Weighting pattern

A weighting pattern for a linear dynamical system describes the relationship between an input $u$ and output Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 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

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

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

Discrete

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