What is a transfer function in linear control systems?

• A. G(s) = Output(s) / Input(s) in Laplace domain; stability is assessed by pole locations.
• B. The time-domain convolution of input and impulse response.
• C. It is defined as the difference between output and input.
• D. It equates to the input signal itself.

Answer: (A)

At its heart, a transfer function describes how an LTI system maps input signals to output signals in the Laplace domain. It is defined as the ratio of the output transform to the input transform, assuming zero initial conditions. This lets you analyze the system’s behavior by treating the input and output in the complex frequency domain: Y(s) = G(s)U(s), where G(s) = Y(s)/U(s). The transfer function also equals the Laplace transform of the impulse response, so the time-domain output is the convolution of the input with that impulse response. The poles of G(s) (the roots of its denominator) tell you about stability: if all poles lie in the left half of the complex plane, the system is BIBO stable. The related time-domain relationship and impulse response help connect intuition about how a system reacts to inputs, but the transfer function itself is the ratio in the Laplace domain, not the time-domain convolution or a difference between output and input.