Transfer function
= Transfer function
{title2=$G(s)=Y(s)/U(s)$}
{wiki}
A <transfer function> packages the response of a <linear time-invariant system> into a rule for turning a <plant input> into a <plant output>. Instead of solving a <differential equation> separately for every <signal>, take a <Laplace transform>, multiply by $G(s)$, then transform back. This also makes it possible to compare slow and fast oscillations through the <continuous-time frequency response>.
Here consider a continuous-time, causal, finite-dimensional <linear time-invariant system> with one <plant input> $u(t)$ and one <plant output> $y(t)$. With every initial state set to zero, its <transfer function> is defined by
$$
U(s)=\mathcal L\{u\}(s),\qquad Y(s)=\mathcal L\{y\}(s),\qquad Y(s)=G(s)U(s).
$$
The ratio $G(s)=Y(s)/U(s)$ is used where $U(s)\ne0$ in a common <Laplace convergence region>; $G$ describes the system, independently of that particular <plant input>. Zero initial state matters: stored motion or another nonzero <initial condition> can contribute to the <plant output> even when the <plant input> vanishes. https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/941349ccabc51e8c116a86db62ba6566_MIT18_03SCF11_s30_1text.pdf[MIT 18.03SC, The Transfer Function: rest initial conditions and transformed input/output]{ref}
The <first-order low-pass filter> below shows the same <transfer function> predicting both a step and a sustained <sinusoid>. Its <pole> gives a decay rate, its value at zero gives the eventual gain, and its values along the imaginary axis give the <continuous-time frequency response>. These descriptions make a useful bridge between a physical relaxation model and <control theory> calculations.