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) = 4(s + 4)/(s²(s + 5)). Find the acceleration error constant K_a.

G(s) = 4(s + 4)/(s²(s + 5))

K_a =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 2(s + 3)/(s²(s + 2)). Find the acceleration error constant K_a.

G(s) = 2(s + 3)/(s²(s + 2))

K_a =

3.Medium

A process G(s) = 4/(1s + 1) is controlled by a proportional gain K = 19 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/(1s + 1)K = 19

e_ss = %

4.Medium

A process G(s) = 4/(1s + 1) is controlled by a proportional gain K = 20 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/(1s + 1)K = 20

e_ss = %

5.Hard

A PD controller C(s) = 20 + K_d·s drives the plant G(s) = 2/(s(s + 4)) in a unity-feedback loop. Find the derivative gain K_d that gives the closed loop a damping ratio ζ = 0.707.

G(s) = 2/(s(s + 4))K_p = 20ζ = 0.707

K_d = s

6.Hard

A PI controller C(s) = 2 + 0.5/s drives the plant G(s) = 20/((s + 2)(s + 5)) in a unity-feedback loop. Find the steady-state error e_ss for the ramp input r(t) = 5t.

C(s) = 2 + 0.5/sG(s) = 20/((s + 2)(s + 5))r(t) = 5t

e_ss =

Answer key

1. K_a = 3.2
2. K_a = 3
3. e_ss = 1.299 %
4. e_ss = 1.235 %
5. K_d = 2.471 s
6. e_ss = 5

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