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)/(s(s + 1)(s + 2)). Find the velocity error constant K_v.

G(s) = 4(s + 4)/(s(s + 1)(s + 2))

K_v =

2.Easy

A proportional controller K = 4 drives the plant G(s) = 2/(s + 4) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 2/(s + 4)K = 4r = 10

y(∞) =

3.Medium

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

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

e_ss =

4.Medium

The input r(t) = 5 + 1t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 2(s + 3)/(s(s + 2)(s + 5)). Find the steady-state error e_ss.

G(s) = 2(s + 3)/(s(s + 2)(s + 5))r(t) = 5 + 1t

e_ss =

5.Hard

A proportional controller K = 20 drives the plant G(s) = 4/(s(s + 5)) in a unity-feedback loop. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 4/(s(s + 5))K = 20

t_s = s

6.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 10/(s + 2) in a unity-feedback loop. The closed loop must have ωₙ = 8 rad/s and ζ = 1. Find the integral gain K_i.

G(s) = 10/(s + 2)ωₙ = 8 rad/sζ = 1

K_i =

Answer key

1. K_v = 8
2. y(∞) = 6.667
3. e_ss = 50
4. e_ss = 1.667
5. t_s = 1.6 s
6. K_i = 6.4

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