A weighting pattern for a linear dynamical system describes the relationship between an input and output . Given the time-variant system 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 if there exists a realization for this weighting pattern then there exist many that do so.[1]

Linear time invariant system

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In a LTI system then the weighting pattern is:

Continuous
 

where   is the matrix exponential.

Discrete
 .

References

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