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

The controller C(s) = 4(s + 5)/s drives the plant G(s) = 5/(s + 3) in a unity-feedback loop. What is the system type N of the loop?

C(s) = 4(s + 5)/sG(s) = 5/(s + 3)

N =

2.Easy

A step of size 1 is applied to a unity-feedback loop with G(s) = 2/((s + 1)(s + 10)). Find the steady-state error e_ss.

G(s) = 2/((s + 1)(s + 10))r(t) = 1·u(t)

e_ss =

3.Medium

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

G(s) = 2(s + 4)/(s(s + 4)(s + 10))r(t) = 5 + 2t

e_ss =

4.Medium

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

G(s) = 4(s + 1)/(s(s + 5))r(t) = 10 + 2t

e_ss =

5.Hard

A PI controller C(s) = 5 + 4/s drives the plant G(s) = 2/((s + 4)(s + 10)) in a unity-feedback loop. Find the steady-state error e_ss for the ramp input r(t) = 5t.

C(s) = 5 + 4/sG(s) = 2/((s + 4)(s + 10))r(t) = 5t

e_ss =

6.Hard

A proportional controller K = 40 drives the plant G(s) = 5/(s(s + 4)) in a unity-feedback loop. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 5/(s(s + 4))K = 40

t_s = s

Answer key

1. N = 1
2. e_ss = 0.8333
3. e_ss = 10
4. e_ss = 2.5
5. e_ss = 25
6. t_s = 2 s

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