Weighting pattern

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A weighting pattern for a linear dynamical system describes the relationship between an input u {\displaystyle u} and output y {\displaystyle y} . Given the time-variant system described by

x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) {\displaystyle {\dot {x}}(t)=A(t)x(t)+B(t)u(t)}

y ( t ) = C ( t ) x ( t ) {\displaystyle 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 {\displaystyle y(t)=y(t_{0})+\int _{t_{0}}^{t}T(t,\sigma )u(\sigma )d\sigma } ,

where T ( ⋅ , ⋅ ) {\displaystyle T(\cdot ,\cdot )} 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 ) {\displaystyle T(t,\sigma )=C(t)\phi (t,\sigma )B(\sigma )} such that ϕ {\displaystyle \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]

In a LTI system then the weighting pattern is:

Continuous

T ( t , σ ) = C e A ( t − σ ) B {\displaystyle T(t,\sigma )=Ce^{A(t-\sigma )}B}

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

Discrete

T ( k , l ) = C A k − l − 1 B {\displaystyle T(k,l)=CA^{k-l-1}B} .

• Control theory