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.

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