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

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

${\displaystyle y(t)=C(t)x(t)}$,

then the output can be written as

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

where ${\displaystyle T(\cdot ,\cdot )}$ is the weighting pattern for the system. For such a system, the weighting pattern is ${\displaystyle T(t,\sigma )=C(t)\phi (t,\sigma )B(\sigma )}$ 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.^[1]

Linear time invariant system

In a LTI system then the weighting pattern is:

Continuous

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

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

Discrete

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

References

1. Brockett, Roger W. (1970). Finite Dimensional Linear Systems. John Wiley & Sons. ISBN 978-0-471-10585-5.