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 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

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

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

Discrete

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

References

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