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.

New worksheet Practise online with hints

For teachers: free to print, copy and use in class. Every new worksheet has new values and its own answer key. A link to ExamProblems on your course page is appreciated.

Name: ____________________ Date: ____________

1.Easy

A proportional controller K = 4 drives the plant G(s) = 5/(s + 10) in a unity-feedback loop. A step of size 10 is applied. Find the final value of the output y(∞).

G(s) = 5/(s + 10)K = 4r = 10

y(∞) =

2.Easy

A unity-feedback loop has the open-loop transfer function G(s) = 2/((s + 4)(s + 10)). Find the position error constant K_p.

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

K_p =

3.Medium

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

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

e_ss =

4.Medium

The input r(t) = 2 + 0.5t (a step plus a ramp) is applied to a unity-feedback loop with G(s) = 20/(s(s + 1)(s + 5)). Find the steady-state error e_ss.

G(s) = 20/(s(s + 1)(s + 5))r(t) = 2 + 0.5t

e_ss =

5.Hard

A proportional controller K = 5 drives the plant G(s) = 10/(s(s + 2)) in a unity-feedback loop. Find the 2 % settling time t_s of the step response, using t_s ≈ 4/(ζωₙ).

G(s) = 10/(s(s + 2))K = 5

t_s = s

6.Hard

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

G(s) = 1/(s + 1)ωₙ = 2 rad/sζ = 1

K_i =

Answer key

1. y(∞) = 6.667
2. K_p = 0.05
3. e_ss = 50
4. e_ss = 0.125
5. t_s = 4 s
6. K_i = 4

Full step-by-step solutions: practise this topic on examproblems.com.