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 proportional controller K = 5 drives the plant G(s) = 1/(s + 5) in a unity-feedback loop. A step of size 5 is applied. Find the final value of the output y(∞).

G(s) = 1/(s + 5)K = 5r = 5

y(∞) =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 10(s + 1)/(s(s + 10)). Find the velocity error constant K_v.

G(s) = 10(s + 1)/(s(s + 10))

K_v =

3.Medium

The input r(t) = 10 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20/(s(s + 4)(s + 5)). Find the steady-state error e_ss.

G(s) = 20/(s(s + 4)(s + 5))r(t) = 10 + 0.5t

e_ss =

4.Medium

The input r(t) = 2 + 3t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 50/(s(s + 1)(s + 10)). Find the steady-state error e_ss.

G(s) = 50/(s(s + 1)(s + 10))r(t) = 2 + 3t

e_ss =

5.Hard

A PI controller C(s) = K_p + K_i/s drives the plant G(s) = 2/(s + 5) in a unity-feedback loop. The closed loop must have ωₙ = 10 rad/s and ζ = 0.7. Find the proportional gain K_p.

G(s) = 2/(s + 5)ωₙ = 10 rad/sζ = 0.7

K_p =

6.Hard

A proportional controller K = 8 drives the plant G(s) = 4/(s(s + 6)) 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 + 6))K = 8

t_s = s

Answer key

1. y(∞) = 2.5
2. K_v = 1
3. e_ss = 0.5
4. e_ss = 0.6
5. K_p = 4.5
6. t_s = 1.333 s

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