Control Engineering Basics
A step of size 1 is applied at time 0 to a first-order system with the transfer function G(s) = 2/(1 + 0.5s). Which of the following is closest to the output at t = 1 s?
- AAbout 1.26
- BAbout 1.73
- C2
Show answer and explanation
Answer: B. About 1.73
The step response of a first-order system K/(1 + Ts) is y(t) = K(1 − e^(−t/T)), here with K = 2 and T = 0.5 s. Since t = 1 s is twice the time constant, y = 2 × (1 − e^(−2)) ≈ 2 × 0.865 ≈ 1.73. The response reaches about 63.2% of its final value at the time constant T, about 95% at 3T and about 99.3% at 5T.
- A. This takes t/T as 1, the value at t = 0.5 s. At t = 1 s, twice the time constant has passed.
- B. Correct. This is about 86.5% of the final value 2.
- C. This is the final value. The output of a first-order system only approaches it with time and does not reach it in finite time.