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

$y(t)=y(t_{0})+\int _{t_{0}}^{t}T(t,\sigma )u(\sigma )d\sigma$

where $T(\cdot ,\cdot )$ is the weighting pattern for the system. For such a system the weighting pattern is $T(t,\sigma )=C(t)\phi (t,\sigma )B(\sigma )$ such that $\phi$ is the state transition matrix.

The weighting pattern will determine a system, but note that if there exists a realization for this weighting pattern then there exists many that do so.

Linear time invariant system

In a LTI system then the weighting pattern is:

Continuous

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

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

Discrete

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