Performance Analysis of EWMA Controllers Subject to Metrology Delay
description
Transcript of Performance Analysis of EWMA Controllers Subject to Metrology Delay
Performance Analysis of EWMA Controllers Subject to
Metrology Delay
報告者:碩研工管二甲 蔡依潾
原著: Ming-Feng Wu中華民國 2008年 8 月
Main point PrefacePreface
Questions of this study
EWMA controller◦ Single EWMA controller
Literature Reviews
Metrology delay
The Single EWMA Controller subject to Metrology delay
Example◦ Analysis
conclusion
PrefacePreface
半導體產業需要很高的投資成本
先進製程控制量測延遲是自然存在的問題
這篇研究主要分成二個部份
◦1.Exponentrally weighted moving average( EWMA) 控制器存在量測延遲下做探討
◦2.利用虛擬量測系統 (virtual metrology system, VM) 來探討量測延遲的問題
Questions of this study1.Is the investment in advanced metrology justified?
2.How do we retune the controller parameters
if the metrology delay is changed?
3.Can virtual metrology be used?
4.Do the above guidelines apply in case of variable delays?
EWMA EWMA Controller
The Single EWMA Controller(1/6)
Model:
is the observed process output at the run.
is the process input at the run.
is the intercept parameter.
is the slope parameter.
is the process disturbance.
tY
1tX
t
ttt XY 1
The Single EWMA Controller(2/6)
The Single EWMA Controller(3/6)
The Single EWMA Controller(4/6)
The Single EWMA Controller(5/6)
調整製程
: is the target of the process
: the estimate of
: the discount factor,
b
aTX t
t
)(
11 1 tttt abXYa
10
T
b
調整截距
11 )( tt aTY
The Single EWMA Controller(6/6)
穩定條件
必要條件
0b
lim ( ) is bounded.t tVar Y
TYE tt )(lim
Literature Reviews
ProcessI-O model
EWMA feedback control scheme
Without metrology delay With metrology delay
MISO(SISO) system
With no linear drift
Ingolfsson & Sachs (1993)Tseng et al. (2003)
With a linear drift
Butler & Stefani (1994)Chen & Guo (2001)
Tseng et al. (2002) Su & Hsu (2004)
Tseng et al. (2007)
MIMOsystem
With no linear drifts
Tseng et al. (2002) Good & Qin (2006)
With linear drifts
Del Castillo & Rajagopal (2002, 2003)
Tseng et al. (2007)Lee et al. (2008)
Metrology delay
741 32 85 6Production Line
d=0 1 2 3 4 5 6 7 8
d=1 1 2 3 4 5 6 7
d=2 1 2 3 4 5 6
d=3 1 2 3 4 5
Lot-to-Lot metrology delay
The Single EWMA Controller subject to Metrology delay(1/2)
Model:
is the observed process output at the run.
is the process input at the run.
is the initial bias of process.
is the process gain.
is the disturbance input.
tY
1tX
1t t tY x
t
Process predicted model
is the model offset parameter.
is the model gain parameter.
Disturbance is estimated to be
tt bXaY ˆ
a
b
111 ˆ)1()(ˆ tdtdtt bXaY
b
aTX t
t̂
delay
The Single EWMA Controller subject to Metrology delay(2/2)
Model mismatch Bias correction
Time-correction noise reduce
1r 1s
Formula of Process Output
1 ( 1)
01 ( 1)
1 (1 )( )
1 (1 ) ( )
d
t t t td
z zY a W
z z
11t tz Y Y
/ b
2
1 1
1, 1
0, [2, 1]
( 1) , [ 2, 2 2]
, [2 3, )
i i d
i i d
i
i dp
r s r i d d
rp sp i d
00
( )t
it i
i
p z a
0
ti
t i ti
W p z
Backshift operator
( 公式1 )
Bias subject to Metrology delay
0Let ( )t tB a
1 ( 1)
01 ( 1)
1 (1 )( )
1 (1 ) ( )
d
t td
z zY a
z z
1
1 1
1, [1, ]
( 1)(1 )1 , [ 1, 2 1]
1, [2 2, )
t d
t
t t d
t d
r s rB t d d
rrB sB t d
( 公式2 )
tt pppB 10
Proof 1-1:◦ Give
◦ From 公式 2 when
1
10
22 dt
Proof (1/2) TYE tt )(lim
Proof 1-2:◦ For any , We can find some value of
◦ So that
From 公式 2 when
1 10
0tB
12 dt
Proof (2/2) TYE tt )(lim
The first property of bias subject to Metrology delay
For an overestimated process gain (i.e., )
◦ is a monotonic decreasing sequence.
For an underestimated process gain (i.e., )
◦ There exists some values of larger than which is oscillatory.
1
tB
1
tB
Proof
The following can be easily derived from公式 1 :
Proof 2:
lim ( ) is bounded.t tVar Y
The second property of bias subject to Metrology delay
In case of , for all t>d
Then
Therefore
In case of and
The optimal value of is equal to one
1 21 )10(
),,(),,( 21 dBdB tt
),,(),,( 21 dSSEdSSE
1 )10(
),1,(),,(min dSSEdSSE
The effects of time delay don the optimalλand SSE(1/2)
We can found that the optimal value ofλdoes not change much with different run numbers.
N=20 N=2004,1 d
The effects of time delay don the optimalλand SSE(2/2)
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Delay
λO
PT
ξ ≦ 1.0
ξ =2.0 ξ =1.8 ξ =1.6
ξ =1.4
ξ =1.2
0 1 2 3 4 50
2
4
6
8
10
12
14
16
18
20
Delay
SS
E = 1.4 = 1.0
= 1.6 = 1.2
= 0.6
= 0.4
The effects of time delay d on the optimalλfor different noise to initial bias
ratios
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Delay
OP
T
= 0.4
= 0.6
= 0.8
= 1.0
= 1.2 = 1.4 = 1.6 = 1.8 = 2.0
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
Delay O
PT
= 0.4
= 0.6
= 0.8
= 1.0
= 1.2 = 1.4 = 1.6 = 1.8 = 2.0
0.1 0.2 2
2( )a
A ratio of metrology noise to the magnitude of error initial bias estimate
The effects of time delay d on the optimalSSE for different noise to initial bias
ratios
0 1 2 3 4 50
2
4
6
8
10
12
14
16
18
20
Delay
SS
E
= 1.4 = 1.0
= 1.6 = 1.2
= 0.6
= 0.4
0 1 2 3 4 50
2
4
6
8
10
12
14
16
18
20
Delay
SS
E = 1.4 = 1.0 = 1.6
= 1.2
= 0.6 = 0.4
0.1 0.2
Time-correlated noise reduction
1
1
Consider that follows a ARMA(1,1) time series model with a metrology noise,
that is,
1 .
1
Note that when 1 the process disturbance becomes a non-s
t
t t t
z
z
tationary process
disturbance IMA(1,1); when 1 the process disturbance is stationary process
disturbance.
1 ( 1) 1
1 ( 1) 1
1 (1 ) 1( , , , , )
1 (1 ) ( ) 1
d
t t t td
z z zY d W
z z z
Non-stationary process disturbance IMA(1,1)
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
λO
PT
Delay
ξ =0.4 ξ =0.6 ξ =1.0 ξ =1.4 ξ =1.8
0 1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
0.8
1
Delay
o
pt = 1
= 0.4
= 0.6
= 0.8
= 1.2
= 1.4
= 1.6 = 1.80 1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
Delay
o
pt
1
= 1.2
= 1.4
= 1.6
= 1.8
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
Delay
AM
SE
= 0.4
= 0.6
= 1
= 1.4 = 1.6
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
Delay
AM
SE
= 0.4 = 0.6
= 1
= 1.4 = 1.6
0 1 2 3 4 5 6 7 8 9 100
2
4
6
8
10
12
14
16
18
20
DelayA
MS
E
= 0.4 = 0.6 = 1 = 1.4 = 1.6
0 0.5 1
2 0
Stationary process disturbance ARMA(1,1)
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
= 1.5
= 0.5
= 1
Delay
opt
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
= 1.5
= 0.5
= 1
Delay
opt
0 1 2 3 4 5 6 7 8 9 100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
= 1.5 = 0.5 = 1
Delay
opt
0 1 2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
Delay
AM
SE
= 1 = 0.5 = 1.5
0 1 2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
Delay
AM
SE
= 1
= 0.5 = 1.5
0 1 2 3 4 5 6 7 8 9 101
1.5
2
2.5
3
3.5
Delay
AM
SE
= 1.5
= 0.5
= 1
0.8, 0 0.8, 1 0.8, 0.5
2 0
Example
Tungsten CVD process
沈澱速率 溫度 partial pressure of hydrogen d=3
)4200ln(
5.0
8800
)10*2ln(
2
1
80
T
C
C
C
assume
]ln[1
)ln( 2210 HCT
CCRw wR
T2H
ttt z
z
IMANoise
1
1
1
5.01
)1,1(:
Analysis(1/3)
Analysis(2/3)
We find H2 pressure is unchanged.
Due to the fact that C2 is so much smaller than C1
Analysis(3/3)
Optimal AMSE of the tungsten CVD process at difference metrology delay
conclusion1.Is the investment in advanced metrology justified?
2.How do we retune the controller parameters if the metrology delay is changed?
心得心得這篇報告主要探討 SEWMA有 delay的問題◦探討有 delay狀況下, 的影響◦探探討有 delay狀況下,分別探討穩定與不穩定干擾之情況控制器如何調整
我覺得新的發展方向可以針對 DEWMA做 delay的探討