Control Engineering Basics
A step of size 1 is applied to a system with the transfer function G(s) = 5/(s + 2). Which of the following is the final value of the output (its value as t → ∞)?
- A0.4
- B2.5
- C5
Show answer and explanation
Answer: B. 2.5
The output is Y(s) = G(s)·(1/s), and by the final value theorem y(∞) = lim[s→0] sY(s) = G(0) = 5/2 = 2.5. G(0) is the gain at DC (zero frequency), also called the steady-state gain (DC gain). Rewriting it as G(s) = 2.5/(1 + 0.5s) shows that it is a first-order system with a steady-state gain of 2.5 and a time constant of 0.5 s. The final value theorem can be used only when the system is stable and the output settles to a constant value.
- A. This inverts numerator and denominator. The steady-state gain is G(0) = 5/2.
- B. Correct. The final value is the value of G(s) as s → 0.
- C. The numerator 5 is not the steady-state gain. It must be divided by the constant term 2 of the denominator.