What is the purpose of a transfer function and how do you obtain it from a step response?

• A. It describes the time-domain impulse response.
• B. It models the input-output relationship in the Laplace domain; obtain by fitting the dynamic response or performing system identification.
• C. It is used only for nonlinear systems.
• D. It is identical to the time-domain response.

Answer: (B)

A transfer function is a Laplace-domain model that describes how the output of a linear time-invariant system responds to a given input. When you drive the system with a unit step, the input in Laplace domain is 1/s, so the relationship in Laplace domain is Y(s) = G(s) * U(s) = G(s)/s. That means you can obtain the transfer function from a step response by multiplying the Laplace transform of the observed step response by s to get G(s). In practice, you can either transform the measured step response to the Laplace domain and compute G(s) = s·Y(s), or you can perform system identification by fitting a parametric transfer function to the step response data. This approach captures how the system behaves in terms of poles and zeros and allows prediction for other inputs. The impulse response is the time-domain counterpart (the inverse Laplace of G(s)); the transfer function itself lives in the Laplace domain and is not simply the time-domain response. It’s not restricted to nonlinear systems, and it’s not identical to the time-domain response.