Weighting pattern - Alchetron, The Free Social Encyclopedia

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 ( t 0 ) + ∫ t 0 t T ( t , σ ) u ( σ ) d σ ,

where T ( ⋅ , ⋅ ) is the weighting pattern for the system. For such a system, the weighting pattern is T ( t , σ ) = C ( t ) ϕ ( t , σ ) B ( σ ) such that ϕ 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 , σ ) = C e A ( t − σ ) B

where e A ( t − σ ) is the matrix exponential.

Discrete

T ( k , l ) = C A k − l − 1 B .

References

Weighting pattern Wikipedia

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