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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)
Spring 2008
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Control of Manufacturing
Processes
Subject 2.830
Spring 2007
Lecture #20
“Cycle To Cycle Control:
The Case for using Feedback and SPC”
May 1, 2008
Manufacturing
Desired
Product
The General Process Control
Problem
CONTROLLER
CONTROLLER
EQUIPMENT
EQUIPMENT
MATERIAL
MATERIAL
Product
Equipment loop
Material loop
Process output loop
Control of Equipment:
Control of Material
Forces,
Strains
Velocities
Stresses
Temperatures,
Temperatures,
, ..
Control of Product:
Geometry
and
Properties
Pressures, ..
Lecture 20 © D.E. Hardt.
5/1/08
2
Manufacturing
Output Feedback Control
∂Y
ΔY =
Δα
∂α
∂Y
+
Δu
∂u
∂Y
∂Y
Δu = −
Δα
∂u
∂α
Manipulate
Actively
Such that
Compensate for Disturbances
Lecture 20 © D.E. Hardt.
5/1/08
3
Manufacturing
Process Control Hierarchy
• Reduce Disturbances
–
–
–
–
Good Housekeeping
Standard Operations (SOP’s)
Statistical Analysis and Identification of Sources (SPC)
Feedback Control of Machines
• Reduce Sensitivity (increase “Robustness”)
– Measure Sensitivities via Designed Experiments
– Adjust “free” parameters to minimize
• Measure output and manipulate inputs
– Feedback control of Output(s)
Lecture 20 © D.E. Hardt.
5/1/08
4
The Generic Feedback “Regulator”
Problem
Manufacturing
D1(s)
R(s)
-
Gp(s)
Gc(s)
D2(s)
Y(s)
H(s)
•Minimize the Effect of the “D’s”
•Minimize Effect of Changes in Gp
•Follow R exactly
Lecture 20 © D.E. Hardt.
5/1/08
5
Effect of Feedback on
Random Disturbances
Manufacturing
• Feedback Minimizes Mean Shift (SteadyState Component)
• Feedback Can Reduce Dynamic
Disturbances
0.45
0.45
0.4
0.4
0.35
0.35
?
0.3
0.25
0.3
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
-4
-3
-2
-1
0
0
1
2
3
-4 4 -3
-2
-1
0
1
2
3
4
Lecture 20 © D.E. Hardt.
5/1/08
6
Typical Disturbances
Manufacturing
• Equipment Control
– External Forces Resisting Motion
– Environment Changes (e.g Temperature)
– Power Supply Changes
• Material Control
– Constitutive Property Changes
•
•
•
•
Hardness
Thickness
Composition
...
Lecture 20 © D.E. Hardt.
5/1/08
7
The Dynamics of
Disturbances
Manufacturing
•
•
•
•
Slowly Varying Quantities
Cyclic
Infrequent Stepwise
Random
Lecture 20 © D.E. Hardt.
5/1/08
8
Manufacturing
Example: Material Property
Changes
• A constitutive property change from
workpiece to workpiece
– In-Process Effect?
• A new constant parameter
• Different outcome each cycle
– Cycle to Cycle Effect
• Discrete random outputs over time
Lecture 20 © D.E. Hardt.
5/1/08
9
Manufacturing
What is Cycle to Cycle?
• Ideal Feedback is the Actual Product
Output
• This Measurement Can Always be
made After the Cycle
• Equipment Inputs can Always be
Adjusted Between Cycles
• Within the Cycle Inputs Are Fixed
Lecture 20 © D.E. Hardt.
5/1/08
10
What is Cycle to Cycle?
Manufacturing
• Measure and Adjust Once per Cycle
d
y
r
Controller
Process
-
Execute the Loop Once Per Cycle
Discrete Intervals rather then Continuous
Lecture 20 © D.E. Hardt.
5/1/08
11
Manufacturing
Run by Run Control
• Developed from an SPC Perspective
• Primarily used in Semiconductor
Processing
• Similar Results, Different Derivations
• More Limited in Analysis and
Extension to Larger problems
Box, G., Luceno, A., “Discrete Proportional-Integral Adjustment and Statistical Process
Control,” Journal of Quality Technology, vol. 29, no. 3, July 1997. pp. 248-260.
Sachs, E., Hu, A., Ingolfsson, A., “Run by Run Process Control: Combining SPC and
Feedback Control.” IEEE Transactions on Semiconductor Manufacturing, 1995, vol. 8, no.
1, pp. 26-43.
Lecture 20 © D.E. Hardt.
5/1/08
12
Manufacturing
Cycle to Cycle Feedback
Objectives
• How to Reduce E(L(x)) & Increase Cpk with
Feedback?
• Bring Output Closer to Target
– Minimize Mean or Steady - State Error
• Decrease Variance of Output
– Reject Time Varying Disturbances
Lecture 20 © D.E. Hardt.
5/1/08
13
A Model for Cycle to Cycle
Feedback Control
Manufacturing
• Simplest In-Process Dynamics:
d(t)
u(t))
y(t)
4τp
Cycle Time Tc > 4τp
d(t) = disturbances seen at the output (e.g. a Gaussian noise)
τp = Equivalent Process Time Constant
Lecture 20 © D.E. Hardt.
5/1/08
14
Manufacturing
Discrete Product Output
Measurement
d(t)
y(t)
u(t))
?
yi
Continuous variable y(t) to sequential variable yi
i = time interval or cycle number
Lecture 20 © D.E. Hardt.
5/1/08
15
The Sampler
Manufacturing
d(t)
y(t)
u(t)
y(t)
TS
yi
yi
TS
i
Lecture 20 © D.E. Hardt.
5/1/08
16
Manufacturing
A Cycle to Cycle Process Model
y(t)
y2
y3
y4
y1
1
yi
2
3
4
…
i
With a Long Sample Time, The Process has
no Apparent Dynamics, i.e. a Very Small Time Constant
Lecture 20 © D.E. Hardt.
5/1/08
17
Manufacturing
A Cycle to Cycle Process Model
a discrete input
sequence at
interval Tc
ui
yi
Kp
a discrete output
sequence
at time intervals Tc
- or ui
yi
Equipment
αe
Material
αm
Lecture 20 © D.E. Hardt.
5/1/08
18
Manufacturing
Cycle to Cycle Output Control
Process
Uncertainty
Desired
Part
Controller
Process
Part
-
Output Sampling
Lecture 20 © D.E. Hardt.
5/1/08
19
Manufacturing
Delays
• Measurement Delays
– Time to acquire and gage
– Time to reach equilibrium
• Controller Delays
– Time to “decide”
– Time to compute
• Process Delay
– Waiting for next available machine cycle
Lecture 20 © D.E. Hardt.
5/1/08
20
Delays
Manufacturing
z = n - step time advance operator
e.g.
n
yi-1
z * yi = yi +1
1
z * yi = yi + 2
2
yi
yi+1
yi-2
yi+2
and
−1
z * yi = yi −1
i
Lecture 20 © D.E. Hardt.
5/1/08
21
Manufacturing
A Pure Delay Process Model
ui-1
Kp
yi
yi = K p ui −1
u
y
z-1 Kp
Lecture 20 © D.E. Hardt.
5/1/08
22
Modeling Randomness
Manufacturing
• Recall the Output of a “Real Process”
Width 2 In)
1.025
1.02
d(t)
u(t)
1.015
y(t)
TS
yi
1.01
1.005
1
1
4
7
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88
• Random even with inputs held constant
Lecture 20 © D.E. Hardt.
5/1/08
23
Manufacturing
Output Disturbance Model
d(t)
u(t)
y(t)
TS
yi
Model:
d(t) is a continuous random
variable that we sample every
cycle (Tc)
Lecture 20 © D.E. Hardt.
5/1/08
24
Manufacturing
Or In Cycle to Cycle Control Terms
d
r
y
-
Gc(z)
Gp(z)
Where:
d(t) is a sequence of random numbers governed by
a stationary normal distribution function
Lecture 20 © D.E. Hardt.
5/1/08
25
Manufacturing
Gaussian White Noise
• A continuous random variable that at
any instant is governed by a normal
distribution
• From instant to instant there is no
correlation
• Therefore if we sample this process we
get:
• A NIDI random number
Normal
Identically Distributed
Independent
Lecture 20 © D.E. Hardt.
5/1/08
26
Manufacturing
The Gaussian “Process”
...
i+2
i+1
i
Lecture 20 © D.E. Hardt.
5/1/08
27
Constant (Mean Value) Disturbance
Rejection- P control
Manufacturing
di ~NIDI(μ,σ2)
r
-
Kc
ui
Kp/z
yi = di + K p K c (r − yi −1 )
yi
yi = di + K pui −1
ui −1 = K c (r − yi −1 )
if di = μ (a constant), we can look at steady - state behavior:
K pKc
di
yi ⇒ yi −1 ⇒ y∞ =
+r
1 + K pKc
1 + K pKc
Lecture 20 © D.E. Hardt.
5/1/08
28
Manufacturing
And For Example
Thus if we want to eliminate the constant
(mean) component of the disturbance
y∞
1
1
=
=
di 1 + K p K c 1 + K
Higher loop gain K improves “rejection”
rejection
but only
K = ∞ eliminates mean shifts
Lecture 20 © D.E. Hardt.
5/1/08
29
Error: Try an Integrator
Manufacturing
i
ui = Kc ∑ ei
running sum of all errors
j =1
recursive form (ei = r − yi )
ui+1 = ui + Kc ei+1
z
u
Gc (z) = K c
=
z −1 e
zU = U + Kc zE
di
r
-
z ui
Kc
z −1
Kp
yi
z
Lecture 20 © D.E. Hardt.
5/1/08
30
Constant Disturbance Integral Control
Manufacturing
D
r
-
z ui
Kc
z −1
Kp
yi
z
z −1
Y (z) =
D
z − 1 + KcK p
(Assume r=0)
or yi +1 + (1 − K c K p )yi = di +1 − di
Again at steady state yi +1 = yi = y∞
And since D is a constant y∞ (2 − K c K p ) = 0
Zero error
regardless
of loop
gain
Lecture 20 © D.E. Hardt.
5/1/08
31
Effect of Loop Gain K on Time Response:
I-Control
Manufacturing
Step Response
Step Response
1.0
Amplitude
1.0
K=0.5
TextEnd
0
0
2
4
6
K=1.5
TextEnd
8
10
0
12
2
4
6
8
10
12
Time (samples)
Time (samples)
Step Response
Step Response
1.0
Amplitude
1.0
K=0.9
K=2
TextEnd
TextEnd
0
0
0
2
4
6
Time (samples)
8
10
0
10
20
Time (samples)
30
40
Lecture 20 © D.E. Hardt.
5/1/08
32
Effect of Loop Gain K = KcKp
Manufacturing
Step Response
Amplitude
1.0
K=1.0
TextEnd
0
0
2
4
6
Time (samples)
8
10
Best performance at Loop Gain K= 1.0
Stability Limits on Loop Gain
0<K<2
Lecture 20 © D.E. Hardt.
5/1/08
33
Manufacturing
What about random component of d?
• di is defined as a NIDI sequence
• Therefore:
– Each successive value of the sequence is
probably different
– Knowing the prior values: di-1, di-2, di-3,…
will not help in predicting the next value
e.g. di ≠ a1di −1 + a2 di − 2 + a3di − 3 + ...
Lecture 20 © D.E. Hardt.
5/1/08
34
Thus
Manufacturing
This implies that with our cycle to cycle
process model under proportional control:
r
e
-
Kc
u
Kp/z
x
di
y
The output of the plant xi will at best represent the error
from the previous value of di-1
xi = − Kc K p di −1
will not cancel di
Lecture 20 © D.E. Hardt.
5/1/08
35
Variance Change with Loop
Gain
Manufacturing
2
σ CtC
σ2
2.4
2.2
2
1.8
1.6
1.4
1.2
1
0
0.2
0.4
0.6
0.8
1
1.2
Loop Gain
Lecture 20 © D.E. Hardt.
5/1/08
36
Conclusion - CtC with Un-Correlated
(Independent) Random Disturbance
Manufacturing
•
•
•
Mean error will be zero using “I” control
Variance will increase with loop gain
Increase in σ at K=1 ~ 1.5 * σ open loop
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
-4
-3
-2
-1
0
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
Lecture 20 © D.E. Hardt.
5/1/08
37
Manufacturing
What if the Disturbance is not NIDI?
di
NIDI
Sequence
a
z-a
cdi
Filter
(Correlator)
CD(Z )(z − a) = aD(z)
cdi +1 = a(di − cdi )
unknown
expect some correlation,
therefore ability to
counteract some of the
disturbances
known
Lecture 20 © D.E. Hardt.
5/1/08
38
What if the Disturbance is
not NIDI?
Manufacturing
Proportional Control
Simulation
0.7
z+0.7
NIDI
Sequence
+
Sum
Kc
Gain
n
Disturbance
Filter
(Correlator)
1/z
Process
+
+
Sum1
y
Ouput
Kc
σ2Ctc /σ2o
0
1
0.1
0.89
0.25
0.77
0.5
0.69
0.9
1.39
Lecture 20 © D.E. Hardt.
5/1/08
39
Gain - Variance Reduction
Manufacturing
2
σ CtC
σ2
2.0
Increasing Correlation
1.0
0
0.5
1.0
Loop Gain
1.5
Lecture 20 © D.E. Hardt.
5/1/08
40
Conclusion - CtC with Correlated
(Dependent) Random Disturbance
Manufacturing
•
•
•
Mean error will be zero using “I” control
Variance will decrease with loop gain
Best Loop Gain is still KcKp =1
0.45
0.45
0.4
0.4
0.35
0.35
0.3
0.3
0.25
0.25
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
-4
-3
-2
-1
0
0
1
2
3
4
-4
-3
-2
-1
0
1
2
3
4
Lecture 20 © D.E. Hardt.
5/1/08
41
Manufacturing
Conclusions from
Cycle to Cycle Control Theory
• Feedback Control of NIDI Disturbance
will Increase Variance
– Variance Increases with Gain
• BUT: If Disturbance is NID but not I;
We CAN Decrease Variance
– Higher Gains -> Lower Variance
– Design Problem: Low Error and Low
Variance
Lecture 20 © D.E. Hardt.
5/1/08
42
Manufacturing
How to Tell if Disturbance
is Independent
• Correlation of output data
– Look at the Autocorrelation
– Effect of Filter on Autocorrelation
• Reaction of Process to Feedback
– If variance decreases data has dependence
Lecture 20 © D.E. Hardt.
5/1/08
43
Manufacturing
Is Disturbance
is Independent?
• Correlation of output data
∞
– Look at the Autocorrelation Φ xx (τ ) = ∫ x(t)x(t − τ )dt
−∞
– Effect of Filter on Autocorrelation
• Reaction of Process to Feedback
– If variance decreases then data must have
some dependence
Lecture 20 © D.E. Hardt.
5/1/08
44
Manufacturing
But Does It Really Work?
– Let’s Look at Bending and Injection
Molding
Lecture 20 © D.E. Hardt.
5/1/08
45
Manufacturing
Experimental Data
Cycle to Cycle Feedback Control
of
Manufacturing Processes
by
George Tsz-Sin Siu
SM Thesis
Massachusetts Institute of Technology
February 2001
Lecture 20 © D.E. Hardt.
5/1/08
46
Manufacturing
Experimental Results
• Bending
– Expect NIDI Noise
– Can Have Step Mean Changes
• Injection Molding
– Could be Correlated owing to Thermal
Effects
– Step Mean Changes from Cycle
Disruption
Lecture 20 © D.E. Hardt.
5/1/08
47
Manufacturing
Process Model for Bending
ui
yi
Kp
yi = K p ui −1
Y (z) =
Kp
z
Kp=?
Lecture 20 © D.E. Hardt.
5/1/08
48
Process Model for Bending
Manufacturing
Local Response Curve (Mat'l I)
60
50
Angle
Angle
40
0.025” steel
30
20
y = 150.85x - 156
2
R = 0.9992
10
0
1.15
1.2
1.25
1.3
Punch depth
1.35
1.4
Lecture 20 © D.E. Hardt.
5/1/08
49
Results for Kc=0.7;Δμ=0
Manufacturing
36.5
Open-loop
Closed-loop
36
35.5
Target
35
34.5
34
33.5
0
10
20
30
40
2
σ CtC
=1.67
2
50
60
80
σ70
Run number Theoretical
=1.96
90
100
Lecture 20 © D.E. Hardt.
5/1/08
50
I-Control Δμ≠0
Manufacturing
36
Tar
t
35
34
33
32
2
σ CtC
= 1.01
2
σ
31
30
29
28
Kc=0.5
Angle ss= 34.95=0.14%
27
26
0
Material
Shift
5
10
Run number
15
20
Lecture 20 © D.E. Hardt.
5/1/08
51
Minimum Expected Loss
Integral-Controller
Manufacturing
Calculated
Expected
Loss
vs. Gain
Performance
Index
versus
Feedback Gain
(σ^2 = 0.0618,
S = 7.2581, N = Verification
20)
Experimental
Performance Index, V
16
14
Kc
0.5
0.98
1.5
12
10
8
6
V
4.47
3.11
4.89
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
Fe e dbac k Ga in, Kc
1.4
1.6
1 .8
2
Lecture 20 © D.E. Hardt.
5/1/08
52
Disturbance Response for
“Optimal” Integral Control Gain
Manufacturing
39
37
Angle
Target
35
33
K=1
31
29
0.025” steel to 0.02” steel
27
25
0
5
Material
Shift
10
15
20
25
Run number
Lecture 20 © D.E. Hardt.
5/1/08
53
Injection Molding:
Process Model
Manufacturing
Yˆ = β 0 + β 2 ⋅ X2 + β 3 ⋅ X 3 + β 23 ⋅ X2 ⋅ X 3 Initial Model
Process inputs
X2 = Hold time (seconds)
X3 = Injection speed (in/sec)
Levels
5 sec
0.5 in/sec
20 sec
6 in/sec
ANOVA on model terms
Effect
1
X2 (Hold time)
X3 (Injection speed)
X2X3
(Hold time*Injection speed)
Error
Total
beta
1.437
-1.04E-03
-3.75E-04
SS
49.6
0
0
2.92E-04
0
0
49.6
Yˆ = β 0 + β 2 ⋅ X2
DOF
MS
1
49.568
1
2.60E-05
1
3.38E-06
1
20
24
2.04E-06
2.46E-06
F
2E+07
10.593
1.373
Fcrit
4.35
4.35
4.35
p-value
0
0.004
0.255
0.831
4.35
0.373
Final Model
Lecture 20 © D.E. Hardt.
5/1/08
54
P-Control Injection Molding
Manufacturing
1.45
Open Loop
Closed Loop, Gain = 0.5
1.445
Diameter
1.44
Hot
1.435
1.43
Cold
1.425
1.42
1.415
Experiment
1.41
0
Open-loop
20
Closed-loop
Mean ( Hot
measurement)
Variance
1.437
9.97E-06
40
60
Run
1.439
2.34E-06
Variance
Ratio
80 0.234
100
Lecture 20 © D.E. Hardt.
5/1/08
55
Output Autocorrelation
Manufacturing
0.8
0.8
0.7
0.6
0.6
0.4
0.5
0.2
0.4
0
0.3
0.2
-0.2
0.1
-0.4
0
-0.6
-0.8
-0.1
-0.2
0
5
10
15
20
0
5
10
15
20
Injection Molding
Bending
Matlab function XCORR
Lecture 20 © D.E. Hardt.
5/1/08
56
Manufacturing
1.455
1.45
P-control: Moving Target
Closed Loop
Target = 1.436
Closed Loop
Target = 1.44
Open Loop
Diameter
1.445
1.44
1.435
1.43
1.425
1.42
1.415
1.41
Mean ( Hot
measurement)
Experiment
0 Closed-loop,
20 target = 1.436Ó
40
Closed-loop, target = 1.440Ó
Open-loop
Variance
601.438
80 6.79E-06
100
Run
1.440
4.54E-06
1.437
1.44E-05
Variance Ratio
0.471
120
0.315
140
-
Lecture 20 © D.E. Hardt.
5/1/08
57
Injection Molding:
Integral Control
Manufacturing
1.4 5
1 .4 4 5
Diameter
1.4 4
1 .4 3 5
1.4 3
1 .4 2 5
1.4 2
1 .4 1 5
1.4 1
Mean (Hot
measurement)
Experiment
0Closed-loop,
1.436
10 =
target
20
1.438
Run
Variance
30
3.94E-06
Open-loop, from first
1.437
9.97E-06
experiment
Lecture 20 © D.E. Hardt, all rights reserved
5/1/08
Variance
Ratio
40
0.395
-
58
Manufacturing
Conclusion
• Model Predictions and Experiment are
in Good Agreement
– Delay - Gain Process Model
– Normal - Additive Disturbance
– Effect of Correlated vs. Uncorrelated
(NIDI) Disturbances
Lecture 20 © D.E. Hardt.
5/1/08
59
Conclusion
Manufacturing
• Cycle to Cycle Control
–
–
–
–
–
–
Obeys Root Locus Prediction wrt Dynamics
Amplifies NIDI Disturbance as Expected
Attenuate non-NIDI Disturbance
Can Reduce Mean Error (Zero if I-control)
Can Reduce “Open Loop” Expected Loss
Correlation Sure Helps!!!!
• Can be Extended to Multivariable Case
– PhD by Adam Rzepniewski (5/5/05)
• Developed Theory and demonstrated on 100X100
problem (discrete die sheet forming)
Lecture 20 © D.E. Hardt.
5/1/08
60