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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 Control of Manufacturing ProcessesProcesses
Subject 2.830Subject 2.830Spring 2007Spring 2007Lecture #20Lecture #20
““Cycle To Cycle Control: Cycle To Cycle Control: The Case for using Feedback and SPCThe Case for using Feedback and SPC””
May 1, 2008May 1, 2008
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Lecture 20 © D.E. Hardt.2
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The General Process Control The General Process Control ProblemProblem
EQUIPMENTEQUIPMENT MATERIALMATERIALCONTROLLERCONTROLLER
Desired Product Product
Equipment loop
Control of Equipment:
Forces,
Velocities
Temperatures,
, ..
Material loop
Control of Material
Strains
Stresses
Temperatures,
Pressures, ..
Process output loop
Control of Product:
Geometry
and
Properties
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Lecture 20 © D.E. Hardt.3
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Output Feedback Control
ΔY = ∂Y∂α
Δα +∂Y∂u
Δu
Manipulate Actively Such that
∂Y∂u
Δu = −∂Y∂α
Δα
Compensate for Disturbances
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Lecture 20 © D.E. Hardt.4
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Process Control HierarchyProcess Control Hierarchy
•• Reduce DisturbancesReduce Disturbances–– Good HousekeepingGood Housekeeping–– Standard Operations (SOPStandard Operations (SOP’’s)s)–– Statistical Analysis and Identification of Sources (SPC)Statistical Analysis and Identification of Sources (SPC)–– Feedback Control of MachinesFeedback Control of Machines
•• Reduce Sensitivity (Reduce Sensitivity (increase increase ““RobustnessRobustness””))–– Measure Sensitivities via Designed ExperimentsMeasure Sensitivities via Designed Experiments–– Adjust Adjust ““freefree”” parameters to minimizeparameters to minimize
•• Measure output and manipulate inputsMeasure output and manipulate inputs–– Feedback control of Output(s)Feedback control of Output(s)
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Lecture 20 © D.E. Hardt.5
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The Generic Feedback The Generic Feedback ““RegulatorRegulator””ProblemProblem
Y(s)
D2(s)
Gp(s)Gp(s)Gc(s)Gc(s)
H(s)H(s)
D1(s)
R(s)-
••Minimize the Effect of the Minimize the Effect of the ““DD’’ss””
••Minimize Effect of Changes in GMinimize Effect of Changes in Gpp
••Follow R exactlyFollow R exactly
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Lecture 20 © D.E. Hardt.6
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Effect of Feedback on Effect of Feedback on Random DisturbancesRandom Disturbances
•• Feedback Minimizes Mean Shift (SteadyFeedback Minimizes Mean Shift (Steady--State Component)State Component)
•• Feedback Can Reduce Dynamic Feedback Can Reduce Dynamic DisturbancesDisturbances
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 4
?
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Lecture 20 © D.E. Hardt.7
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Typical DisturbancesTypical Disturbances
•• Equipment ControlEquipment Control–– External Forces Resisting MotionExternal Forces Resisting Motion–– Environment Changes (e.g Temperature)Environment Changes (e.g Temperature)–– Power Supply ChangesPower Supply Changes
•• Material ControlMaterial Control–– Constitutive Property ChangesConstitutive Property Changes
•• HardnessHardness•• ThicknessThickness•• CompositionComposition•• ......
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Lecture 20 © D.E. Hardt.8
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The Dynamics of The Dynamics of DisturbancesDisturbances
•• Slowly Varying QuantitiesSlowly Varying Quantities•• CyclicCyclic•• Infrequent StepwiseInfrequent Stepwise•• RandomRandom
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Lecture 20 © D.E. Hardt.9
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Example: Material Property Example: Material Property ChangesChanges
•• A constitutive property change from A constitutive property change from workpiece to workpieceworkpiece to workpiece–– InIn--Process Effect?Process Effect?
•• A new A new constantconstant parameterparameter•• Different outcome each cycleDifferent outcome each cycle
–– Cycle to Cycle EffectCycle to Cycle Effect•• Discrete random outputs over timeDiscrete random outputs over time
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Lecture 20 © D.E. Hardt.10
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What is Cycle to Cycle?What is Cycle to Cycle?
•• Ideal Feedback is the Actual Product Ideal Feedback is the Actual Product OutputOutput
•• This Measurement Can Always be This Measurement Can Always be made After the Cyclemade After the Cycle
•• Equipment Inputs can Always be Equipment Inputs can Always be Adjusted Between CyclesAdjusted Between Cycles
•• Within the Cycle Inputs Are Fixed Within the Cycle Inputs Are Fixed
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Lecture 20 © D.E. Hardt.11
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What is Cycle to Cycle?What is Cycle to Cycle?
•• Measure and Adjust Once per CycleMeasure and Adjust Once per Cycle
y
d
ProcessProcessControllerControllerr
-
Execute the Loop Once Per Cycle
Discrete Intervals rather then Continuous
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Run by Run ControlRun by Run Control
•• Developed from an SPC PerspectiveDeveloped from an SPC Perspective•• Primarily used in Semiconductor Primarily used in Semiconductor
ProcessingProcessing•• Similar Results, Different DerivationsSimilar Results, Different Derivations•• More Limited in Analysis and More Limited in Analysis and
Extension to Larger problemsExtension 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.
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Lecture 20 © D.E. Hardt.13
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Cycle to Cycle Feedback Cycle to Cycle Feedback ObjectivesObjectives
•• How to Reduce E(L(x)) & Increase CHow to Reduce E(L(x)) & Increase Cpkpk with with Feedback?Feedback?
•• Bring Output Closer to TargetBring Output Closer to Target–– Minimize Mean or Steady Minimize Mean or Steady -- State ErrorState Error
•• Decrease Variance of OutputDecrease Variance of Output–– Reject Time Varying DisturbancesReject Time Varying Disturbances
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Lecture 20 © D.E. Hardt.14
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A Model for Cycle to Cycle A Model for Cycle to Cycle Feedback ControlFeedback Control
•• SimplestSimplest InIn--Process Dynamics:Process Dynamics:
u(t)) y(t)
d(t)
Cycle Time Tc > 4τp
d(t) = disturbances seen at the output (e.g. a Gaussian noise)
τp = Equivalent Process Time Constant
4τp
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Lecture 20 © D.E. Hardt.15
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Discrete Product Output Discrete Product Output MeasurementMeasurement
yi
Continuous variable y(t) to sequential variable yi
?u(t)) y(t)
d(t)
i = time interval or cycle number
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Lecture 20 © D.E. Hardt.16
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The SamplerThe Sampler
u(t) y(t)
d(t)
yi
TS
y(t) yi
iTS
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Lecture 20 © D.E. Hardt.17
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A Cycle to Cycle Process ModelA Cycle to Cycle Process Model
y(t)
With a Long Sample Time, The Process has no Apparent Dynamics, i.e. a Very Small Time Constant
1 2 3 4 i
y1
y2
…
y3
y4yi
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Lecture 20 © D.E. Hardt.18
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A Cycle to Cycle Process ModelA Cycle to Cycle Process Model
Kpui yi
- or -
Equipment Materialui yi
αe αm
a discrete output sequenceat time intervals Tc
a discrete inputsequence atinterval Tc
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Lecture 20 © D.E. Hardt.19
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Cycle to Cycle Output ControlCycle to Cycle Output Control
PartController Process
Output Sampling
DesiredPart
Process Uncertainty
-
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Lecture 20 © D.E. Hardt.20
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DelaysDelays
•• Measurement DelaysMeasurement Delays–– Time to acquire and gageTime to acquire and gage–– Time to reach equilibriumTime to reach equilibrium
•• Controller DelaysController Delays–– Time to Time to ““decidedecide””–– Time to computeTime to compute
•• Process DelayProcess Delay–– Waiting for next available machine cycleWaiting for next available machine cycle
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Lecture 20 © D.E. Hardt.21
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DelaysDelays
zn = n - step time advance operatore.g.z1 * yi = yi+1
z2 * yi = yi+2
andz−1 * yi = yi−1
yi-2
yi-1 yi
yi+1yi+2
i
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Lecture 20 © D.E. Hardt.22
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A Pure Delay Process ModelA Pure Delay Process Model
Kpui-1 yi
yi = Kpui −1
z-1 Kp
u y
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Lecture 20 © D.E. Hardt.23
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Modeling RandomnessModeling Randomness
•• Recall the Output of a Recall the Output of a ““Real ProcessReal Process””
•• Random even with inputs held constantRandom even with inputs held constant
u(t) y(t)
d(t)
yi
TS
Width 2 In)
1
1.005
1.01
1.015
1.02
1.025
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
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Output Disturbance ModelOutput Disturbance Model
u(t) y(t)
d(t)
yi
TS
Model: d(t) is a continuous random variable that we sample every cycle (Tc)
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Or In Cycle to Cycle Control TermsOr In Cycle to Cycle Control Terms
d
Gp(z)-r y
Gc(z)
Where:
d(t) is a sequence of random numbers governed by
a stationary normal distribution function
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Gaussian White NoiseGaussian White Noise
•• A continuous random variable that at A continuous random variable that at any instant is governed by a normal any instant is governed by a normal distributiondistribution
•• From instant to instant there is no From instant to instant there is no correlation correlation
•• Therefore if we sample this process we Therefore if we sample this process we get:get:
•• A NIDI random numberA NIDI random number
Normal Identically Distributed Independent
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The Gaussian The Gaussian ““ProcessProcess””
i
i+1
i+2
...
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Lecture 20 © D.E. Hardt.28
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Constant (Mean Value) Disturbance Constant (Mean Value) Disturbance RejectionRejection-- P controlP control
Kp/z-r yi
Kc yi = di + K pui−1
if di = μ (a constant), we can look at steady - state behavior:
ui−1 = Kc (r − yi−1)
ui
yi = di + K pKc (r − yi−1)
yi ⇒ yi−1 ⇒ y∞ =di
1 + K pKc
+ rK pKc
1 + K pKc
di ~NIDI(μ,σ2)
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And For ExampleAnd For Example
Higher loop gain K improves Higher loop gain K improves ““rejectionrejection”
but only
K = ∞ eliminates mean shifts
Thus if we want to eliminate the constant (mean) component of the disturbance
y∞
di
=1
1+ K pKc
=1
1+ K
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Error: Try an IntegratorError: Try an Integrator
Gc (z) = Kcz
z − 1=
ue
ui = Kc eij =1
i
∑ running sum of all errors
recursive formui+1 = ui + Kcei+1
zU = U + KczE
(ei = r − yi )
di
-r yiui K p
zKc
zz − 1
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Constant Disturbance Constant Disturbance --Integral ControlIntegral Control
Zero error Zero error regardless regardless of loop of loop gaingain
yi+1 + (1− KcK p )yi = di+1 − di
Y (z) =z − 1
z − 1+ KcK p
D
-r yiui K p
zKc
zz − 1
(Assume r=0)
or
Again at steady state yi+1 = yi = y∞
And since D is a constant y∞ (2 − KcK p ) = 0
D
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Effect of Loop Gain K on Time Response: Effect of Loop Gain K on Time Response: II--ControlControl
Time (samples)
TextEnd
Step Response
0 10 20 30 40
Time (samples)
TextEnd
Step Response
0 2 4 6 8 10 12
K=1.5
K=2
Time (samples)
Ampl
itude
TextEnd
Step Response
2 4 6 8 10 12
K=0.5
1.0
0
Time (samples)
Ampl
itude
TextEnd
Step Response
0 2 4 6 8 10
K=0.9
1.0
0
1.0
0
1.0
0
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Effect of Loop Gain K = KEffect of Loop Gain K = KccKKpp
Time (samples)
Ampl
itude
TextEnd
Step Response
0 2 4 6 8 10
K=1.0
1.0
0
Best performance at Loop Gain K= 1.0
Stability Limits on Loop Gain 0<K<2
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What about random component of What about random component of dd? ?
•• ddi i is defined as a NIDI sequenceis defined as a NIDI sequence•• Therefore:Therefore:
–– Each successive value of the sequence is Each successive value of the sequence is probably differentprobably different
–– Knowing the prior values: Knowing the prior values: ddii--11, d, dii--22, d, dii--33,,……will not help in predicting the next valuewill not help in predicting the next value
e.g. di ≠ a1di−1 + a2di−2 + a3di− 3 + ...
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ThusThus
di
Kp/z-r y
Kc
This implies that with our cycle to cycle process model under proportional control:
xue
The output of the plant xi will at best represent the errorfrom the previous value of di-1
xi = −KcKpdi −1will not cancel will not cancel ddii
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Variance Change with Loop Variance Change with Loop GainGain
1
1.2
1.4
1.6
1.8
2
2.2
2.4
0 0.2 0.4 0.6 0.8 1 1.2
Loop Gain
σCtC
σ
2
2
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Conclusion Conclusion -- CtC with UnCtC with Un--Correlated Correlated (Independent) Random Disturbance(Independent) Random Disturbance
•• Mean error will be zero using Mean error will be zero using ““II”” controlcontrol•• Variance will increase with loop gainVariance will increase with loop gain•• Increase in Increase in σσ at at KK=1 ~ 1.5 * =1 ~ 1.5 * σσ open loopopen loop
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 4
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NIDI Sequence
az-a
Filter(Correlator)
What if the Disturbance is not NIDI?What if the Disturbance is not NIDI?
di cdi
CD(Z )(z − a) = aD(z)
cdi +1 = a(di − cdi )
unknown known
expect some correlation, expect some correlation, therefore ability to therefore ability to counteract some of the counteract some of the disturbancesdisturbances
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What if the Disturbance is What if the Disturbance is not NIDI?not NIDI?
++
Sum1
y
Ouput
+-
Sum
0.7z+0.7
Filter(Correlator)
NIDI Sequence
n
Disturbance
1/z
Process
Kc
Gain
Kc σ2Ctc /σ2
o
0 1
0.1 0.89
0.25 0.77
0.5 0.69
0.9 1.39
Proportional ControlProportional Control
SimulationSimulation
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Gain Gain -- Variance ReductionVariance Reduction
Loop Gain
σCtC2
σ 2
0 0.5 1.0 1.5
1.0
2.0Increasing Correlation
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Conclusion Conclusion -- CtC with Correlated CtC with Correlated (Dependent) Random Disturbance(Dependent) Random Disturbance
•• Mean error will be zero using Mean error will be zero using ““II”” controlcontrol•• Variance will decrease with loop gainVariance will decrease with loop gain•• Best Loop Gain is still Best Loop Gain is still KKccKKpp =1=1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 40
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-4 -3 -2 -1 0 1 2 3 4
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Conclusions from Conclusions from Cycle to Cycle Control TheoryCycle to Cycle Control Theory
•• Feedback Control of NIDI Disturbance Feedback Control of NIDI Disturbance will Increase Variancewill Increase Variance–– Variance Increases with GainVariance Increases with Gain
•• BUT: If Disturbance is BUT: If Disturbance is NIDNID but not but not II; ; We CAN Decrease VarianceWe CAN Decrease Variance–– Higher Gains Higher Gains --> Lower Variance> Lower Variance–– Design Problem: Low Error and Low Design Problem: Low Error and Low
VarianceVariance
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How to Tell if DisturbanceHow to Tell if Disturbanceis Independentis Independent
•• Correlation of output dataCorrelation of output data–– Look at the Autocorrelation Look at the Autocorrelation –– Effect of Filter on AutocorrelationEffect of Filter on Autocorrelation
•• Reaction of Process to FeedbackReaction of Process to Feedback–– If variance decreases data has dependenceIf variance decreases data has dependence
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Is DisturbanceIs Disturbanceis Independent?is Independent?
•• Correlation of output dataCorrelation of output data–– Look at the Autocorrelation Look at the Autocorrelation –– Effect of Filter on AutocorrelationEffect of Filter on Autocorrelation
•• Reaction of Process to FeedbackReaction of Process to Feedback–– If variance decreases then data must have If variance decreases then data must have
some dependencesome dependence
Φxx (τ) = x(t)x( t −τ )dt−∞
∞
∫
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But Does It Really Work?But Does It Really Work?–– LetLet’’s Look at Bending and Injection s Look at Bending and Injection
MoldingMolding
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Experimental DataExperimental Data
Cycle to Cycle Feedback Control of
Manufacturing Processesby
George Tsz-Sin SiuSM Thesis
Massachusetts Institute of Technology
February 2001
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Experimental ResultsExperimental Results
•• BendingBending–– Expect NIDI NoiseExpect NIDI Noise–– Can Have Step Mean ChangesCan Have Step Mean Changes
•• Injection MoldingInjection Molding–– Could be Correlated owing to Thermal Could be Correlated owing to Thermal
EffectsEffects–– Step Mean Changes from Cycle Step Mean Changes from Cycle
DisruptionDisruption
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Process Model for BendingProcess Model for Bending
Kpui yi
yi = Kpui −1
Y (z) =Kp
zKp=?
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Local Response Curve (Mat'l I)
y = 150.85x - 156R2 = 0.9992
0
10
20
30
40
50
60
1.15 1.2 1.25 1.3 1.35 1.4Punch depth
Ang
le
Process Model for BendingProcess Model for BendingA
n gle 0.025” steel
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33.5
34
34.5
35
35.5
36
36.5
0 10 20 30 40 50 60 70 80 90 100
Run number
Target
Open-loop Closed-loop
Results for KResults for Kcc=0.7;=0.7;ΔμΔμ=0 =0
=1.67
Theoretical =1.96
σCtC2
σ 2
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II--Control Control ΔμΔμ≠≠0 0
2627282930313233343536
0 5 10 15 20Run numberMaterial
Shift
Tart
σCtC2
σ 2 = 1.01
Angle ss= 34.95=0.14%
Kc=0.5
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Minimum Expected Loss Minimum Expected Loss IntegralIntegral--ControllerController
Performance Index versus Feedback Gain(σ 2̂ = 0.0618, S = 7.2581, N = 20)
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Feedback Gain, Kc
Perf
orm
ance
Inde
x, V
Calculated Expected Loss vs. Gain
Kc V0.5 4.47
0.98 3.111.5 4.89
Experimental Verification
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Disturbance Response for Disturbance Response for ““OptimalOptimal”” Integral Control GainIntegral Control Gain
25
27
29
31
33
35
37
39
0 5 10 15 20 25
Run number
Ang
le
MaterialShif t
Target
0.025” steel to 0.02” steel
K=1
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Injection Molding: Injection Molding: Process ModelProcess Model
ˆ Y = β0 + β2 ⋅ X2 + β3 ⋅ X3 + β23 ⋅ X2 ⋅ X3 Initial ModelProcess inputs Levels
X2 = Hold time (seconds) 5 sec 20 sec X3 = Injection speed (in/sec) 0.5 in/sec 6 in/sec
ˆ Y = β0 + β2 ⋅ X2 Final Model
Effect beta SS DOF MS F Fcrit p-value1 1.437 49.6 1 49.568 2E+07 4.35 0
X2 (Hold time) -1.04E-03 0 1 2.60E-05 10.593 4.35 0.004X3 (Injection speed) -3.75E-04 0 1 3.38E-06 1.373 4.35 0.255
X2X3(Hold time*Injection speed) 2.92E-04 0 1 2.04E-06 0.831 4.35 0.373
Error 0 20 2.46E-06Total 49.6 24
ANOVA on model terms
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PP--Control Injection MoldingControl Injection Molding
1.41
1.415
1.42
1.425
1.43
1.435
1.44
1.445
1.45
0 20 40 60 80 100Run
Dia
met
er
Open Loop Closed Loop, Gain = 0.5
Hot
Cold
Experiment Mean ( Hotmeasurement) Variance Variance
Ratio
Open-loop 1.437 9.97E-06 -
Closed-loop 1.439 2.34E-06 0.234
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Output AutocorrelationOutput Autocorrelation
0 5 10 15 20-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 5 10 15 20-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Bending Injection Molding
Matlab function XCORR
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1.41
1.415
1.42
1.425
1.43
1.435
1.44
1.445
1.45
1.455
0 20 40 60 80 100 120 140Run
Dia
met
er
Closed LoopTarget = 1.436
Closed LoopTarget = 1.44
Open Loop
PP--control: Moving Targetcontrol: Moving Target
Experiment Mean ( Hotmeasurement) Variance Variance Ratio
Closed-loop, target = 1.436Ó 1.438 6.79E-06 0.471
Closed-loop, target = 1.440Ó 1.440 4.54E-06 0.315
Open-loop 1.437 1.44E-05 -
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Injection Molding:Injection Molding:Integral ControlIntegral Control
1.41
1.415
1.42
1.425
1.43
1.435
1.44
1.445
1.45
0 10 20 30 40Run
Dia
met
er
Experiment Mean (Hot measurement) Variance Variance
Ratio
Closed-loop, target = 1.436 1.438 3.94E-06 0.395
Open-loop, from first experiment 1.437 9.97E-06 -
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ConclusionConclusion
•• Model Predictions and Experiment are Model Predictions and Experiment are in Good Agreementin Good Agreement–– Delay Delay -- Gain Process ModelGain Process Model–– Normal Normal -- Additive DisturbanceAdditive Disturbance–– Effect of Correlated vs. Uncorrelated Effect of Correlated vs. Uncorrelated
(NIDI) Disturbances(NIDI) Disturbances
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ConclusionConclusion
•• Cycle to Cycle ControlCycle to Cycle Control–– Obeys Root Locus Prediction wrt DynamicsObeys Root Locus Prediction wrt Dynamics–– Amplifies NIDI Disturbance as ExpectedAmplifies NIDI Disturbance as Expected–– Attenuate nonAttenuate non--NIDI DisturbanceNIDI Disturbance–– Can Reduce Mean Error (Zero if ICan Reduce Mean Error (Zero if I--control)control)–– Can Reduce Can Reduce ““Open LoopOpen Loop”” Expected LossExpected Loss–– Correlation Sure Helps!!!!Correlation Sure Helps!!!!
•• Can be Extended to Multivariable CaseCan be Extended to Multivariable Case–– PhD by Adam Rzepniewski (5/5/05)PhD by Adam Rzepniewski (5/5/05)
•• Developed Theory and demonstrated on 100X100 Developed Theory and demonstrated on 100X100 problem (discrete die sheet forming)problem (discrete die sheet forming)