Transcript
Page 1: Journal of the Korean Data

Journal of the Korean Data & ํ•œ๊ตญ๋ฐ์ดํ„ฐ์ •๋ณด๊ณผํ•™ํšŒ์ง€Information Science Society

2009, 20(1), 11โ€“24

ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ๊ณผ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์˜ ๊ท ํ˜•์„ ์œ„ํ•œ ๋‹จ์ผ ๋ฐ ์ด์ค‘์ง€์ˆ˜๊ฐ€์ค‘์ด๋™ํ‰๊ท  ํ”ผ๋“œ๋ฐฑ ์ˆ˜์ •๊ธฐ์˜ ์ˆ˜์ •โ€ 

๋ฐ•์ฐฝ์ˆœ1 ยท ๊ถŒ์„ฑ๊ตฌ2

1์ค‘์•™๋Œ€ํ•™๊ต ์ˆ˜ํ•™ํ†ต๊ณ„ํ•™๋ถ€ ยท 2์ค‘์•™๋Œ€ํ•™๊ต ์ผ๋ฐ˜๋Œ€ํ•™์› ํ†ต๊ณ„ํ•™๊ณผ

์ ‘์ˆ˜ 2008๋…„ 12์›” 5์ผ, ์ˆ˜์ • 2009๋…„ 1์›” 5์ผ, ๊ฒŒ์žฌํ™•์ • 2009๋…„ 1์›” 8์ผ

์š” ์•ฝ

์ˆ˜์ •์ ˆ์ฐจ์—์„œ ๊ณต์ •์ˆ˜์ •๊ธฐ๋Š” ์žก์Œ์ด ์กด์žฌํ•˜์ง€๋งŒ ์ œ๊ฑฐํ•  ์ˆ˜ ์—†์„ ๋•Œ ๊ณต์ •์ˆ˜์ค€์„ ๋ชฉํ‘œ์น˜์— ๊ฐ€๊น๊ฒŒ ์ˆ˜

์ •ํ•˜๋Š”๋ฐ ์ข…์ข… ์œ ์šฉํ•˜๊ฒŒ ์‚ฌ์šฉ๋œ๋‹ค. ๊ฐ•๊ฑด ์ˆ˜์ •๊ธฐ์˜ ์˜ˆ๋กœ๋Š” ๋‹จ์ผ ๋ฐ ์ด์ค‘ ์ง€์ˆ˜๊ฐ€์ค‘์ด๋™ํ‰๊ท  ์ˆ˜์ •๊ธฐ๊ฐ€์žˆ๋‹ค. ์ด์ค‘ ์ง€์ˆ˜๊ฐ€์ค‘์ด๋™ํ‰๊ท  ์ˆ˜์ •๊ธฐ๋Š” ๋‹จ์ผ ์ง€์ˆ˜๊ฐ€์ค‘์ด๋™ํ‰๊ท  ์ˆ˜์ •๊ธฐ๊ฐ€ ์ œ๊ฑฐํ•  ์ˆ˜ ์—†๋Š” ๊ณต์ •ํŽธ์ฐจ์˜์น˜์šฐ์นจ์„ ์ค„์ผ ์ˆ˜ ์žˆ๋„๋ก ๊ณ ์•ˆ๋˜์—ˆ๋‹ค. ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ์ด ๋‘ ๊ฐ€์ง€ ์ˆ˜์ •๊ธฐ๊ฐ€ ์ ์šฉ๋  ๋•Œ ๊ณผ๋„ํ•˜๊ฒŒ ์ปค์งˆ

์ˆ˜ ์žˆ๋Š” ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„ ์ค„์ผ ์ˆ˜ ์žˆ๋„๋ก ์›๋ž˜์˜ ์ˆ˜์ •๊ธฐ์— ์ง€์ˆ˜๊ฐ€์ค‘์ด๋™ํ‰๊ท ์„ ์ ์šฉํ•จ์œผ๋กœ์จ ์ˆ˜์ •๋˜์—ˆ๋‹ค. ์ฃผ์–ด์ง€ ์ˆ˜์ •๊ธฐ์— ๋Œ€ํ•œ ์ง€์ˆ˜๊ฐ€์ค‘์ด๋™ํ‰๊ท  ์ˆ˜์ •์€ ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์€ ์กฐ๊ธˆ ์ฆ๊ฐ€์‹œํ‚ค์ง€๋งŒ, ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ

์„์ค„์ด๋Š”๋ฐ์„ฑ๊ณต์ ์ž„์„๋ณด์ด๊ณ ์žˆ๋‹ค.

์ฃผ์š”์šฉ์–ด: ๊ฐ€์ค‘์ด๋™ํ‰๊ท , ๊ณต์ •์ˆ˜์ •, ๊ณต์ •ํŽธ์ฐจ, ์ˆ˜์ •๊ธฐ, ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์ง€์ˆ˜, ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท .

1. ์„œ๋ก 

๊ณต์ •์‚ฐ์—…์—์„œ ๊ณต์ •์˜ ํ’ˆ์งˆํŠน์„ฑ์„ ๋‚˜ํƒ€๋‚ด๋Š” ์š”์ธ์œผ๋กœ ๊ณต์ •ํŽธ์ฐจ (process deviation)๊ฐ€ ์ฃผ๋กœ ์‚ฌ์šฉ๋˜๊ณ 

์žˆ๋‹ค. ์ด๋Š” ๊ณต์ •๊ด€์ธก๊ฐ’์—์„œ ๊ณต์ •๋ชฉํ‘œ์น˜ (target)์„ ๋บ€ ๊ฐ’์œผ๋กœ์„œ ๊ณต์ •๊ด€๋ฆฌ์ž๋Š” ์ด ๊ฐ’์ด 0์— ๊ฐ€๊นŒ์šด ๊ฐ’

์œผ๋กœ ๋ถ„ํฌ๋˜๊ธฐ๋ฅผ ํฌ๋งํ•œ๋‹ค. ๊ณต์ •๊ด€์ธก๊ฐ’์€ ์ข…์ข… ์ž๊ธฐ์ƒ๊ด€๊ด€๊ณ„๋ฅผ ๊ฐ€์ง„ ์žก์Œ (noise)์— ์˜ํ•ด ๋ชฉํ‘œ์น˜๋กœ๋ถ€

ํ„ฐ ๋ฒ—์–ด๋‚˜๊ฒŒ ๋˜๋ฉฐ, ์ด๋Š” ์ž์—ฐ๋ฐœ์ƒ์  ํ˜„์ƒ ๋˜๋Š” ๋ช…ํ™•ํžˆ ์•Œ๋ ค์ง€์ง€ ์•Š์€ ์ด์œ ์— ์˜ํ•ด ๋ฐœ์ƒํ•˜์—ฌ ๊ธฐ์ˆ ์  ์ธก

๋ฉด์ด๋‚˜ ๊ฒฝ์ œ์  ์ธก๋ฉด์—์„œ ์ œ๊ฑฐํ•  ์ˆ˜ ์—†๋‹ค๊ณ  ํŒ๋‹จ๋œ๋‹ค. ์ด๋Ÿฌํ•œ ์žก์Œ (noise)์€ ๊ด€๋ฆฌ๋„์—์„œ ์ฐพ์•„๋‚ด๊ณ ์ž ํ•˜๋Š” ์ด์ƒ์›์ธ (special cause)๊ณผ๋Š” ๊ด€๋ฆฌ๋ฐฉ๋ฒ•์ด ๋‹ค๋ฅด๋‹ค. ๊ด€๋ฆฌ๋„๋ฅผ ์‚ฌ์šฉํ•˜์—ฌ ์ด์ƒ์›์ธ์„ ๋ฐœ๊ฒฌํ•˜๋ฉด ๊ทธ ์›์ธ์„ ์ฐพ์€ ๋‹ค์Œ ๊ณต์ •์—์„œ ์ œ๊ฑฐํ•  ์ˆ˜ ์žˆ๋‹ค๋Š” ๊ฐ€์ •์„ ํ•˜๊ณ  ์žˆ๋‹ค. ๋งŒ์•ฝ ๊ณต์ •ํŽธ์ฐจ์˜ ์›์ธ์„ ์ฐพ์„ ์ˆ˜ ์—†๊ฑฐ

๋‚˜, ์ฐพ์•„๋„ ์ œ๊ฑฐํ•  ์ˆ˜ ์—†๋‹ค๋ฉด ๊ด€๋ฆฌ๋„๋ฅผ ์‚ฌ์šฉํ•˜๋Š” ๊ฒƒ์€ ๋น„ํšจ์œจ์ ์ด ๋˜๋ฉฐ, ์ด๋•Œ์—๋Š” ๊ณต์ •์ˆ˜์ • (process

adjustment)์„ํ†ตํ•˜์—ฌํ•ด๊ฒฐํ•˜๊ฒŒ๋œ๋‹ค.

๊ณต์ •์ˆ˜์ •์€๊ณตํ•™์ ๊ณต์ •๊ด€๋ฆฌ (engineering process control)์—์„œ๋„๋ฆฌ์‚ฌ์šฉํ•˜๋Š”๋ฐฉ๋ฒ•์œผ๋กœ์„œ, ๊ณต์ •ํŽธ์ฐจ

๋ฅผ๊ด€์ธกํ•œํ›„์—๋‹ค์Œ๊ณต์ •์ƒ์‚ฐ์„์œ„ํ•˜์—ฌ์ˆ˜์ •ํ•˜๊ฒŒ๋˜๋ฉด์ด๋ฅผํ”ผ๋“œ๋ฐฑ์กฐ์ • (feedback adjustment)์ด๋ผํ•œ

๋‹ค. ๋ฐ˜๋ฉด์— ๊ณต์ •ํŽธ์ฐจ๋ฅผ ๊ด€์ธกํ•˜๊ธฐ ์ „ ์ž…๋ ฅ์š”์ธ (input factor)์˜ ๊ฐ’์„ ํ™•์ธํ•˜๊ณ  ๊ณต์ •ํŽธ์ฐจ๋ฅผ ์•Œ ์ˆ˜ ์žˆ๋‹ค๋ฉด

์ž…๋ ฅ์š”์ธ์„ ์ˆ˜์ •ํ•˜์—ฌ ๊ณต์ •ํŽธ์ฐจ๋ฅผ ์กฐ์ •ํ•˜๊ฒŒ ๋˜๋ฉฐ, ์ด๋ฅผ ํ”ผ๋“œํฌ์›Œ๋“œ์กฐ์ • (feedforward adjustment)์ด๋ผ

ํ•œ๋‹ค. ์‹ค์ œ ์ƒ์‚ฐํ˜„์žฅ์—์„œ๋Š” ๊ณต์ •์ž…๋ ฅ์š”์ธ์„ ๋งค ์‹œ๊ฐ„ ์ฃผ๊ธฐ์ ์œผ๋กœ ์ธก์ •ํ•˜๋Š” ๊ฒƒ์— ์–ด๋ ค์›€์ด ๋งŽ์ด ์žˆ์„ ์ˆ˜

โ€ ์ด ๋…ผ๋ฌธ์€ 2007๋…„๋„ ์ค‘์•™๋Œ€ํ•™๊ต ์—ฐ๊ตฌ์žฅํ•™๊ธฐ๊ธˆ ์ง€์›์— ์˜ํ•œ ๊ฒƒ์ž„.

1 ๊ต์‹ ์ €์ž: (156-756) ์„œ์šธํŠน๋ณ„์‹œ ๋™์ž‘๊ตฌ ํ‘์„๋™ 221, ์ค‘์•™๋Œ€ํ•™๊ต ์ˆ˜ํ•™ํ†ต๊ณ„ํ•™๋ถ€, ๊ต์ˆ˜.E-mail: [email protected]

2 (156-756) ์„œ์šธํŠน๋ณ„์‹œ ๋™์ž‘๊ตฌ ํ‘์„๋™ 221, ์ค‘์•™๋Œ€ํ•™๊ต ์ผ๋ฐ˜๋Œ€ํ•™์› ํ†ต๊ณ„ํ•™๊ณผ, ์„์‚ฌ๊ณผ์ •.

Page 2: Journal of the Korean Data

12 Changsoon Park ยท Sunggu Kwon

์žˆ์œผ๋ฉฐ, ๋˜ํ•œ ์ž…๋ ฅ ํ›„ ์ถœ๋ ฅ๊นŒ์ง€์˜ ๊ธฐํƒ€ ์ƒํ™ฉ๋ณ€ํ™”๋ฅผ ๊ณ ๋ คํ•˜๋ฉด, feedback adjustment๊ฐ€ ๊ฐ•๊ฑดํ•œ (robust)

๊ฒฐ๊ณผ๋ฅผ๊ฐ€์ ธ์˜ฌ์ˆ˜์žˆ๋‹ค.

ํ˜„์žฌ๊นŒ์ง€์˜ ๊ณต์ •ํŽธ์ฐจ๋ฅผ ๊ด€์ธกํ•˜์—ฌ ๋‹ค์Œ ์‹œ์ ์˜ ๊ณต์ •ํŽธ์ฐจ๋ฅผ ์˜ˆ์ธกํ•˜๊ธฐ ์œ„ํ•ด์„œ๋Š” ๊ณต์ •ํŽธ์ฐจ์— ๋Œ€ํ•œ ์‹œ๊ณ„์—ด๋ชจํ˜•์„ ์ •์˜ํ•˜๊ณ , ๊ทธ ๋‹ค์Œ์€ ์ˆ˜์ •๋Ÿ‰์„ ๊ฒฐ์ •ํ•˜๊ฒŒ ๋œ๋‹ค. ์ด ๋•Œ ์ˆ˜์ •์— ์‚ฌ์šฉ๋˜๋Š” ์š”์ธ์„ ์ˆ˜์ •๊ธฐ (con-

troller)๋ผ ํ•œ๋‹ค. ์˜ˆ๋ฅผ ๋“ค๋ฉด, ๋ฐ˜๋„์ฒด ์›จ์ดํผ ์ œ์กฐ๊ณผ์ •์˜ ํด๋ฆฌ์‹ค๋ฆฌ์ฝ˜ (polysilicon)์นจ์ „๊ณต์ •์—์„œ ์นจ์ „๋‘๊ป˜

( Qt )๋Š”ํ’ˆ์งˆํŠน์„ฑ์น˜์—ํ•ด๋‹นํ•˜๊ณ , ๋ชฉํ‘œ์น˜ ( T )๋ฅผ๋บ€๊ฐ’์€๊ณต์ •ํŽธ์ฐจ

Yt = Qt โˆ’ T

๊ฐ€ ๋œ๋‹ค. ๊ณต์ •ํŽธ์ฐจ๊ฐ€ ์–ด๋–ค ์‹œ๊ณ„์—ด ๋ชจํ˜• Zt๋ฅผ ๋”ฐ๋ฅธ๋‹ค๊ณ  ํ•˜์ž. ์ด๋•Œ t โˆ’ 1์‹œ์ ์—์„œ ๋‹จ์œ„ ์‹œ๊ฐ„๋‹น ์œ ์ž…๋˜๋Š”

์‹ค๋ฆฌ์ฝ˜๊ฐ€์Šค์˜ ์–‘ Xtโˆ’1์— ์˜ํ•ด t์‹œ์ ์—์„œ ์ƒ์‚ฐ๋˜๋Š” ์นจ์ „๋‘๊ป˜๊ฐ€ ฮฒXtโˆ’1๋งŒํผ ์ฆ๊ฐ€ํ•œ๋‹ค๊ณ  ํ•˜๋ฉด ์‹ค์ œ ๊ด€์ธก๋˜๋Š”๊ณต์ •ํŽธ์ฐจ๋Š”

Yt = ฮฒXtโˆ’1 + Zt

๊ฐ€ ๋œ๋‹ค. ์ด๋•Œ, ฮฒ๋ฅผ ์‹œ์Šคํ…œ ๊ฒŒ์ธ์ด๋ผ ํ•œ๋‹ค. ์œ„ ๋ชจํ˜•์—์„œ๋Š” ๋ฐ”๋กœ ํ•œ ์‹œ์  ์ „์˜ ์กฐ์ •์ด ๋‹ค์Œ ์‹œ์ ์—์„œ ๋ชจ

๋‘ ๋ฐ˜์˜๋˜๋Š” ๊ฒƒ์œผ๋กœ์„œ ๋ฐ˜์‘๋ชจํ˜•์ด๋ผ ํ•œ๋‹ค[Box์™€ Luceno (1997)]. ๋ฐ˜์‘๋ชจํ˜•์—์„œ๋Š” ๊ณต์ •์ง€์—ฐ์‹œ๊ฐ„ (sys-

tem delay)์ด 1์ธ๊ฒฝ์šฐ์ด๋ฉฐ, ์ด๋Š”๋Œ€๋ถ€๋ถ„์˜๊ณต์ •์‚ฐ์—…์—์„œํ”ํžˆ๋‚˜ํƒ€๋‚˜๋Š”ํ˜„์ƒ์ด๋‹ค.

๊ณต์ •ํŽธ์ฐจ๋ชจํ˜• Zt๊ฐ€ ์„ค์ •๋˜๋ฉด ์ˆ˜์ •๊ธฐ Xt๋Š” ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑ์ด ์ตœ์†Œํ™”ํ•  ์ˆ˜ ์žˆ๋„๋ก ์„ ํƒํ•˜๊ฒŒ ๋˜๋Š”๋ฐ ์ด๋Ÿฌ

ํ•œ์ˆ˜์ •๊ธฐ๋ฅผ MMSE (minimum mean squared error) ์ˆ˜์ •๊ธฐ๋ผํ•œ๋‹ค. ํŽธ์ฐจ๋ชจํ˜•์„

Zt = ฮตt + f(ฮตtโˆ’1, ฮตtโˆ’2, ยท ยท ยท )

๋‹จ, ฮตt๋Š”๋ฐฑ์ƒ‰์žก์Œ, N(0, ฯƒ2ฮต )๋กœ์ •์˜ํ•˜๋ฉด, E(Z2

t ) โ‰ฅ ฯƒ2ฮต์„๋งŒ์กฑํ•˜์—ฌ๊ณต์ •ํŽธ์ฐจ๋Š”

Yt = ฮตt

์ผ๋•Œ์ตœ์†ŒํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์ด๋œ๋‹ค. ์ด๋ฅผ์œ„ํ•ด์ˆ˜์ •๊ธฐ๋ฅผ

Xtโˆ’1 = โˆ’1

ฮฒf(ฮตtโˆ’1, ฮตtโˆ’2, ยท ยท ยท )

๋กœ ์„ค์ •ํ•˜๊ฒŒ ๋˜๋ฉด MMSE ์ˆ˜์ •๊ธฐ๊ฐ€ ๋œ๋‹ค. ์ด๋Ÿฌํ•œ MMSE ์ˆ˜์ •๊ธฐ๋Š” ๊ณต์ •ํŽธ์ฐจ๋ฅผ ๋ฐฑ์ƒ‰์žก์Œ (white

noise)์ด ๋˜๊ฒŒ ํ•œ๋‹ค๋Š” ์ ์—์„œ ์ตœ์ ์˜ ์ˆ˜์ •๊ธฐ๊ฐ€ ๋˜์ง€๋งŒ, ์‹ค์ œ๋กœ ๊ณต์ •๋ชจํ˜• Zt์˜ ์„ค์ •๊ณผ ๋ชจํ˜•์— ํฌํ•จ๋œ

๋‹ค์–‘ํ•œ ๋ชจ์ˆ˜์˜ ์ •๋ฐ€ํ•œ ์ถ”์ •์ด ์šฉ์ดํ•˜์ง€ ์•Š์€ ๋ฌธ์ œ์ ์ด ์žˆ๋‹ค. ๋ชจํ˜•์„ค์ •๊ณผ ๋ชจ์ˆ˜์ถ”์ •์ด ๋ถ€์ •ํ™•ํ•˜๊ฒŒ ๋˜๋ฉด

MMSE ์ˆ˜์ •๊ธฐ๋Š” ๋” ์ด์ƒ ์ตœ์ ์ด ๋˜์ง€ ์•Š์„ ๋ฟ ์•„๋‹ˆ๋ผ ์˜คํžˆ๋ ค ๋” ๋‚˜์œ ๊ฒฐ๊ณผ๋ฅผ ์ดˆ๋ž˜ํ•  ๊ฐ€๋Šฅ์„ฑ์ด ํฌ๊ฒŒ ๋œ

๋‹ค. ์ฆ‰, ๊ณต์ •๋ชจ์ˆ˜์˜ ๊ฐ’์ด ์ถ”์ •๋œ ๊ฐ’๊ณผ ์ฐจ์ด๊ฐ€ ๋‚˜๊ฑฐ๋‚˜, ๋˜๋Š” ๊ณต์ • ์ง„ํ–‰ ์ค‘์— ๋ชจ์ˆ˜์˜ ๊ฐ’์ด ๋ณ€ํ•˜๋ฉด ์• ์ดˆ์˜

MMSE ์ˆ˜์ •๊ธฐ์— ์˜ํ•œ ๊ณต์ •ํŽธ์ฐจ๋Š” ๋” ์ด์ƒ white noie๊ฐ€ ๋˜์ง€ ์•Š๊ณ  ํŽธ์ฐจ๋ถ„์‚ฐ์ด ์ฆ๊ฐ€ํ•  ๋ฟ๋งŒ ์•„๋‹ˆ๋ผ ์—ฐ

์†์ƒ๊ด€์„ ๊ฐ€์ง€๊ฒŒ ๋œ๋‹ค. ์ด๋Ÿฌํ•œ ํ˜„์ƒ์€ ์‹œ๊ฐ„์ด ํ๋ฆ„์— ๋”ฐ๋ผ ๋” ๊ทน๋‹จ์ ์ธ ๋ฐฉํ–ฅ์œผ๋กœ ํ๋ฅด๊ฒŒ ๋˜์–ด ์‹ฌ๊ฐํ•˜๊ฒŒํฐ๊ณต์ •ํŽธ์ฐจ๊ฐ€๋ฐœ์ƒํ• ์ˆ˜์žˆ๋‹ค.

์ด๋Ÿฌํ•œMMSE์ˆ˜์ •๊ธฐ์˜๋‹จ์ ์„๋ณด์™„ํ•˜๊ธฐ์œ„ํ•ด์ฃผ๋กœ์‚ฌ์šฉ๋˜๋Š”๊ฒƒ์ด EWMA (exponentially weighted

moving average) ์ˆ˜์ •๊ธฐ์ด๋‹ค (del Castillo, 1999; 2001). ์ด EWMA ์ˆ˜์ •๊ธฐ๋Š”๋ฐ˜๋„์ฒด์ œ์กฐ์—…์—์„œ๋„๋ฆฌ

์‚ฌ์šฉ๋˜๋Š” ๋‹จ์ˆœํ•˜๋ฉด์„œ๋„, ๋‹ค์–‘ํ•˜๊ฒŒ ๋ฐœ์ƒํ•˜๋Š” ๊ณต์ •์žก์Œ ํ™˜๊ฒฝ์—์„œ robustํ•œ ๊ฒฐ๊ณผ๋ฅผ ์–ป์„ ์ˆ˜ ์žˆ๋‹ค๊ณ  ์•Œ๋ ค์ ธ

์žˆ๋‹ค (Garcia์™€ Morari, 1985; Morari์™€ Zafiriou, 1989; Butler์™€ Stefani, 1994; Sachs ๋“ฑ, 1995; del

Castillo์™€ Hurwitz, 1997; Edgar ๋“ฑ, 2000; Moyne ๋“ฑ, 2000).

์ˆ˜์ •๊ธฐ์˜ ํŠน์„ฑ์€ ๋‘ ๊ฐ€์ง€๋กœ ๋‚˜ํƒ€๋‚œ๋‹ค. ์ฒซ์งธ๋Š” ๊ณต์ •ํŽธ์ฐจ์˜ ํฌ๊ธฐ์ด๊ณ , ๋‘˜์งธ๋Š” ์ˆ˜์ •๋Ÿ‰์˜ ๋ณ€ํ™”๋Ÿ‰์ด๋‹ค.

์ฆ‰, ๊ณต์ •ํŽธ์ฐจ๋Š” 0์— ๊ฐ€๊นŒ์ด ๋ถ„ํฌํ• ์ˆ˜๋ก ๋ฐ”๋žŒ์งํ•˜๋ฉฐ, ์ˆ˜์ •๋Ÿ‰์€ ๋ณ€ํ™”๊ฐ€ ํฌ์ง€ ์•Š์€ ๊ฒƒ์ด ์ข‹๋‹ค. ๊ณต์ •ํŽธ

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Modifications of single and double EWMA feedback controllers 13

์ฐจ์˜ ํŠน์„ฑ์€ ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท  (mean squared deviation, MSD)์œผ๋กœ ๋‚˜ํƒ€๋‚ด๊ณ , ์ˆ˜์ •๋Ÿ‰์˜ ๋ณ€ํ™”๋Š” ์ˆ˜์ •๋Ÿ‰

๋ถ„์‚ฐ (adjustment variance)์œผ๋กœ ๋‚˜ํƒ€๋‚ธ๋‹ค. ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ์œ„์ฃผ๋กœ ์ œ์•ˆ๋œ EWMA ์ˆ˜์ •๊ธฐ์—๋Š” ๋‹จ์ผ

EWMA ์ˆ˜์ •๊ธฐ์™€ ์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ๊ฐ€ ์žˆ๋‹ค. ์ด ๋…ผ๋ฌธ์—์„œ๋Š” ์ด ๋‘๊ฐ€์ง€ ์ˆ˜์ •๊ธฐ์— ๋Œ€ํ•ด ์•Œ์•„๋ณด๊ณ , ์ด

๋ฅผ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ ์ธก๋ฉด์—์„œ ๊ฐœ์„ ํ•  ์ˆ˜ ์žˆ๋„๋ก ๊ณ ์•ˆ๋œ ์ˆ˜์ •๊ธฐ๋ฅผ ์ œ์•ˆํ•œ๋‹ค. ์ด ์ œ์•ˆ๋œ ์ˆ˜์ •๊ธฐ๋Š” ์ˆ˜์ •๊ธฐ ์ž์ฒด

์— EWMA๋ฅผ์ ์šฉํ•˜๊ฒŒ๋œ๋‹ค.

2. EWMA ์ˆ˜์ •๊ธฐ

๊ณต์ •์žก์Œ Zt์™€์ˆ˜์ •๊ธฐ Xt ์—๋Œ€ํ•ด๊ณต์ •ํŽธ์ฐจ๋Š”์ผ๋ฐ˜์ ์œผ๋กœ๋‹ค์Œ๊ณผ๊ฐ™์ดํ‘œํ˜„๋œ๋‹ค.

Yt = ฮฒXtโˆ’1 + Zt (2.1)

๋‹จ, ฮฒ๋Š”์‹œ์Šคํ…œ๊ฒŒ์ธ.

๊ทธ๋ฆผ 2.1 ๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ

๊ทธ๋ฆผ 2.2 ์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ

์‹œ์Šคํ…œ ๊ฒŒ์ธ์€ ๊ณต์ •๊ฒฝํ—˜์„ ํ†ตํ•ด ์ถ”์ •๋  ์ˆ˜ ์žˆ์œผ๋ฉฐ, ํŠนํžˆ ์ˆ˜์ •๊ธฐ Xt์˜ ๊ฐ’์ด 0์— ๊ฐ€๊นŒ์šธ์ˆ˜๋ก ์ •๋ฐ€ํ•œ ์ถ”

์ •์ด๊ฐ€๋Šฅํ•˜๋‹ค. ์‹œ์Šคํ…œ๊ฒŒ์ธ์˜์ถ”์ •๊ฐ’์„ b๋ผํ•˜๋ฉด๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ๋Š”๋‹ค์Œ๊ณผ๊ฐ™์ด์ •์˜ํ•œ๋‹ค.

Xst = โˆ’

at

b(2.2)

๋‹จ, at = ฮป(Yt โˆ’ bXtโˆ’1) + (1 โˆ’ ฮป)atโˆ’1 , 0 โ‰บ ฮป โ‰ค 1 ,

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14 Changsoon Park ยท Sunggu Kwon

๊ทธ๋ฆผ 2.3 ์ˆ˜์ •๊ธฐ XXst

๊ทธ๋ฆผ 2.4 ์ˆ˜์ •๊ธฐ XXdt

a0 = 0 .

๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ์˜ ์ง„ํ–‰๊ณผ์ •์„ ๋ธ”๋ก ๊ทธ๋ฆผ์œผ๋กœ ํ‘œํ˜„ํ•˜๋ฉด ๊ทธ๋ฆผ 2.1๊ณผ ๊ฐ™๋‹ค. ๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ

๋Š” EWMA ์ˆ˜์ •๊ธฐ์˜์žฅ์ ์ธ๊ฐ•๊ฑด์„ฑ์„์œ ์ง€ํ•˜์ง€๋งŒ์ ๊ทผํŽธ์ฐจ (asymptotic bias)๊ฐ€์‚ฌ๋ผ์ง€์ง€์•Š๋Š”๋ฌธ์ œ์ 

์„๊ฐ€์ง€๊ณ ์žˆ๋‹ค. ์ด๋ฅผ๊ฐœ์„ ํ•˜๊ธฐ์œ„ํ•ด์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ๊ฐ€์ œ์•ˆ๋˜์—ˆ์œผ๋ฉฐ๋‹ค์Œ๊ณผ๊ฐ™์ด์ •์˜ํ•œ๋‹ค.

Xdt = โˆ’

at + Dt

b(2.3)

๋‹จ, at = ฮป1(Yt โˆ’ bXtโˆ’1) + (1 โˆ’ ฮป1)atโˆ’1 , 0 โ‰บ ฮป1 โ‰ค 1

Dt = ฮป2(Yt โˆ’ bXtโˆ’1 โˆ’ atโˆ’1) + (1 โˆ’ ฮป2)Dtโˆ’1 , 0 โ‰บ ฮป2 โ‰ค 1

a0 = 0 , D0 = 0 .

์ด์ค‘ EWMA์ˆ˜์ •๊ธฐ์˜์ง„ํ–‰๊ณผ์ •์„๋ธ”๋ก๊ทธ๋ฆผ์œผ๋กœํ‘œํ˜„ํ•˜๋ฉด๊ทธ๋ฆผ 2.2์™€๊ฐ™๋‹ค (del Castillo, 2002). ์ด

์ค‘ EWMA ์ˆ˜์ •๊ธฐ๋ฅผ ์‚ฌ์šฉํ•˜๋ฉด ์ ์šฉ๊ณผ์ •์€ ์ข€ ๋” ๋ณต์žกํ•ด์ง€์ง€๋งŒ ์ ๊ทผํŽธ์ฐจ๊ฐ€ ์ค„์–ด๋“ค์–ด ๋‹จ์ผ EWMA ์ˆ˜

์ •๊ธฐ์˜๋‹จ์ ์„๋ณด์™„ํ•จ์ด์•Œ๋ ค์ ธ์žˆ๋‹ค.

์ด ๋‘๊ฐ€์ง€ EWMA ์ˆ˜์ •๊ธฐ๋Š” ๊ณต์ •ํŽธ์ฐจ๋ฅผ ์ค„์—ฌ ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์ด ์ž‘์•„์ง€๋„๋ก ๊ณ ์•ˆ๋œ ๋„๊ตฌ์ด๋‹ค. ๊ณต์ •์ˆ˜

์ •ํ™œ๋™์—์„œ ๊ณ ๋ คํ•ด์•ผํ•˜๋Š” ํŠน์„ฑ์€ ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท  ์ด์™ธ์— ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์ด ์žˆ๋‹ค. ์ˆ˜์ •๋Ÿ‰์€ ๋ฐ”๋กœ ์ „ ์‹œ์ ๊นŒ์ง€

์˜์ด์ˆ˜์ •๋Ÿ‰์—๊ฐ์‹œ์ ์—์„œ์ถ”๊ฐ€ํ•˜๋Š”์ˆ˜์ •๋Ÿ‰์œผ๋กœ์„œ, t์‹œ์ ์—์„œ์˜์ˆ˜์ •๋Ÿ‰์€

โ–ฝXt = Xt โˆ’ Xtโˆ’1

์ด ๋œ๋‹ค. ์ฆ‰, ์ˆ˜์ •๋Ÿ‰์ด๋ผ ํ•จ์€ ๋งค ์‹œ์ ๋งˆ๋‹ค ์ ์šฉํ•˜๋Š” ๊ณต์ •์ž…๋ ฅ์š”์ธ์˜ ๋ณ€ํ™”๋Ÿ‰์„ ๋‚˜ํƒ€๋‚ด๋ฉฐ ์ด๋Š” ์ž…๋ ฅ์š”์ธ

์—์˜ํ•œ๋น„์šฉ์„์œ ๋ฐœํ•˜๊ฒŒ๋œ๋‹ค.

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Modifications of single and double EWMA feedback controllers 15

๋‹จ์ผ EWMA์™€์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ๋Š”๊ฒฝ์šฐ์—๋”ฐ๋ผ์„œ๋Š”์ˆ˜์ •๋Ÿ‰ | โ–ฝ Xt|๊ฐ€์ƒ๋‹นํžˆ์ปค์งˆ์ˆ˜์žˆ๋Š”๊ฐ€๋Šฅ์„ฑ์„๊ฐ€์ง€๊ณ ์žˆ์–ดํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์—์˜ํ•œ๋น„์šฉ์€์ค„์ด๋”๋ผ๋„, ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์—์˜ํ•œ๋น„์šฉ์ด์ปค์ง€๊ฒŒ๋˜์–ด๊ถ๊ทน์ ์œผ๋กœ ๊ณต์ •๋น„์šฉ์„ ์ค„์ด๋Š”๋ฐ ๋น„ํšจ๊ณผ์ ์ผ ์ˆ˜ ์žˆ๋‹ค. ์ด๋Ÿฌํ•œ ๋ฌธ์ œ์ ์„ ๊ฐœ์„ ํ•˜๊ธฐ ์œ„ํ•ด ๋‹ค์Œ๊ณผ ๊ฐ™์€ ์ˆ˜์ •๋œ

์ˆ˜์ •๊ธฐ๋ฅผ์ •์˜ํ•œ๋‹ค.

๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ์—๋Œ€ํ•ด์ˆ˜์ •๋œ์ˆ˜์ •๊ธฐ๋Š”๋‹ค์Œ๊ณผ๊ฐ™์ด์ •์˜ํ•œ๋‹ค.

XXst = l ยท Xs

t + (1 โˆ’ l)XXstโˆ’1 (2.4)

๋‹จ, Xst ๋Š”๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ, XXs

0 = 0.

์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ์—๋Œ€ํ•ด์ˆ˜์ •๋œ์ˆ˜์ •๊ธฐ๋Š”๋‹ค์Œ๊ณผ๊ฐ™์ด์ •์˜ํ•œ๋‹ค.

XXdt = l ยท Xd

t + (1 โˆ’ l)XXdtโˆ’1 (2.5)

๋‹จ, Xdt๋Š”์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ, XXd

0 = 0 .

๋‘ ์ˆ˜์ •๊ธฐ XXst ์™€ XXd

t ๋Š” Xst์™€ Xd

t์— ๊ฐ๊ฐ EWMA๋ฅผ ์ทจํ•œ ๊ฒƒ์œผ๋กœ Xst์™€ Xd

t๋ณด๋‹ค ๋ถ€๋“œ๋Ÿฌ์šด ๋ณ€ํ™”๊ฐ€

์˜ˆ์ƒ๋˜์–ด ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์ด ์ค„์–ด๋“ค ๊ฒƒ์ž„์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค. ์ˆ˜์ •๊ธฐ XXst , XXd

t ์˜ ์ง„ํ–‰๊ณผ์ •์€ ๊ทธ๋ฆผ 2.3๊ณผ ๊ทธ๋ฆผ2.4์˜ ๋ธ”๋ก ๊ทธ๋ฆผ์œผ๋กœ ํ‘œํ˜„๋œ๋‹ค. ๋‹ค์Œ์ ˆ๋ถ€ํ„ฐ๋Š” ๋‹ค์–‘ํ•œ ๊ณต์ •ํŽธ์ฐจ๋ชจํ˜•์— ๋Œ€ํ•ด ์œ„์—์„œ ์†Œ๊ฐœํ•œ ๋„ค ์ข…๋ฅ˜์˜ ์ˆ˜

์ •๊ธฐ Xst , Xd

t , XXst , XXd

t ๋ฅผ ์‚ฌ์šฉํ•  ๋•Œ ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ๊ณผ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์˜ ํŠน์„ฑ์„ ์‹œ๊ฐ„์˜ ํ๋ฆ„์— ๋”ฐ๋ผ ๋น„๊ต๋ถ„์„ํ•˜์˜€๋‹ค. ๊ณต์ •ํŽธ์ฐจ๋ชจํ˜•์œผ๋กœ๋Š” Shewhart ๋ชจํ˜•, ํ๋ฆ„ (drift)์ด ์žˆ๋Š” Shewhart ๋ชจํ˜•, IMA (1, 1) ๋ชจ

ํ˜•, ํ๋ฆ„์ด ์žˆ๋Š” IMA (1, 1)๋ชจํ˜•์œผ๋กœ ๋„ค๊ฐ€์ง€ ๋ชจํ˜•์„ ์„ค์ •ํ•˜์˜€๋‹ค. ํŠนํžˆ IMA (1, 1) ๋ชจํ˜•์€ ์‚ฐ์—…์ƒ์‚ฐ๊ณต

์ •์—์„œ๋ฐœ์ƒํ•˜๋Š”๊ณต์ •ํŽธ์ฐจ๋ฅผ์ž˜๋‚˜ํƒ€๋‚ด๋Š”๊ฒƒ์œผ๋กœ์•Œ๋ ค์ ธ์žˆ๋‹ค (Box๊ณผ Kramer, 1992).

3. Shewhart ๊ณต์ •๋ชจํ˜•

Shewhart ๋ชจํ˜•์„๋”ฐ๋ฅด๋Š”๊ณต์ •์žก์Œ์€๋‹ค์Œ๊ณผ๊ฐ™๋‹ค.

Zt = ฮฑ + ฮตt (3.1)

๋‹จ, ฮตt๋Š”๋ฐฑ์ƒ‰์žก์Œ N(0, ฯƒ2ฮต ).

Shewhart ๊ณต์ •๋ชจํ˜•์—์ˆ˜์ •๊ธฐ๋ฅผ์‚ฌ์šฉํ•œ๊ณต์ •ํŽธ์ฐจ๋Š”๋‹ค์Œ๊ณผ๊ฐ™์ดํ‘œํ˜„๋œ๋‹ค.

Yt = ฮฑ + ฮฒXtโˆ’1 + ฮตt (3.2)

์ด๋•Œ ฮฑ๋Š” ๊ณต์ •์˜ ์น˜์šฐ์นจ (offset)์„ ๋‚˜ํƒ€๋‚ด๋Š” ์ƒ์ˆ˜์ด๋‹ค. ์ฃผ์–ด์ง„ ์‹œ๊ฐ„ t๊นŒ์ง€์˜ ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ๊ณผ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์€๋ชจ๋‘๋ชจ์˜์‹คํ—˜์„ํ†ตํ•ด๊ณ„์‚ฐํ•˜์˜€์œผ๋ฉฐ, ์‚ฌ์šฉ๋œ๊ณต์ •๋ชจ์ˆ˜๋Š”๋‹ค์Œ๊ณผ๊ฐ™๋‹ค.

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1

์œ„์˜ ๋ชจ์ˆ˜๊ฐ’์€ ๊ณต์ •์‚ฌ์ •์— ๋”ฐ๋ผ ๋‹ค์–‘ํ•œ ๊ฐ’์„ ๊ฐ€์งˆ ์ˆ˜ ์žˆ์œผ๋‚˜ ์—ฌ๊ธฐ์„œ๋Š” ๊ทธ ์ค‘ ํ•œ ๊ฐ€์ง€ ์˜ˆ๋ฅผ ์„ ํƒํ•˜์˜€๋‹ค.

์‹œ์Šคํ…œ ๊ฒŒ์ธ์˜ ์ถ”์ •๊ฐ’์€ b = (1.2)ฮฒ๋กœ ๊ฐ€์ •ํ•˜์˜€๊ณ  (์ถ”์ • ๊ฐ’์ด 20%์ •๋„์˜ ์˜ค์ฐจ๋ฅผ ํฌํ•จํ•œ๋‹ค๊ณ  ๊ฐ€์ •), ๊ฐ

์ˆ˜์ •๊ธฐ์—์„œ์‚ฌ์šฉํ•˜๋Š”์ˆ˜์ •๊ธฐ๋ชจ์ˆ˜๋Š”

ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๋กœ ์„ค์ •ํ•˜์˜€๋‹ค. EWMA ์ˆ˜์ •๊ธฐ์˜ ๊ฐ€์ค‘์น˜ ฮป1, ฮป2, l์€ 0์— ๊ฐ€๊นŒ์šธ์ˆ˜๋ก ์ˆ˜์ •๋Ÿ‰๋ณ€๋™์ด ์ž‘์•„์ง€๋Š” ๊ฒฝํ–ฅ์ด ์žˆ

์œผ๋ฉฐ๊ทธ์ค‘๋Œ€ํ‘œ์ ์œผ๋กœ์‚ฌ์šฉ๋˜๋Š” 0.1 ๋˜๋Š” 0.2๋ฅผ์‚ฌ์šฉํ•˜์˜€๋‹ค.

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๊ทธ๋ฆผ 3.1 ๊ณต์ •ํŽธ์ฐจ ( Yt )ํ‰๊ท  : ๋ชจํ˜• (3. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 3.2 ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑ ( Y 2t )ํ‰๊ท  : ๋ชจํ˜• (3. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 3.1์™€ ๊ทธ๋ฆผ 3.2์€ ๋„ค ์ข…๋ฅ˜์˜ ์ˆ˜์ •๊ธฐ์— ๋Œ€ํ•œ t์‹œ์ ๊นŒ์ง€์˜ ๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค. ๋˜ํ•œ๊ทธ๋ฆผ 3.3์€์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค.

๊ทธ๋ฆผ 3.1์™€ ๊ทธ๋ฆผ 3.2์€ ์‹œ๊ฐ„์˜ ํ๋ฆ„์— ๋”ฐ๋ผ ๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ๋‚˜ํƒ€๋‚ธ ๊ทธ๋ฆผ์ด๋‹ค.

0์— ๊ฐ€๊นŒ์šธ์ˆ˜๋ก ํšจ๊ณผ์ ์ธ ๊ณต์ •์ด๋ผ ํ•  ์ˆ˜ ์žˆ๋Š”๋ฐ, ๊ณต์ •์ด ์˜ค๋ž˜ ์ง€์†๋˜๋ฉด (t โ‰ฅ 500) ์ˆ˜์ •๊ธฐ๊ฐ„์˜ ์ฐจ์ด

๋Š” ๋ฏธ๋ฏธํ•จ์„ ๋ณผ ์ˆ˜ ์žˆ๋‹ค. ๋‹ค๋งŒ, ๊ณต์ •์ดˆ๊ธฐ์˜ ๊ฒฝ์šฐ ์ˆ˜์ •๊ธฐ Xdt๊ฐ€ ๊ฐ€์žฅ ํšจ์œจ์ ์ž„์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค. ๊ทธ๋ฆผ 3.3์€

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Modifications of single and double EWMA feedback controllers 17

๊ทธ๋ฆผ 3.3 ์ˆ˜์ •๋Ÿ‰ ( โ–ฝXt )๋ถ„์‚ฐ : ๋ชจํ˜• (3. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

์ˆ˜์ •๊ธฐ๋ณ„ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„ ๋‚˜ํƒ€๋‚ธ ๊ฒƒ์ธ๋ฐ, 0์— ๊ฐ€๊นŒ์šธ์ˆ˜๋ก ์ข‹์€ ์ˆ˜์ •๊ธฐ๋ผ ํ•  ์ˆ˜ ์žˆ๋‹ค. ๊ทธ๋ฆผ 3.3์—์„œ ์ˆ˜์ •

๋Ÿ‰ ๋ถ„์‚ฐ์€ ์ผ์ •๊ฐ’์— ์ ๊ทผ์ ์œผ๋กœ ์ˆ˜๋ ดํ•จ์„ ์•Œ ์ˆ˜ ์žˆ์œผ๋‚˜, ๊ทธ ์ˆ˜๋ ดํ•˜๋Š” ๊ฐ’์ด ๊ฐ ์ˆ˜์ •๊ธฐ์— ๋”ฐ๋ผ ํฌ๊ฒŒ ์ฐจ์ด๊ฐ€ ๋‚จ์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค. ์ˆ˜์ •๊ธฐ Xd

t์˜ ๊ฒฝ์šฐ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์ด ๋‹ค๋ฅธ ์ˆ˜์ •๊ธฐ์— ๋น„ํ•ด ๋งค์šฐ ํฐ ๊ฒƒ์„ ๋ณผ ์ˆ˜ ์žˆ๋‹ค. ๋ฐ˜

๋ฉด ์ˆ˜์ •๊ธฐ Xst , XXs

t , XXdt ์˜ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์€ ๋น„์Šทํ•˜์ง€๋งŒ XXs

t ๊ฐ€ ๊ฐ€์žฅ ์ž‘๊ณ , ๊ทธ ๋‹ค์Œ์€ XXdt , Xs

t ์ˆœ์œผ๋กœ

์กฐ๊ธˆ์”ฉ์ปค์ง์„์•Œ์ˆ˜์žˆ๋‹ค.

4. ํ๋ฆ„์ด ์žˆ๋Š” Shewhart ๊ณต์ •๋ชจํ˜•

Shewhart ๊ณต์ •๋ชจํ˜•์—์„œํ๋ฆ„์ด์ถ”๊ฐ€๋œ๋ชจํ˜•์œผ๋กœ์ˆ˜์ •๊ธฐ์—์‚ฌ์šฉํ•œ๊ณต์ •ํŽธ์ฐจ๋Š”๋‹ค์Œ๊ณผ๊ฐ™๋‹ค.

Yt = ฮฑ + ฮฒXtโˆ’1 + ฮดt + ฮตt (4.1)

์ด๋•Œ, ฮด๋Š”๋‹จ์œ„์‹œ๊ฐ„๋‹น๋ณ€ํ™”๋Ÿ‰์œผ๋กœ์„œํ๋ฆ„์—ํ•ด๋‹น๋œ๋‹ค. ๋ชจ์˜์‹คํ—˜์—์‚ฌ์šฉ๋œ๊ณต์ •๋ชจ์ˆ˜์™€์ˆ˜์ •๊ธฐ๋ชจ์ˆ˜๋Š” 3์ ˆ

๊ณผ๋™์ผํ•˜๋ฉฐํ๋ฆ„์€

ฮด = 0.1

๋กœ ์„ค์ •ํ•˜์˜€๋‹ค. ํ๋ฆ„์€ ๋งค์‹œ๊ฐ„ ๊พธ์ค€ํžˆ ๋ฐœ์ƒํ•˜๋Š” ๊ณต์ •์ˆ˜์ค€์˜ ๋ณ€ํ™”๋Ÿ‰์„ ์˜๋ฏธํ•˜๋ฏ€๋กœ ฯƒฮต = 1์— ๋น„ํ•ด ์ž‘์€

๊ฐ’์„ ์„ ํƒํ•˜์˜€๋‹ค. ๊ทธ๋ฆผ 4.1๊ณผ ๊ทธ๋ฆผ 4.2๋Š” ๋„ค ์ข…๋ฅ˜์˜ ์ˆ˜์ •๊ธฐ์— ๋Œ€ํ•œ t์‹œ์ ๊นŒ์ง€์˜ ๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค. ๋˜ํ•œ๊ทธ๋ฆผ 4.3์€์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค.

๊ทธ๋ฆผ 4.1๊ณผ ๊ทธ๋ฆผ 4.2์—์„œ๋Š” ์‹œ๊ฐ„์˜ ํ๋ฆ„์— ๋”ฐ๋ผ ๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ๋‚˜ํƒ€๋‚ธ ๊ทธ๋ฆผ์ด๋‹ค. ๊ณต์ •์ด ์˜ค๋ž˜ ์ง€์†๋˜์–ด๋„ (t โ‰ฅ 500) ๊ฐ ์ˆ˜์ •๊ธฐ๊ฐ„ ์ผ์ •ํ•œ ์ฐจ์ด๋ฅผ ์œ ์ง€ํ•˜๋Š” ๊ฒƒ์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค. ๊ทธ๋ฆผ 4.3์—์„œ ์ˆ˜์ •๊ธฐ๋ณ„ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์€ Shewhart๋ชจํ˜•์˜ ๊ฒฐ๊ณผ์™€ ๋™์ผํ•˜๊ฒŒ ์ˆ˜์ •๊ธฐ Xd

t๊ฐ€ ๊ฐ€์žฅ ํฌ๋ฉฐ, ์ˆ˜์ •

๊ธฐ XXst , XXd

t ์˜๋ถ„์‚ฐ์ด์ž‘์Œ์„์•Œ์ˆ˜์žˆ๋‹ค.

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๊ทธ๋ฆผ 4.1 ๊ณต์ •ํŽธ์ฐจ ( Yt )ํ‰๊ท  : ๋ชจํ˜• (4. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮด = 0.1, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 4.2 ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑ ( Y 2t )ํ‰๊ท  : ๋ชจํ˜• (4. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮด = 0.1, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

5. IMA (1, 1) ๋ชจํ˜•

์ˆ˜์ •๊ธฐ๋ฅผ์‚ฌ์šฉํ•œ๊ณต์ •ํŽธ์ฐจ๋Š”๋‹ค์Œ๊ณผ๊ฐ™๋‹ค.

Yt = ฮฑ + ฮฒXtโˆ’1 + Nt (5.1)

๋‹จ, Nt = Ntโˆ’1 โˆ’ ฮธฮตtโˆ’1 + ฮตt .

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Modifications of single and double EWMA feedback controllers 19

๊ทธ๋ฆผ 4.3 ์ˆ˜์ •๋Ÿ‰ ( โ–ฝXt )๋ถ„์‚ฐ : ๋ชจํ˜• (4. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮด = 0.1, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 5.1 ๊ณต์ •ํŽธ์ฐจ ( Yt )ํ‰๊ท  : ๋ชจํ˜• (5. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮธ = 0.7, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

์ด๋•Œ, Nt๋Š” IMA (1, 1)๋ชจํ˜•์„ ๋”ฐ๋ฅด๋ฉฐ ฮธ๋Š” ์ด๋™ํ‰๊ท  (moving average)๋ชจ์ˆ˜์ด๋‹ค. ๋ชจ์˜์‹คํ—˜์—๋Š” ์•ž ์ ˆ

๊ณผ๋™์ผํ•œ๋ชจ์ˆ˜๋“ค์ด์‚ฌ์šฉ๋˜์—ˆ์œผ๋ฉฐ, ์ด๋™ํ‰๊ท ๋ชจ์ˆ˜ ฮธ๋Š”

ฮธ = 0.7

๋กœ ์„ค์ •ํ•˜์˜€๋‹ค. ์ด๋™ํ‰๊ท ๋ชจ์ˆ˜๋Š” Box์™€ Luceno (1997)์— ์˜ํ•˜๋ฉด ์ผ๋ฐ˜์ ์œผ๋กœ (0.3, 0.7)์‚ฌ์ด์˜ ๊ฐ’์„ ๊ฐ€

์ง€๋Š” ๊ฒƒ์œผ๋กœ ์•Œ๋ ค์ ธ ์žˆ์œผ๋ฉฐ ฮธ = 1 ์ด๋ฉด ๋ฐฑ์ƒ‰์žก์Œ์ด ๋˜๊ณ  ฮธ = 0 ์ด๋ฉด ํ™•๋ฅ ๋ณดํ–‰ (random walk)์ด ๋œ๋‹ค.

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20 Changsoon Park ยท Sunggu Kwon

๊ทธ๋ฆผ 5.2 ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑ ( Y 2t )ํ‰๊ท  : ๋ชจํ˜• (5. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮธ = 0.7, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 5.3 ์ˆ˜์ •๋Ÿ‰ ( โ–ฝXt )๋ถ„์‚ฐ : ๋ชจํ˜• (5. 1)

ฮฑ = 5, ฮฒ = 0.2, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮธ = 0.7, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

์—ฌ๊ธฐ์„œ๋Š”ํ™•๋ฅ ๋ณดํ–‰๋ณด๋‹ค๋Š”์ƒ๋Œ€์ ์œผ๋กœ๋ณ€๋™์ด์ ์€๊ฒฝ์šฐ๋ฅผ๊ณ ๋ คํ•˜์˜€๋‹ค.

๊ทธ๋ฆผ 5.1๊ณผ ๊ทธ๋ฆผ 5.2๋Š” ๋„ค ์ข…๋ฅ˜์˜ ์ˆ˜์ •๊ธฐ์— ๋Œ€ํ•œ t์‹œ์ ๊นŒ์ง€์˜ ๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค. ๋˜ํ•œ๊ทธ๋ฆผ 5.3์€์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค.

๊ทธ๋ฆผ 5.1๊ณผ๊ทธ๋ฆผ 5.2๋ฅผ๋ณด๋ฉด๊ณต์ •์˜์‹œ๊ฐ„์ด์ง€๋‚ ์ˆ˜๋ก์ˆ˜์ •๊ธฐ๋ณ„์ฐจ์ด๊ฐ€๊ฑฐ์˜์—†์–ด์ง์„๋ณผ์ˆ˜์žˆ์œผ๋ฉฐ, ๊ทธ๋ฆผ 5.3์—์„œ ์ˆ˜์ •๊ธฐ Xd

t๊ฐ€ ๋ถ„์‚ฐ์ด ๊ฐ€์žฅ ํฐ ๊ฒƒ์„ ์•Œ ์ˆ˜ ์žˆ๋‹ค. ์ „์ฒด์ ์ธ ๊ฒฝํ–ฅ์€ Shewhart ๋ชจํ˜•๊ณผ ์œ ์‚ฌํ•จ์„

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Modifications of single and double EWMA feedback controllers 21

์•Œ ์ˆ˜ ์žˆ๋Š”๋ฐ, ์ด๋Š” ๊ฐ ์ˆ˜์ •๊ธฐ๊ฐ€ ์žก์Œ๋ชจํ˜•์ด ๋ณ€ํ•˜์—ฌ๋„ ์ˆ˜์ •์„ ํ†ตํ•ด ์žก์Œ์˜ ํšจ๊ณผ๋ฅผ ์ ์ ˆํžˆ ์ œ๊ฑฐํ•˜๊ณ  ์žˆ์Œ

์„์•Œ์ˆ˜์žˆ๋‹ค.

6. ํ๋ฆ„์ด ์žˆ๋Š” IMA (1, 1) ๋ชจํ˜•

์ˆ˜์ •๊ธฐ๋ฅผ์‚ฌ์šฉํ•œ๊ณต์ •ํŽธ์ฐจ๋Š”๋‹ค์Œ๊ณผ๊ฐ™๋‹ค.

Yt = ฮฑ + ฮฒXtโˆ’1 + Nt (6.1)

์ฃผ์–ด์ง„ ์‹œ๊ฐ„ t๊นŒ์ง€์˜ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ๊ณผ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์€ ๋ชจ๋‘ ๋ชจ์˜์‹คํ—˜์„ ํ†ตํ•ด ๊ณ„์‚ฐ ํ•˜์˜€์œผ๋ฉฐ, ์‚ฌ์šฉ๋œ

๊ณต์ •๋ชจ์ˆ˜๋Š”์•ž์ ˆ์—์„œ์™€๋™์ผํ•˜๊ฒŒ์‚ฌ์šฉ๋˜์—ˆ๋‹ค. ์ฆ‰

ฮฑ = 5, ฮฒ = 0.2, ฮด = 0.1, ฮธ = 0.7, ฯƒฮต = 1

b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 6.1์™€๊ทธ๋ฆผ 6.2๋Š”๋„ค์ข…๋ฅ˜์˜์ˆ˜์ •๊ธฐ์—๋Œ€ํ•œ t ์‹œ์ ๊นŒ์ง€์˜๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค. ๋˜ํ•œ๊ทธ๋ฆผ 6.3๋Š”์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„๋‚˜ํƒ€๋‚ด๊ณ ์žˆ๋‹ค.

๊ทธ๋ฆผ 6.1 ๊ณต์ •ํŽธ์ฐจ ( Yt )ํ‰๊ท  : ๋ชจํ˜• (6. 1)

ฮฑ = 5, ฮฒ = 0.2, ฮด = 0.1, ฮธ = 0.7, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 6.1์™€ ๊ทธ๋ฆผ 6.2์— ์˜ํ•˜๋ฉด ์ˆ˜์ •๊ธฐ Xdt๋Š” ๊ณต์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ์ค„์ด๋Š”๋ฐ ํšจ๊ณผ์ ์ด

์ง€๋งŒ, ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์ด ๋‹ค๋ฅธ ์„ธ๊ฐ€์ง€ ์ˆ˜์ •๊ธฐ์— ๋น„ํ•ด ์ปค์ง€๋Š” ๋ฌธ์ œ์ ์„ ๊ฐ€์ง€๊ณ  ์žˆ๋‹ค. ๋˜ํ•œ ์ˆ˜์ •๊ธฐ XXdt ๋Š” ๊ณต

์ •ํŽธ์ฐจํ‰๊ท ๊ณผ ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„ ์กฐ๊ธˆ ์ฆ๊ฐ€์‹œํ‚ค๋Š” ๋ฌธ์ œ๊ฐ€ ์žˆ์ง€๋งŒ, ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„ ์ค„์ด๋Š” ํšจ๊ณผ๊ฐ€ ์žˆ์Œ์„

์•Œ์ˆ˜์žˆ๋‹ค. ์ด๊ฒฐ๊ณผ๋Š”ํ๋ฆ„์ด์žˆ๋Š” Shewhart ๋ชจํ˜•์˜๊ฒฐ๊ณผ์™€๋งค์šฐ์œ ์‚ฌํ•จ์„์•Œ์ˆ˜์žˆ๋‹ค. ํŠนํžˆํ๋ฆ„์˜ํšจ

๊ณผ๋Š”์‹œ๊ฐ„์ด์ง€๋‚˜๋„์—ฌ์ „ํžˆ๊ณต์ •์—๋‚จ์•„์žˆ์Œ์„์•Œ์ˆ˜์žˆ๋‹ค.

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22 Changsoon Park ยท Sunggu Kwon

๊ทธ๋ฆผ 6.2 ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑ ( Y 2t )ํ‰๊ท  : ๋ชจํ˜• (6. 1)

ฮฑ = 5, ฮฒ = 0.2, ฮด = 0.1, ฮธ = 0.7, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

๊ทธ๋ฆผ 6.3 ์ˆ˜์ •๋Ÿ‰ ( โ–ฝXt )๋ถ„์‚ฐ : ๋ชจํ˜• (6. 1)

ฮฑ = 5, ฮฒ = 0.2, ฮด = 0.1, ฮธ = 0.7, ฯƒฮต = 1, b = (1.2)ฮฒ, ฮป = ฮป1 = ฮป2 = 0.1, l = 0.2

7. ๊ฒฐ๋ก 

์ด ๋…ผ๋ฌธ์—์„œ๋Š” ๊ธฐ์กด์˜ ๋‹จ์ผ EWMA ์ˆ˜์ •๊ธฐ์™€ ์ด๋ฅผ ๋ณด์™„ํ•œ ์ด์ค‘ EWMA ์ˆ˜์ •๊ธฐ์— ๋Œ€ํ•ด ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ

์„ ์ค„์ผ ์ˆ˜ ์žˆ๋„๋ก ์ˆ˜์ •๋œ ์ˆ˜์ •๊ธฐ๋ฅผ ์ œ์•ˆํ•˜์˜€๋‹ค. ์ˆ˜์ •๋œ ์ˆ˜์ •๊ธฐ๋Š” ์ˆ˜์ •๋˜์ง€ ์•Š์€ ๊ธฐ์กด์˜ ์ˆ˜์ •๊ธฐ๋ณด๋‹ค ๊ณต

์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์ด ์ฆ๊ฐ€ํ•˜์ง€๋งŒ, ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์€ ์ค„์ผ ์ˆ˜ ์žˆ์Œ์„ ๋ณด์˜€๋‹ค. ์ˆ˜์ •ํ™œ๋™์— ์†Œ์š”๋˜๋Š” ๊ณต์ •๋น„์šฉ์€

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Modifications of single and double EWMA feedback controllers 23

๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ๊ณผ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์˜์„ ํ˜•๊ฒฐํ•ฉ์œผ๋กœ

E(Y 2t ) + ฮณ ยท V ar(โ–ฝXt)

์™€ ๊ฐ™์ด ํ‘œํ˜„๋œ๋‹ค. ์ด๋•Œ ฮณ๋Š” ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์— ๋Œ€ํ•œ ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์˜ ์ƒ๋Œ€์  ๋น„์šฉํฌ๊ธฐ๋ฅผ ์˜๋ฏธํ•œ๋‹ค. ๋”ฐ

๋ผ์„œ ฮณ๊ฐ€ ํฌ๋ฉด ์ˆ˜์ •๋Ÿ‰๋ถ„์‚ฐ์„ ์ค„์ผ ์ˆ˜ ์žˆ๋Š” ๊ณต์ •์„ค๊ณ„๊ฐ€ ํ•„์š”ํ•˜๊ณ , ๋ฐ˜๋ฉด์— ฮณ๊ฐ€ ์ž‘์œผ๋ฉด ๊ณต์ •ํŽธ์ฐจ์ œ๊ณฑํ‰๊ท ์„์ค„์ด๋„๋ก์„ค๊ณ„ํ•˜์—ฌ์•ผ๊ณต์ •๋น„์šฉ์„์ตœ์†Œํ™”ํ• ์ˆ˜์žˆ๋‹ค. ์ฆ‰, ์ˆ˜์ •๊ธฐ์˜์„ ํƒ์€์ƒ๋Œ€์ ๋น„์šฉํฌ๊ธฐ ฮณ์—๋”ฐ๋ผ

์„ ํƒ์ ์œผ๋กœ์‚ฌ์šฉํ• ํ•„์š”๊ฐ€์žˆ๋‹ค๊ณ ํŒ๋‹จํ•˜๊ฒŒ๋œ๋‹ค.

์ด ์—ฐ๊ตฌ์—์„œ๋Š” ์ˆ˜์ •๊ธฐ์˜ ์ ๊ทผ์  ํŠน์„ฑ ๋Œ€์‹  ์œ ํ•œํ•œ ์‹œ๊ฐ„๋ฒ”์œ„์—์„œ ํŠน์„ฑ์„ ๋น„๊ตํ•˜์˜€๋Š”๋ฐ, ์‹ค์ œ ์ƒ์‚ฐ ๊ณต

์ •์—์„œ๋Š” ๋™์ผ ๊ทœ๊ฒฉ์˜ ์ œํ’ˆ์„ ์ผ์ •๊ธฐ๊ฐ„ ๋™์•ˆ ์ƒ์‚ฐํ•˜๊ณ , ๋˜ ๋‹ค๋ฅธ ๊ทœ๊ฒฉ์˜ ์ œํ’ˆ์„ ์ƒ์‚ฐํ•˜๋Š” ๊ณผ์ •์„ ๋ฐ˜๋ณตํ•˜๋Š”๊ฒƒ์ด์ผ๋ฐ˜์ ์ด๋‹ค. ๋”ฐ๋ผ์„œ์ ๊ทผ์ ํŠน์„ฑ๋ณด๋‹ค์œ ํ•œํ•œ์‹œ๊ฐ„์˜ํŠน์„ฑ์ด๋”์œ ์šฉํ•˜๊ฒŒ์‚ฌ์šฉ๋ ์ˆ˜์žˆ์Œ์„๊ณ ๋ คํ•˜์˜€๋‹ค. ์ฐธ๊ณ ๋กœ ๊ณต์ •์žก์Œ๋ชจํ˜•์—์„œ ํ๋ฆ„์ด ์žˆ๋Š” ๊ฒฝ์šฐ์—๋Š” ์—†๋Š” ๊ฒฝ์šฐ๋ณด๋‹ค ์ˆ˜์ •๊ธฐ๊ฐ„ ํŠน์„ฑ์˜ ์ฐจ์ด๊ฐ€ ์‹œ๊ฐ„์ด

ํ˜๋Ÿฌ๋„๋ณ€ํ•˜์ง€์•Š์Œ์„์•Œ๊ฒŒ๋˜์–ด์ ์ ˆํ•œ์ˆ˜์ •๊ธฐ์˜์„ ํƒ์ด๋”์ค‘์š”ํ•œ์š”์ธ์ด๋จ์„์•Œ์ˆ˜์žˆ์—ˆ๋‹ค.

์ฐธ๊ณ ๋ฌธํ—Œ

Box, G. E. P. and Kramer, T. (1992). Statistical process control and feedback adjustment-a discussion.Technometrics, 34, 251-285.

Box, G. E. P. and Luceno, A. (1997). Statistical control by monitoring and feedback adjustment, New York,NY: John Wiley and Sons.

Butler, S. W. and Stefani, J. A. (1994). Supervisory run-to-run control of a polysilicon gate etch using insitu ellipsometry. IEEE Transactions on Semiconductor Manufacturing, 7, 193-201.

Del Castillo, E. (1999). Long-run and transient analysis of a double EWMA feedback controller. IIETransactions, 31, 1157-1169.

Del Castillo, E. (2001). Some properties of EWMA feedback quality adjustment schemes for drifting dis-turbances. Journal of Quality Technology, 33, 153-166.

Del Castillo, E. (2002). Statistical process adjustment for quality control, Wiley.Del Castillo, E., and Hurwitz, A. (1997). Run to run process control: a review and some extensions.

Journal of Quality Technology, 29, 184-196.Edgar, T. F., Butler, S. W., Campbell, W. J., Pfeiffer, C., Bode, C., Hwang, S. B., Balakrishnan, K. S.

and Hahn, J. (2000). Automatic control in microelectronics manufacturing: practices, challenges andpossibilities. Automatica, 36, 1567-1603.

Garcia, C. E. and Morari, M. (1985). Internal model control. 2. design procedure for multivariable systems.Industrial and Engineering Chemistry, Process Design and Development, 24, 472-484.

Morari, M. and Zafiriou, E. (1989). Robust process control , Upper Saddle River, N.J. : Prentice Hall.Moyne, J., del Castillo, E. and Hurwitz, A., eds (2000). Run to run control in semiconductor manufacturing,

Boca Raton, Fla.: CRC Press.Sachs, E., Hu, A. and Ingolfsson, A. (1995). Run by run process control: combining SPC and feedback

control. IEEE Transactions on Semiconductor Manufacturing, 8, 26-43.

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Journal of the Korean Data & ํ•œ๊ตญ๋ฐ์ดํ„ฐ์ •๋ณด๊ณผํ•™ํšŒ์ง€Information Science Society2009, 20(1), 11โ€“24

Modifications of single and double EWMA feedback

controllers for balancing the mean squared deviation

and the adjustment varianceโ€ 

Changsoon Park1ยท Sunggu Kwon2

12Department of Statistics, Chung-Ang University

Received 5 December 2008, revised 5 January 2009, accepted 8 January 2009

Abstract

The process controller in the adjustment procedure is often used effectively to con-

trol the process level close to target when noise is present and unremovable. Examples

of the robust controller are single EWMA controller and double EWMA controller.

Double EWMA controller is designed to reduce the offset of the process deviation,

which single EWMA can not eliminate. In this paper, the two controllers are modified

by taking EWMA of the original controller to reduce the adjustment variance, which

may become excessively large when the two given controllers are implemented. It is

shown that the EWMA modification of the given controllers is successful in reducing

the adjustment variance, while the mean squared deviation increases slightly.

Keywords: Adjustment variance, controller, EWMA, mean squared deviation, process

adjustment, process deviation.

โ€  This research was supported by the Chung-Ang University Research Scholarship Grants in 2007.1 Corresponding author: Professor, Department of Statistics, Chung-Ang University, Seoul 156-756,

Korea. E-mail: [email protected] Masterโ€™s Course, Department of Statistics, Chung-Ang University, Seoul 156-756, Korea.


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