Transfer Function
The Laplace-domain ratio of output to input for a linear time-invariant system under zero initial conditions.
A transfer function describes the dynamic relationship between an input and an output in the Laplace domain without exposing every internal physical detail of the system.
Under zero initial conditions:
G(s) = Y(s) / U(s)The roots of the numerator and denominator define zeros and poles. Poles are particularly important for natural dynamics, transient behavior, and stability.
A transfer function is not the universal representation of every system. Its classical form assumes linear time-invariant behavior; state-space models can be more expressive for multi-input/multi-output systems, explicit state reasoning, and other control problems.
The concept is developed alongside time response and stability in Automatic Control.