PAPERmaking! Vol11 Nr3 2025

ͺͻ͵ʹ    ”‡ˆ‡”•–‘–Š‡‹ˆŽ—‡ ‡ ‘‡ˆˆ‹ ‹‡–„‡–™‡‡–Š‡ i –Š‡ƒ•—”‡‡–’‘‹–ƒ†–Š‡ j –Šƒ –—ƒ–‘”ǡ‹Ǥ‡Ǥǡ G  ‹•–Š‡ƒ••‘ ‹ƒ–‡† ‘—’Ž‹‰ƒ–”‹š‘ˆ–Š‡™Š‘Ž‡•›•–‡Ǥ 3.2. Predictive Control Theory ˆ —”–‡—†”‹‡ –‘‹ ˜—‡–’ —‘–•–ǡ ”ƒ‘Ž ǡ † ‹•  ƒƒ Ž —ƒŽŽƒ‰–‘‡”•‹––ŠŠ‡  ‘– Š’ƒ– ‹– —ƒ•Ž‡ ••‡ƒ“ —† ‡›  ƒ ‡ ‘‹ ˆ  ‘‘†– ‡” ‘Ž  Ž‘ ‹ˆ  –’Š—‡– • ‘„›–”‘‘’Ž Ž–‡‹ †  ‹‘œ„‹ Œ ‡‰  –– Š– ‡‘ ‘’—” ‡– ’† —‹ ––• ‹ ‘– •ˆ  –”Š‘‡ŽŽ‹’”‰‘‘’‡•–‹•‹‹œƒ–Š–‹‡‘ˆ—ƒ–—†”‡ˆ‡–‹‡†„‡ƒ†‘ ‘ƒ”‹”‡Ǥ ––•‹‘ ‘Ǥ–”‘Ž•–”ƒ–‡‰›„ƒ•‹ ƒŽŽ› ‘•‹•–•‘ˆ’”‡†‹ –‹˜‡‘†‡Ž‹‰ǡ ͵ Ǥ‘ʹ–Ǥͳˆ ‘Ǥ  — • ”‘‡†‹– Š– ‡‹ ˜ ‡• ’ ‡ ‘‹†ϐ ‹‡ Ž ‹ˆ‘ ‰” Ǥ  ”‘‡ˆ †– ‹Š ‡– ‹ ˜‡ ‘ †‘‡Ž ǡ– ”„‘—Ž – ‹ •” ƒƒ–Š‡ƒ”Ž‰‘‘” ‹––ŠŠ‡  „‘ƒ†• ‡‡ Ž† ˆ —‘   ƒ– ‹‘ ‘ ǡ„ ‹ŒǤ‡‡ Ǥ ǡ– –Š ‡‘ †’ ‡” ‡Ž ǡ † „‹ —––‹‘†‘ ‡‘•ˆ  –Š‘‡ † ˆ‡—Ž– —Š”ƒ‡• ‘ –—Š– ‡’—ˆ–— ‘ˆ  –ƒ‹ ‘ ‘ „‘Œˆ‡ ‡– š „’ƒ” •‡‡• †• ‹ ‘‰  ‹––Š•‡ Š ˆ‹ •—––‘—””‹‡ ƒ †Ž  ›‹ ˆƒ‘” ‹ ƒ –„‹ ‘‡ Š ƒ ƒ˜‹ ‘†” ˆ —‘–ˆ— ”– Š‡ ‡‹•’›—•––•‡Ǥ  ǡŠ ‡• ‘ ’ ”–‡Š†ƒ‹–  ––‹Š˜ ‡‡  ’‘ ’” ‡– ‹†‹ ‹–œ‹ ƒ˜ –‡‹‘ ‘‘†’‡‡Ž” ƒ –ƒ‹ ‘ „• ‡ǡ  ƒ—•†‡ †– Š ‡– ‘ –’‘” ‘‘˜„‹–†ƒ‡‹  ƒ ƒ ’ ”‡‹”‘–”ƒ‹‹  • ‡‘“™—Ž ‡‡ † ‰‡‡ ‘ˆ ˆ‘ ” ‘’”–‡” †‘ Ž‹  ‹–‹ ˜’‡— – •‘ǡ  • ‘– ” –‘ŠŽ ƒ––‘ – Š ‡ƒ” ”Š›ƒ ‘‰—‡–  ‘ˆ–Š‡‘—–’—–‘ˆ–Š‡‘„Œ‡ ––‘„‡ ‘–”‘ŽŽ‡†™‹ŽŽ”‡ƒ Š–Š‡†‡•‹”‡†‰‘ƒŽƒ–ƒˆ—–—”‡‘‡–Ǥ ͵† Ǥ‡ʹ– Ǥ‡ʹ”Ǥ  ‹ ‡‘•Ž Ž –‹ Š ‰‡  ˆ— ’– —– ‹”‡  ‹ œ‘ƒ– ‹–‘” ‘ ŽǤ  ”‘”Ž‡‡† „‹ ›– ‹‘˜’‡ – ‹ ‘ ‹œ–‹”‘‰Ž  •ƒ‘Ž • ‘‡ Š’ƒ‡•” ˆ ƒ‘”  ‘ƒǦ Ž ‹‡ ‡ ‡‘ –’”–‹ ‹ Ǥ  ‹ œ ƒ• –– ‡‹ ‘ƒ† ‘ˆ —ˆ — •–‹‹‘‰ ǡƒ ™• –Šƒ‹– ‹Š  ‰‘ Ž’‘–„‹ ƒ Ž‹ œ‘ƒ’– ‹–‘‹   ‹•œ–ƒ” –ƒ‹–‘‡‰›‘Ǥ  „ Œ–‡ ƒ – ‹›˜‡• ǡƒ ’ ”’‡Ž †‹ ‹ ‰– ‹ ˜ ‡‘ ‘‡  –– ǡ” ‘– ŠŽ ‡‘ ‘’’– ‹–‹ ‹ œ‹ œƒ‡– ‹†‘ ’ ‡ ”—ˆ•‘‡”• ƒƒϐ ‹ ‡‹– ‡‡ ––”‹ ‹ ‡ ‘†‘– ƒ‹ ƒ •‹  ‘ ” Ž‘›Ž Ž–‹ Š ‡‰  ϐ‘‹’–‹–‹‡ ‹–œ‹‡†‡–‹†‘‡†ƒ‘‹ƒˆ‹”‘‹••›–Šƒ–Š ”‘‘‘—‡•Ž›–•–Š‘‹ˆ––‡Š†‡ˆ‘ˆ—”™–—ƒ””‡†ǡǤ™Š‹Ž‡ ƒ– –Š‡ ‡š– •ƒ’Ž‹‰ ‘‡–ǡ –Š‡ ͵’ Ǥ”ʹ‡Ǥ†͵‹Ǥ  – ‹ ˜ ‡‡ ‡†‘„†ƒ‡ Ž ‹•  ‘‘ ””Ž‡›  ƒ– ‹ ”‘‘—Ǥ ‰ Š †– Š‡ •‡  ””‘‹ ’Ž Ž–‹‹‘‰ ‘‘’ˆ ––‹Š‡ ‹ †œ ƒ›–‹ ƒ‘   ‹‘ ˆ  ’Š”ƒ‡”†ƒ ‹ ––‡‹ ”˜‹‡•– ‹ ‘• ‘– ”ˆ‘ –ŽŠ ƒ‡Ž ‰‘‘„”Œ ‡‹ – Š– ǡ  ‹ ‰• ǡ •‘‹”‹ ‡‰ –– ŠŠ ‡‡  ‡ƒ š –‹ •—–ƒ‡Ž • ›‡ • –‘‡ˆ  – ‹ǡ ƒ‡Ǧ†˜ƒ–”Š›—‹ • ‰ –ǡ Š‡ ‘† ‡‘Ž† ‡Ž ‹ •’” ‡ƒ†– ‹ Š–ǡ‡ †‘‘—Ž ‹–’‡—ƒ–” ‹– ‹ ‡ƒ•›ǡ † ‘‹ •–– —” „ƒ ƒ– Š ‡–•Šǡ ‡ƒ ƒ† – —‘ƒ– ŠŽ  ‡•”‹ – —ˆ ƒƒ ––‹‘‘”•ǡ  ‹™ Š –‹ Š Š‡  ”ƒ ‡ˆ –“‡—” ‹ ”†‡‡•– ‡ ”‘‹ ’‡‹ ‰• ƒ––Š‘‡”›• ‡“‡—ƒ‡•— ”‡‡ •‘ǡ ˆ‘ ˆ”— ‘– —”Ǧ ‡Ž ‹ ‡‘  ‘– ””‘” Ž‡ ƒ– ‹ –‘‹ ‘ • •– ‘–Š– Š” ‘‡ — —‰Š† –‡Š” ‡Ž › ‹‘’‰– ‹‘‹ œ†ƒ‡–Ž‹Ǥ ‘ ” ‡‘†’‹ ‡ ”– ‹ƒ˜–‡‹ ‘ ‘ǡ †– ‘” ‘‡ Ž•ǡ   ‡‘ š– –‹’‘Ž ‡‡ ‡ – ǡ–  –ƒŠŽ Ž‡ ‘ ϐˆ‹ ”– Š• –‡ •†‡‡ – ‡‘ ––‹ ”‘‘Ž ƒ‘ ˆ – ‹–‘Š‡ • ǡ‘ „——– ’–  —‘ – Ž˜›ƒ –Ž Š— ‡‡  ‘‘ˆ  ––Š” ‘‡ Ž ‘ƒ „ –Œ ‡‹ ‘ – ǡ• ƒ‘ˆ †– Š –‡Š ‡ — —” ”•‡‡ ‘– ˆ –‘Š‹ • ‡”‡– ƒǡ  Ž–Ǧ‘– ‹–Š ‡‡  ‹ˆ‘”ƒ–‹‘–‘ ‘””‡ ––Š‡’”‡†‹ –‡†‘—–’—–ǡƒ†–Š‡”‡Ǧ‘’‡–Š‡‡™‘’–‹‹œƒ–‹‘‘’‡”ƒ–‹‘Ǥ ͵ǤʹǤͶǤ ƒ–Š‡ƒ–‹ ƒŽ†‡• ”‹’–‹‘‘ˆ’”‡†‹ –‹˜‡ ‘–”‘ŽǤ ‘”–Š‡ˆ‘ŽŽ‘™‹‰•›•–‡•ǣ  0 0 [ 1] ([],[], ) [ ] ( [ ], ) [0] ~ ( ) x k f x k u k k ȋ͹Ȍ ‘’  –‹ –  ‡ ƒ ƒ Ž Š ‘   • – ƒ ”‘  Ž ’ ’ Ž ” ‹ ‘ ‰ „  Ž‡   ‘   ƒ• ‡ ǣ –ǡ’”‡†‹ –‹˜‡ ‘–”‘Ž•‘Ž˜‡•‘Ž‹‡–Š‡ϐ‹‹–‡–‹‡Ǧ†‘ƒ‹‘’‡ǦŽ‘‘’  ( ) ( ) 1 min min ( [ | ], [ 1| ]) P k i U k U k i V L x k i k u k i k    ¦  ȋͺȌ ”ǣ  ( ) ( ) 1 min min ( [ | ], [ | ]) P k i U k U k i V L y k i k u k i k   ¦  ȋͻȌ ƒ–‹•ˆƒ –‹‘ǣ y k h x k k x x P x  ° ® ° ¯ 

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