Lineaarsust

Lineaarsust is a concept described in discussions of linear systems to denote the persistence of a system’s response after the input is removed. In this sense, it captures how long the output remains affected by a transient input, independent of the input’s current state. The term combines lineair, meaning linear, with sust, a shorthand for sustain or sustenance.

Formalization is variable in the literature, but a common approach treats lineaarsust as a single-parameter summary of transient behavior in linear time-invariant systems. For a system with impulse response h(t), one definition is the integral from 0 to infinity of the absolute value of h(t), often normalized by the impulse magnitude. An alternative practical definition is the time τ at which the output magnitude falls below a chosen threshold ε relative to the peak response, i.e., |y(t)| ≤ ε|y(0)| for t ≥ τ. These definitions reflect a focus on decay characteristics rather than steady-state performance.

Properties of lineaarsust depend on the system’s damping and pole structure. In stable LTI systems, lineaarsust is finite; greater damping generally reduces it, while poles near the imaginary axis increase it and can introduce oscillatory components that extend persistence. It is invariant to input scale under linearity, making it a robust measure of transient behavior.

Examples often cite a first-order exponential decay, where lineaarsust is proportional to the inverse of the decay rate, and underdamped second-order systems, which exhibit longer persistence due to oscillations.

Usage of lineaarsust remains unofficial and is not standard in formal textbooks, but it serves as a heuristic for comparing transient persistence across designs in control theory and signal processing. See also linear time-invariant systems, impulse response, stability, and transient response.