• Source: Weighting pattern

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.




Linear time invariant system


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}

.




References

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