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

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s²(s + 5)). Find the acceleration error constant K_a.

G(s) = 10(s + 1)/(s²(s + 5))

K_a =

2.Easy

The controller C(s) = 2(s + 1)/s 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) = 2(s + 1)/sG(s) = 2/(s(s + 10))

N =

3.Medium

A process G(s) = 5/(0.5s + 1) is controlled by a proportional gain K = 9 in a unity-feedback loop. Find the steady-state error e_ss for a step, as a percentage of the step size.

G(s) = 5/(0.5s + 1)K = 9

e_ss = %

4.Medium

A process G(s) = 4/(0.5s + 1) is controlled by a proportional gain K = 9 in a unity-feedback loop. Find the steady-state error e_ss for a step, as a percentage of the step size.

G(s) = 4/(0.5s + 1)K = 9

e_ss = %

5.Hard

A proportional controller K = 40 drives the plant G(s) = 2/(s(s + 10)) in a unity-feedback loop. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 2/(s(s + 10))K = 40

t_s = s

6.Hard

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

C(s) = 1 + 1/sG(s) = 5/((s + 1)(s + 10))r(t) = 5t

e_ss =

Answer key

1. K_a = 2
2. N = 2
3. e_ss = 2.174 %
4. e_ss = 2.703 %
5. t_s = 0.8 s
6. e_ss = 10

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