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: $T(t,\sigma )=Ce^{A(t-\sigma )}B$

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

Discrete: $T(k,l)=CA^{k-l-1}B$ .

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References

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

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