Weighting pattern

A weighting pattern for a linear dynamical system describes the relationship between an input $u$ and output $y$. Given the time-variant system described by

$\dot{x}(t)=A(t)x(t)+B(t)u(t)$

$y(t)=C(t)x(t)$,

then the output can be written as

$y(t)=y(t_0)+\underset{t_0}{\overset{t}{\int }}T(t,\sigma )u(\sigma )d\sigma$,

where $T(\cdot ,\cdot )$ is the weighting pattern for the system. For such a system, the weighting pattern is $T(t,\sigma )=C(t)\varphi (t,\sigma )B(\sigma )$ such that $\varphi$ is the state transition matrix. The weighting pattern will determine a system, but if there exists a realization for this weighting pattern then there exist many that do so.

Linear time invariant system

In a LTI system then the weighting pattern is:

Continuous

$T(t,\sigma )=C{e}^{A(t-\sigma )}B$

where $e^{A(t-\sigma )}$ is the matrix exponential.

Discrete

$T(k,l)=C{A}^{k-l-1}B$.