A weighting pattern for a linear dynamical system

Linear dynamical system

Linear dynamical systems are a special type of dynamical system where the equation governing the system's evolution is linear. While dynamical systems in general do not have closed-form solutions, linear dynamical systems can be solved exactly, and they have a rich set of mathematical properties...

 describes the relationship between an input and output . Given the time-variant system

Time-variant system

A time-variant system is a system that is not time invariant . Roughly speaking, characteristics of its output depend explicitly upon time.- Overview :...

 described by

then the output can be written as

where is the weighting pattern for the system. For such a system the weighting pattern is such that is the state transition matrix.

The weighting pattern will determine a system, but note that if there exists a realization

Realization (systems)

Realization, in the system theory context refers to a state space model implementing a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of matrices [A,B,C,D] such that-LTI System:...

 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

where is the matrix exponential

Matrix exponential

In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. Abstractly, the matrix exponential gives the connection between a matrix Lie algebra and the corresponding Lie group....

.

Discrete

.