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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

Preface

Questions of this study

EWMA controller

Literature Reviews

Metrology delay

The Single EWMA Controller subject to Metrology delay

Example

conclusion
◦ Single EWMA controller
◦ Analysis
Preface

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 study

1.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 Controller
The Single EWMA Controller(1/6)



Model:
Yt
Yt    X t 1   t
is the observed process output at the run.
X t 1 is the process input at the run.


is the intercept parameter.


is the slope parameter.

t
is the process disturbance.
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
(T  at )
Xt 
b
調整截距
at   Yt  bX t 1   1   at 1
  (Yt 1  T )  at 1

T
: is the target of the process

b
: the estimate of


: the discount factor,

0   1
The Single EWMA Controller(6/6)

穩定條件
lim t  E (Yt )  T
limt  Var (Yt ) is bounded.

必要條件

b
 0
Literature Reviews
Process
I-O model
MISO
(SISO)
system
MIMO
system
EWMA feedback control scheme
Without
metrology delay
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)
With no
linear
drifts
Tseng et al. (2002)
With
linear
drifts
Del Castillo & Rajagopal (2002,
2003)
Tseng et al. (2007)
Lee et al. (2008)
With metrology delay
Good & Qin (2006)
Metrology delay

d=0
d=1
d=2
d=3
Lot-to-Lot metrology delay
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
1
2
3
4
5
6
1
2
3
4
5
Production Line
The Single EWMA Controller subject to
Metrology delay(1/2)

Model:

Yt

X t 1





t
Yt     xt 1  t
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.
The Single EWMA Controller subject to
Metrology delay(2/2)

Process predicted model
Yˆt  a  bX t
a is the model offset parameter.
b

is the model gain parameter.
Disturbance is estimated to be
ˆt   (Yt d 1  a  bX t d 1 )  (1   )ˆt 1
delay
T  a  ̂t
Xt 
b
Formula of Process Output
(公式1）
 1  (1   ) z 1   z  ( d 1)

Yt  
(   a0  t )  t  Wt
1
 ( d 1) 
1  (1   ) z  (   ) z

z 1Yt  Yt 1
Model mismatch
Bias correction
   /b
Time-correction
noise reduce
Backshift operator
 t
i 
t    pi z  (   a0 )
 i 0

 t
i 
Wt    pi z t
 i 0

r  1 
s   1   
1,
0,

pi  
i d 2
(
r

s

1)
r
,

rpi 1  spi  d 1 ,
i 1
i  [2, d  1]
i  [d  2, 2d  2]
i  [2d  3, )
Bias subject to Metrology delay
 1  (1   ) z 1   z ( d 1)

Yt  
(   a0 )  t
1
 ( d 1) 
1  (1   ) z  (   ) z

Let t  Bt (   a0 )
Bt  p0  p1    pt
1,

 (r  s  1)(1  r t d 1 )
Bt  1 
,
1 r

rBt 1  sBt d 1 ,
t  [1, d ]
t  [d  1, 2d  1]
t  [2d  2, )
(公式2）
Proof lim t  E (Yt )  T (1/2)

Proof 1-1:
◦ Give
0   1
 1
◦ From 公式2 when
t  2d  2
Proof lim t  E (Yt )  T (2/2)


Proof 1-2:
◦ For any
  1 ，We
◦ So that
Bt  0
From 公式2
can find some value of
when t  2d  1
0   1
The first property of bias subject to
Metrology delay

For an overestimated process gain (i.e.,  1 )
◦

Bt
is a monotonic decreasing sequence.
For an underestimated process gain (i.e.,  1 )
◦ There exists some values of
oscillatory.

larger than which
Bt
is
Proof limt  Var (Yt ) is bounded.

The following can be easily derived from
公式1：

Proof 2:
The second property of bias subject to
Metrology delay

In case of   1 , 1  2 (0    1) for all t>d

Then Bt ( , 1 , d )  Bt ( , 2 , d )


Therefore SSE( , 1, d )  SSE( , 2 , d )
In case of   1 and (0    1)

The optimal value of

is equal to one
min SSE ( ,  , d )  SSE ( ,1, d )

The effects of time delay d
on the optimalλand SSE(1/2)
  1, d  4
N=20
N=200
We can found that the optimal value ofλdoes not change much
with different run numbers.
The effects of time delay d
on the optimalλand SSE(2/2)
20
1
18
ξ ≦ 1.0
16
0.8
14
12
SSE
0.6
λ
OPT
ξ =1.2
ξ =1.4
0.4
10
 = 0.4
8
 = 0.6
6
4
0.2
ξ =2.0 ξ =1.8 ξ =1.6
0
0
1
2
3
Delay
 = 1.6
 = 1.0  = 1.2  = 1.4
2
4
5
0
0
1
2
3
Delay
4
5
The effects of time delay d on the optimalλfor
different noise to initial bias ratios
 = 0.4
1
1
 = 0.6
0.8
 = 0.4
0.8
0.6
OPT
OPT
 = 0.8
 = 1.0
0.4
0.6
 = 0.6
 = 0.8
0.4
 = 1.0
0.2
0.2
 = 2.0  = 1.8  = 1.6  = 1.4  = 1.2
0
0
1
2
3
4
5
0
0
 = 2.0  = 1.8  = 1.6  = 1.4  = 1.2
1
Delay
3
Delay
  0.1
A ratio of metrology noise to
the magnitude of error initial
bias estimate
2
  0.2
2

(   a)2
4
5
The effects of time delay d on the optimal
SSE for different noise to initial bias ratios
20
20
18
18
16
16
14
14
 = 0.4
 = 0.6
10
12
SSE
SSE
12
8
 = 1.0  = 1.2  = 1.4
10
 = 0.6
 = 1.6
8
 = 1.6
6
6
 = 1.0  = 1.2  = 1.4
4
4
2
0
0
 = 0.4
2
1
2
3
Delay
  0.1
4
5
0
0
1
2
3
Delay
  0.2
4
5
Time-correlated noise reduction
Consider that t follows a ARMA(1,1) time series model with a metrology noise,
that is,
1   z 1
t 
  t .
1 t
1 z
Note that when   1 the process disturbance becomes a non-stationary process
disturbance IMA(1,1); when   1 the process disturbance is stationary process
disturbance.
 1  (1   ) z 1   z  ( d 1)
  1   z 1

Yt ( ,  , d ,  , )  Wt  



t 
1
 ( d 1)  
1 t
1

(1


)
z

(



)
z
1


z



Non-stationary process disturbance IMA(1,1)
 2  0
  0.5
 0
1
 = 0.4
1
1
1
 = 0.6
0.8
 = 0.8
0.6
0.6
=1
0.4
 = 1.2
 = 1.4
0.4
 = 1.6
 = 1.4
0.2
ξ=0.4
 = 1.8
0
1
2
3
4
5
6
7
8
9
10
0
0.6
0.4
0.2
0.2
 = 1.8
 = 1.6
0
1
2
3
4
Delay
5
6
7
0
8
9
10
0
20
18
18
16
16
14
14
12
12
 = 0.4
 = 0.6
14
AMSE
12
10
 = 1.4
 = 1.6
AMSE
20
18
6
10
 = 0.4
8
 = 1.6
 = 0.6  = 1.4
4
4
4
2
2
2
3
4
5
Delay
4
6
7
8
9
10
0
0
=1
1
2
3
4
5
Delay
5
6
7
8
9
10
8
6
=1
1
3
10
6
0
0
2
Delay
20
8
1
ξ=0.6 ξ=1.0 ξ=1.4 ξ=1.8
Delay
16
AMSE
0.8
λOPT
0.8
 opt
 opt
 = 1.2
0
 1
6
7
8
9
=1
 = 0.4
 = 1.6
 = 0.6  = 1.4
2
10
0
0
1
2
3
4
5
Delay
6
7
8
9
10
Stationary process disturbance ARMA(1,1)
 2  0
1
1
1
0.9
0.9
0.9
0.8
0.8
0.7
0.7
0.7
0.6
0.6
=1
0.6
 = 0.5
 opt
0.5
 opt
 = 0.5
0.8
 opt
  0.8,   1
  0.8,   0.5
  0.8,   0
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.5
0.4
0.3
=1
0.2
 = 1.5
0.1
0
0
0.1
1
2
3
4
5
6
7
8
9
0
0
10
1
2
3
4
Delay
5
6
7
8
9
0
0
10
1
2
3
 = 1.5
=1
 = 0.5
0.1
 = 1.5
4
Delay
5
6
7
8
9
10
8
9
10
Delay
3.5
3.5
3.5
 = 1.5
3
3
3
 = 0.5
=1
2.5
2.5
AMSE
AMSE
AMSE
2.5
2
2
2
 = 1.5  = 0.5
1.5
1.5
1.5
=1
1
0
1
0
1
2
3
4
5
Delay
6
7
8
9
10
1
2
3
4
5
Delay
6
7
8
 = 1.5
9
10
1
0
1
2
3
 = 0.5
=1
4
5
Delay
6
7
Example

Tungsten CVD process
1
ln( Rw )  C 0  C1  C 2 ln[ H 2 ]
T
R

w 沈澱速率
 T
溫度
 H2
d=3
partial pressure of hydrogen
assume
C0  ln( 2 *108 )
Noise：IMA(1,1)
C1  8800
C2  0.5
T  ln( 4200)
1  0.5 z 1
t 
  t  t
1
1 z
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
conclusion

1.Is the investment in advanced metrology
justified？

2.How do we retune the controller parameters if
the metrology delay is changed?
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