Space Synthesis

Contents

Description

Space synthesis tasks are formulated as approximation problems on linear space of continuous functions with the quadratic norm and continuous basis functions as week solutions of describing partial differential equations of controlled technological and manufacturing processes:

$$min\left \| W\left ( x,\infty \right )-\sum_{i=1}^{n}W_{i}\left
(x_{i},\infty \right )\mathcal{H}HR_{i}\left ( x,\infty \right ) \right
\|=\left \| W\left ( x,\infty \right )-\sum_{i=1}^{n}\breve{W_{i}}\left (
x_{i},\infty \right )\mathcal{H}HR_{i}\left ( x,\infty \right ) \right \|
$$

For simplicity in block SS1 continuous basis functions $\left \{ \mathcal{H} HR_{i} \left ( x,\infty \right )\right \}_{i=1}^{n}$ are represented as vectors of discrete quantities selected from week solutions in m nodes of numerical net along with $W\left ( x,\infty \right )$ :

$$min\left \| W\left ( x_{m} \right ) - \sum_{i=1}^{n} W_{i}\left ( x_{i}
\right ) \mathcal{H} HR_{i}\left ( x_{m} \right ) \right \|=\left \|
W\left ( x_{m} \right ) -\sum_{i=1}^{n} \breve{W_{i}}\left ( x_{i} \right
) \mathcal{H} HR_{i}\left ( x_{m} \right ) \right \|$$

where

$W\left ( x_{m} \right ) = \left \{ W \left ( x_{j}, \infty \right ) \right \}_{j=1}^{m}$ , $W_{i}\left ( x_{i} \right ) = \left \{ W_{i} \left ( x_{i}, \infty \right ) \right \}_{i=1}^{n}$ , $\mathcal{H}HR_{i}\left ( x_{m} \right )=\left \{ \mathcal{H}HR_{i}\left ( x_{j}, \infty \right ) \right \}_{j=1}^{m}$

$HR=\left \{  \left \{ \mathcal{H}HR_{i}\left ( x_{m} \right ) \right \}_{j=1}^{m}\right \}_{i=1}^{n}$

and finally

$\breve{W_{i}}\left ( x_{i} \right ) = HR'*HR \backslash HR'W\left ( x_{m} \right )$

in block SS2

$$min\left \| Y\left ( x,k \right ) - \sum_{i=1}^{n} Y_{i}\left ( x_{i},k
\right ) YR_{i} \left ( x,k \right ) \right \|=\left \| Y\left ( x,k
\right ) - \sum_{i=1}^{n} \breve{Y_{i}}\left ( x_{i},k \right ) YR_{i}
\left ( x,k \right ) \right \|$$

in vector form

$$min\left \| Y\left ( x_{m},k \right ) - \sum_{i=1}^{n} Y_{i}\left (
x_{i},k \right ) YR_{i} \left ( x_{m},k \right ) \right \|=\left \|
Y\left ( x_{m},k \right ) - \sum_{i=1}^{n} \breve{Y_{i}}\left ( x_{i},k
\right ) YR_{i} \left ( x_{m},k \right ) \right \|$$

where

$$Y\left ( x_{m},k \right ) = \left \{ Y \left ( x_{j},k \right ) \right
\}_{j=1}^{m} , Y_{i}\left ( x_{i},k \right ) = \left \{ Y_{i} \left (
x_{i},k \right ) \right \}_{i=1}^{n} , YR_{i}\left ( x_{m} \right ) =
\left \{ YR_{i} \left ( x_{j},k \right ) \right \}_{j=1}^{m} , YR = \left
\{ \left \{ YR_{i} \left ( x_{j},k \right ) \right \}_{j=1}^{m} \right
\}_{i=1}^{n}$$

and finally

$$\breve{Y_{i}}\left ( x_{i} \right ) = YR'* YR \backslash YR'Y\left (
x_{m},k \right )$$

in block SS

$$min\left \| E \left ( x,k \right ) - \sum_{i=1}^{n} E_{i} \left (
x_{i},k \right ) \mathcal{H}HR_{i} \left ( x,\infty \right ) \right \| =
\left \| E \left ( x, \infty \right ) - \sum_{i=1}^{n} \breve{E_{i}}
\left ( x_{i},k \right ) \mathcal{H}HR_{i} \left ( x,\infty \right )
\right \|$$

in vector form

$$min\left \| E \left ( x_{m},k \right ) - \sum_{i=1}^{n} E_{i} \left (
x_{i},k \right ) \mathcal{H}HR_{i} \left ( x_{m} \right ) \right \| =
\left \| E \left ( x_{m}, k \right ) - \sum_{i=1}^{n} \breve{E_{i}} \left
( x_{i},k \right ) \mathcal{H}HR_{i} \left ( x_{m} \right ) \right \|$$

where

$$E\left ( x_{m},k \right ) = \left \{ E \left ( x_{j},k \right ) \right
\}_{j=1}^{m} , E_{i}\left ( x_{i},k \right ) = \left \{ E_{i}\left (
x_{i},k \right ) \right \}_{i=1}^{n} , \mathcal{H}HR_{i}\left ( x_{m}
\right ) = \left \{ \mathcal{H}HR_{i}\left ( x_{j}, \infty \right )
\right \}_{j=1}^{m} , HR = \left \{ \left \{ \mathcal{H}HR_{i} \left (
x_{m} \right ) \right \}_{j=1}^{m} \right \}_{i=1}^{n}$$

and finally

$$\breve{E_{i}}\left ( x_{i},k \right ) = HR'*HR \backslash HR'E\left (
x_{m},k \right )$$

In continuous cases instead of scalar products of vectors integral operations should be used.