Weighting pattern explained

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
•
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(t0)+

t
\int
t0

T(t,\sigma)u(\sigma)d\sigma

,where

T( ⋅ , ⋅ )

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)=CeA(t-\sigma)B

where

eA(t-\sigma)

is the matrix exponential.
Discrete

T(k,l)=CAk-l-1B

.

Notes and References

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

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