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+40(s + 1)(s + 5)C(s)−

A step of size 5 is applied to a unity-feedback loop with G(s) = 40/((s + 1)(s + 5)). Find the steady-state error ess.

G(s) = 40/((s + 1)(s + 5))r(t) = 5·u(t)
ess =
2.Easy
r(t) = 2+K = 22s + 10y(t)−

A proportional controller K = 2 drives the plant G(s) = 2/(s + 10) in a unity-feedback loop. A step of size 2 is applied. Find the final value of the output y(∞).

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

A unity-feedback loop has G(s) = K/((s + 4)(s + 10)). Find the gain K that gives a steady-state error of 2 % for a step input.

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

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

G(s) = 10(s + 4)/(s(s + 1)(s + 5))r(t) = 2 + 4t
ess =
5.Hard
r(t) = 1t+5 + 1/s4(s + 5)(s + 10)C(s)−

A PI controller C(s) = 5 + 1/s drives the plant G(s) = 4/((s + 5)(s + 10)) in a unity-feedback loop. Find the steady-state error ess for the ramp input r(t) = 1t.

C(s) = 5 + 1/sG(s) = 4/((s + 5)(s + 10))r(t) = 1t
ess =
6.Hard
R(s)+2 + Kd​s2s(s + 1)C(s)−

A PD controller C(s) = 2 + Kd·s drives the plant G(s) = 2/(s(s + 1)) in a unity-feedback loop. Find the derivative gain Kd that gives the closed loop a damping ratio ζ = 0.6.

G(s) = 2/(s(s + 1))Kp = 2ζ = 0.6
Kd = s

Answer key

  1. 1. ess = 0.5556
  2. 2. y(∞) = 0.5714
  3. 3. K = 1960
  4. 4. ess = 0.5
  5. 5. ess = 12.5
  6. 6. Kd = 0.7 s

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