PAPERmaking! Vol11 Nr3 2025

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ͺͻ͵͵

 ȋͳͲȌ ™Š‡”‡ k V ‹•–Š‡‘’–‹‹œƒ–‹‘‘„Œ‡ –‹˜‡ˆ— –‹‘ǡ f ‹•–Š‡’”‡†‹ –‹‘‘†‡Žǡ–Š‡‡•–‹ƒ–‡†•–ƒ–‡ –”ƒŒ‡ –‘”› [ ] x k ǡ h ‹• –Š‡ ‘’—–‡† ’”‡†‹ –‹‘ ‘—–’—–ǡ , l c  ‹• –Š‡ ‡“—ƒ–‹‘ ƒ† ‹‡“—ƒŽ‹–› ‘•–”ƒ‹–•ƒ––Š‡•ƒ’Ž‹‰‘‡–ǡ P ‹•–Š‡’”‡†‹ –‹‘–‹‡†‘ƒ‹ǡƒ†–Š‡ ‘–”‘Ž–‹‡†‘ƒ‹ ‹• [ 1] ([],[], ) 0 [ ] ( [ ], ) 0 x k f x k u k k y k h x k k l x k u k c x k u k   ­ °  ( [ ], [ ]) 0 ( [ ], [ ]) 0 t ° ® ° ° ¯  M ǡƒ† ( ) U k ‹•–Š‡ ‘’—–‡† ‘–”‘Ž•‡“—‡ ‡‘ˆ–Š‡‘’–‹‹œƒ–‹‘’”‘„Ž‡ƒ––Š‡ k  ‘‡–Ǥ – k ‘‡–•‘Ž›–Š‡ϐ‹”•– ‘–”‘Ž“—ƒ–‹–›‘ˆ ( ) U k ‹•‹’Ž‡‡–‡†‘–Š‡ ‘–”‘ŽŽ‡†‘„Œ‡ –‹ –‹‡ †‘ƒ‹ [ , 1] k k  ǡƒ†ƒ––Š‡‡š–‘‡–ǡ–Š‡’”‡†‹ –‹‘–‹‡†‘ƒ‹ƒ†–Š‡ ‘–”‘Ž–‹‡ †‘ƒ‹ƒ”‡•Š‹ˆ–‡†‘‡•–‡’ˆ‘”™ƒ”†–‘”‡Ǧ•‘Ž˜‡–Š‡ƒ„‘˜‡‘’–‹‹œƒ–‹‘’”‘„Ž‡™‹–Š•–ƒ–‡ [ 1] x k   ȋͳͳȌ Š‡–Š‡ˆ— –‹‘ˆƒ‹Ž› ^ ` ( ) i P x ‹••ƒ‹†–‘„‡‘”–Š‘‰‘ƒŽǤ ‘”‡“—ƒŽŽ›•’ƒ ‡††ƒ–ƒ’‘‹–•ǡƒ †‹• ”‡–‡‘”–Š‘‰‘ƒŽ’‘Ž›‘‹ƒŽ‹• ƒŽŽ‡†ƒ ”ƒ’‘Ž›‘‹ƒŽ•›•–‡™Š‡ ( ( )) 1 w x i Ǥ ƒ•–Š‡‹‹–‹ƒŽ ‘†‹–‹‘ǡˆ‘”‹‰ƒ”‘ŽŽ‹‰‘’–‹‹œƒ–‹‘Ǥ 3.3. Gram polynomials ƒ•–Š‡ˆ‘ŽŽ‘™‹‰†‡ˆ‹‹–‹‘ǣˆ‘”ƒˆƒ‹Ž›‘ˆˆ— –‹‘• ^ ` ( ) i P x ‹ƒ†‹• ”‡–‡•‡–‘ˆ’‘‹–• ( ) x i  ™‹–Š ”‡•’‡ ––‘–Š‡™‡‹‰Š–ˆ— –‹‘ 1 N  ( ) w x  ‹ˆƒ›ǣ  0 0 ( ) ) ( ()) ( ()) ( ()) w x i P x i P x i ( , 0,1, , ) m   ( j k i i O j k j k z j k ­ ® ¯ ¦ 3.4. Intelligent Predictive Design of Controller ‹ – Š ‡ƒ „ƒ ‡– Š •‡‡‡ ƒ ––‹Š ƒƒ Ž–– Š‘‡† ‡ Žƒ ‘– Šˆ  ‡– Š‡ƒ ––”‹ ƒ ƒŽ• ˜ ‡‘” †• ‡‡ Ž“ ‘—ˆƒ –Š–‡‹ – ƒ‘– ‹˜–‡”‘ Ž‘ ‘ „– ”Œ ‘‡ Ž  –• › •‘–‡•‹ • –Š•ƒ ‘• ˆ„ –‡™‡‘’‰ƒ‹ ˜”‡– •ǡ  ‹‘ ‡ Š’ƒƒ’” ––‡ ‹”•͵– ŠǤ ͳ‡ǡ  •ˆ —’ƒ ” –•‹‡‘  ‘ —‘ ˆ’ Ž –‹ Š ‰‡  ’ ƒƒ ’– ”‡‹ š”  –Š ”ƒ‘ —Š‰‹ Š‡‘—ˆ–” ‘– Š‡ ‡– Š ‡– ‹ ”†‡ ‹ „Ž —ƒ–‹ ‘‡”ǡ‘ ƒˆ –†Š ‡– Š ‡™ ‘ƒ –– Š‡ ”‡”  ’ ‡ƒ –”Š– ‘‹ •† – Š‘‡ˆ  Ž–‘Š‡‰ ‹ƒ– —–†—‹ ƒ–ƒ‘Ž ”– ”–ƒ‘ •–ˆŠ‡‡”  ‘‘ ”” ”” ‡‡ •• ’’ ‘‘  †† ‹‹  ‰‰  ‘ ‡ –ƒ”•‘—Ž ” ‡• –” ƒ‡–‡–‰ ‹ ‡’•‘ ‹•Š–‘Ǥ  — Ž †‘  „ ‡‘ ƒ †– ”‘‘’Ž – ‡–†Š ‡ˆ‘ ”“ —– Šƒ ‡ •– ‹‡œ ƒ– –™‹ ‘‘ ’ ƒ‘”ˆ – • – Š” ‡‡ • ’™‡ Š –‘‹ ˜Ž ‡‡ Ž ›„Ǥ  ƒŠ‡‡ ”‘ ˜™‡ ”‡ƒŽ ŽŽ Žǡ  † ‡‘ ‘Ǧ •—“’—Ž ‹ƒ”‰‡ ‹ • ‘ ƒ— ’ŠŽ‹‹‡˜‰‡ † „ƒ ›– ” ƒ‹ š ” ‹ƒ– –‹ ‘‘  ƒƒŽ • –“”—ƒ ƒ”•‡ˆ ‘ ” ƒ –ƒ”–‹‹š‘‘ ˆ‘ ˆ’ –”Š‹ ‡  ‹ ’”ƒƒŽ † ‹’ƒ‘‰Ž‘› ƒ‘Ž ‡‹Žƒ‡Ž• ǡ ‡™Š– •‹ ǡ  Š– Š –—” ƒ• ••‘ˆ ‘Ž ˜”‹ ‰• –– ŠŠ ‡‡  ‹– ” ƒ˜‡ ”• •˜‡‡ ” •†‡‡ “‘——ƒ’Ž–‹ ‹œ‰ƒǤ – ‹ ‘Š‡™ ‹†– ‡Š  –‘Š—‡’ ŽŠ‡‡†Ž ’ • ‘› ˆ•––‡Š‡   ƒ‘†‡ Žˆ ‹ ’ ƒ” ‡Ž Ž†› ‹ ƒ– ‹ ˜Š‡‹‡ ˜‘‡ –ƒ” ‘ Ž• ƒ™– ‹Š•‹ˆ ƒŠ –†‘ ‘” ›‡ •  ‘‘–– ””‘‡Ž“ —‡‹ˆ”ˆ ‡‡  Š– ‹ ‰‘Šˆ  ‘†‡Žƒ —”ƒ ›ƒ† ƒ„‡‡ˆˆ‡ –‹˜‡Ž› ‘•–”ƒ‹‡†Ǥ ͵ƒ Ǥ Ͷ†Ǥ ͳ Ǥ  ‡ ƒ• —‡ ” ‹‘—‰’ ’Ž ‹‘‹‰ – •‘ ƒ – ”” ‘‘ •Ž •„ –ƒŠ•‡‡†’ ƒ‘’‡ ” ”ƒ ƒ Š’‹‘Ž‡›™‘‹ † –‹ŠƒǤŽ •Ǥ – ‡Šƒ‡ ”Š‡  ƒ‘”‡– ”ƒ‘ ŽŽƒ ”›‰ ‡Ž‡ǡ —– Š‡„ ‡‘”  ‘– ˆ”‘†Ž‹Ž“——– ‹ƒ‘– ‹ ˜– ‹ƒ‡Ž•˜‡‘•ˆ  ƒ• ‹Ž Ž  †‹ Ž‹ ƒŽ —” –Ž ›‹ ‘ǡ  – Š ™‡  ƒ– ‡‡”ƒ •˜—ƒ ”Ž ˜‡‡• ‡ ‘ˆ– –•Š‹ ‡‰‡ƒŽ –•‹”‘‡ˆ „ƒƒŽ Ž “—‡ƒ”  ƒ–”‹ –‡ƒ •– ‹‡˜“‡— ‡ ‡–ƒ‹ ƒ• —Ž Ž ›” ‡ Ž‹ ‡‡†– —’’‘‹ƒ– •† ‘˜ˆ‹ ‡–™Š ‡‡†‡ƒ–•‹”ƒ‡ ’„”ƒ‘ϐ ‹ Ž ‡‡ ”ǡ  ƒƒ” †‡  •’‡‘“‹ —–‡•ǡ  – ‹ ƒƒ Ž Ž ›„ Ž‡‹ ”‡ƒ†– ‹ ‘—’ƒ Žƒ‹ œ‡†† ˜™‹ ‡‹™– Š‡ †† ‹ •ƒ •” ‡ƒ – ‡’” ‘”ϐ ƒ‹ Ž‡ Ǥ  ’‘‘Ž–›Š ‘‘ ˆ  –‹Šƒ ‡Ž ••ǡ ‡ – Š’—” •‘ϐ”‹ ‡Ž ‡Ǧ •– ”ǡ  ƒ ‘•ˆ •‘‹”•– ‹ ‹ ‰‰ ‘–ˆŠ ‡†‹ • ƒ” –‡Š– ‡‡ • ‡ƒ––•‹  ‘ƒ ˆŽ   ‘ †‹ ”‡•Ž – ‡ •†– ‡ƒ ϐ„‹ Ž‹ •‡Š ‡ƒ† ‡ ƒƒ”–Ž”‹ ‡‹ š”Ǥ  0 x ™Š‹ Š ‘–ƒ‹•–Š‡ϐ‹”•– m ‘”–Š‘‰‘ƒŽ’‘Ž›‘‹ƒŽ•‘ˆ–Š‡ ”ƒ ’‘Ž›‘‹ƒŽ•›•–‡ǡ‡ƒ Š™‹–Š†‹• ”‡–‡’‘‹–•ǣ

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