Transfer function

A transfer function is the ratio of a linear time-invariant system’s output transform to its input transform under zero initial conditions. It encodes the system’s dynamics as a function of complex frequency.

Laplace transform: An integral transform that represents time-domain signals as functions of complex frequency. Continuous-time transfer functions are commonly formed by transforming differential equations with this method.
Linear time-invariant system: A system whose response obeys superposition and does not change when its input is shifted in time. Transfer functions describe this system class, not arbitrary input-output devices.
Feedback control: A control strategy that measures output and uses its deviation from a target to adjust the input. Open-loop and closed-loop transfer functions quantify how feedback changes system behavior.
Impulse response: A system’s output when its input is a unit impulse, assuming zero initial conditions. It describes the same LTI input-output behavior in time rather than transform form.
Z-transform: A transform that represents discrete-time sequences as functions of a complex variable. Discrete-time transfer functions use this transform in place of the Laplace transform.
Convolution: An operation combining two functions to describe how one signal is distributed and accumulated over time. An LTI system’s output is the convolution of its input with its impulse response.
Control system stability: The property that a control system’s response remains bounded and settles after disturbances. Closed-loop transfer-function poles provide a standard test of stability.
Nonlinear system: A system whose output does not obey superposition with respect to its input. A single transfer function generally cannot represent nonlinear input-output behavior.
Pole–zero plot: A complex-plane diagram marking the poles and zeros of a rational function. Its geometry reveals stability, resonances, and frequency-response features of a transfer function.
Zero initial conditions: The assumption that stored energy and internal state are initially zero. This assumption isolates the input-driven behavior represented by the transfer function.

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