % for a step input.

G(s) = K/((s + 1)(s + 2))ess = 25 %
K =
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 2 % for a step input.

G(s) = K/((s + 1)(s + 5))ess = 2 %
K =
5.Hard
r(t) = 5t+5 + 0.5/s2(s + 2)(s + 4)C(s)−

A PI controller C(s) = 5 + 0.5/s drives the plant G(s) = 2/((s + 2)(s + 4)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 5t.

C(s) = 5 + 0.5/sG(s) = 2/((s + 2)(s + 4))r(t) = 5t
ess =
6.Hard
R(s)+Kp​ + Ki​/s2s + 5C(s)−

A PI controller C(s) = Kp + Ki/s drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 0.6. Find the integral gain Ki.

G(s) = 2/(s + 5)ωₙ = 8 rad/sζ = 0.6
Ki =

Answer key

  1. 1. N = 1
  2. 2. Ka = 8
  3. 3. K = 6
  4. 4. K = 245
  5. 5. ess = 40
  6. 6. Ki = 32

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