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Unformatted text preview: http://ocw.mit.edu ____________ MIT OpenCourseWare 2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit: ________________ http://ocw.mit.edu/terms. Control of Manufacturing Processes Processes Subject 2.830 Spring 2007 Lecture #20 “Cycle To Cycle Control: Cycle The Case for using Feedback and SPC” The May 1, 2008 Manufacturing The General Process Control The Problem Problem CONTROLLER CONTROLLER Equipment loop Material loop Process output loop EQUIPMENT EQUIPMENT MATERIAL MATERIAL Product Desired Product Control of Equipment: Forces, Velocities Temperatures, , .. 5/1/08 Control of Material Strains Stresses Temperatures, Pressures, .. Lecture 20 © D.E. Hardt. Control of Product: Geometry and Properties 2 Manufacturing Output Feedback Control ∂Y ΔY = Δα ∂α ∂Y ∂Y Δu = − Δα ∂u ∂α ∂Y + Δu ∂u Manipulate Actively Such that Compensate for Disturbances Lecture 20 © D.E. Hardt. 5/1/08 3 Manufacturing Process Control Hierarchy Good Housekeeping Standard Operations (SOP’s) Statistical Analysis and Identification of Sources (SPC) Feedback Control of Machines • Reduce Disturbances – – – – • Reduce Sensitivity (iincrease “Robustness”) ncrease – Measure Sensitivities via Designed Experiments – Adjust “free” parameters to minimize Adjust • Measure output and manipulate inputs – Feedback control of Output(s) Lecture 20 © D.E. Hardt. 5/1/08 4 Manufacturing The Generic Feedback “Regulator” The Problem D1(s) D2(s) Y(s) R(s) - Gc(s) H(s) Gp(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 Manufacturing Effect of Feedback on Effect Random Disturbances Random • Feedback Minimizes Mean Shift (SteadyState Component) • Feedback Can Reduce Dynamic Disturbances 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -4 -3 -2 -1 0 1 2 3 -4 4 -3 -2 -1 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 ? Lecture 20 © D.E. Hardt. 5/1/08 6 Manufacturing Typical Disturbances – External Forces Resisting Motion – Environment Changes (e.g Temperature) – Power Supply Changes • Equipment Control • Material Control – Constitutive Property Changes • • • • 5/1/08 Hardness Thickness Composition ... Lecture 20 © D.E. Hardt. 7 Manufacturing The Dynamics of The Disturbances Disturbances • • • • Slowly Varying Quantities Cyclic Infrequent Stepwise Random Lecture 20 © D.E. Hardt. 5/1/08 8 Manufacturing Example: Material Property Example: Changes Changes • A constitutive property change from constitutive workpiece to workpiece workpiece – In-Process Effect? • A new constant parameter new constant • 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 Ideal Output Output • This Measurement Can Always be This made After the Cycle made • Equipment Inputs can Always be Equipment Adjusted Between Cycles Adjusted • Within the Cycle Inputs Are Fixed Within Lecture 20 © D.E. Hardt. 5/1/08 10 Manufacturing What is Cycle to Cycle? • Measure and Adjust Once per Cycle d r - y 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 Primarily Processing Processing • Similar Results, Different Derivations • More Limited in Analysis and More Extension to Larger problems Extension 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 Cycle Objectives Objectives • How to Reduce E(L(x)) & Increase Cpk with with Feedback? Feedback? • Bring Output Closer to Target – Minimize Mean or Steady - State Error Minimize • Decrease Variance of Output – Reject Time Varying Disturbances Lecture 20 © D.E. Hardt. 5/1/08 13 Manufacturing A Model for Cycle to Cycle Model Feedback Control Feedback • Simplest In-Process Dynamics: d(t) u(t)) 4τp Cycle Time Tc > 4τp y(t) 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 Discrete Measurement Measurement d(t) u(t)) y(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 Manufacturing The Sampler d(t) u(t) y(t) TS yi y(t) yi TS Lecture 20 © D.E. Hardt. 5/1/08 i 16 Manufacturing A Cycle to Cycle Process Model y(t) y1 y2 y3 y4 yi 1 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 Kp - or - yi a discrete output sequence at time intervals Tc ui Equipment αe Material αm Lecture 20 © D.E. Hardt. 5/1/08 yi 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 – Time to compute • Process Delay – Waiting for next available machine cycle Lecture 20 © D.E. Hardt. 5/1/08 20 Manufacturing Delays z = n - step time advance operator e.g. n z * yi = yi +1 1 yi-1 yi yi+1 yi+2 z * yi = yi + 2 2 yi-2 and z * yi = yi −1 Lecture 20 © D.E. Hardt. 5/1/08 −1 i 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 Manufacturing Modeling Randomness • Recall the Output of a “Real Process” Recall Width 2 In) 1.025 1.02 d(t) u(t) y(t) TS yi 1.015 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) TS yi u(t) y(t) 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 continuous any instant is governed by a normal distribution distribution • From instant to instant there is no From correlation • Therefore if we sample this process we Therefore get: get: • A NIDI random number Normal 5/1/08 Identically Distributed Lecture 20 © D.E. Hardt. Independent 26 Manufacturing The Gaussian “Process” The ... i+2 i+1 i Lecture 20 © D.E. Hardt. 5/1/08 27 Manufacturing Constant (Mean Value) Disturbance Constant Rejection- P control Rejection di ~NIDI(μ,σ2) ui Kp/z yi r - Kc yi = di + K pui −1 ui −1 = K c (r − yi −1 ) yi = di + K p 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” Higher rejection but only K = ∞ eliminates mean shifts Lecture 20 D.E. Hardt. 5/1/08 29 Manufacturing Error: Try an Integrator i ui = Kc ∑ ei j =1 running sum of all errors recursive form (ei = r − yi ) ui+1 = ui + Kc ei+1 zU = U + Kc zE r z u Gc ( z ) = K c = z −1 e di - z ui Kc z −1 Kp z yi Lecture 20 © D.E. Hardt. 5/1/08 30 Manufacturing Constant Disturbance Constant Integral Control D r - z ui Kc z −1 Kp z yi 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 Lecture 20 © D.E. Hardt. 5/1/08 Zero error Zero regardless of loop gain gain 31 Manufacturing Effect of Loop Gain K on Time Response: I-Control Step Response Step Response 1.0 Amplitude 1.0 TextEnd K=0.5 0 4 6 8 10 12 TextEnd K=1.5 4 6 8 10 12 Time (samples) 0 2 0 2 Time (samples) Step Response Step Response 1.0 Amplitude 1.0 K=0.9 TextEnd TextEnd K=2 10 20 Time (samples) 30 40 0 0 2 4 6 Time (samples) 8 10 0 0 Lecture 20 © D.E. Hardt. 5/1/08 32 Manufacturing Effect of Loop Gain K = KcKp Step Response 1.0 Amplitude 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? What • di is defined as a NIDI sequence is • Therefore: – Each successive value of the sequence is Each probably different probably – Knowing the prior values: di-1, di-2, di-3,… Knowing 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 Manufacturing Thus 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 Lecture 20 © D.E. Hardt. 5/1/08 will not cancel di will 35 Manufacturing Variance Change with Loop Variance Gain Gain 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 2 σ CtC σ2 Loop Gain Lecture 20 © D.E. Hardt. 5/1/08 36 Manufacturing Conclusion - CtC with Un-Correlated Conclusion Correlated (Independent) Random Disturbance (Independent) • • • Mean error will be zero using “I” control Mean Variance will increase with loop gain Increase in σ at K=1 ~ 1.5 * σ open loop Increase open 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 37 Lecture 20 © D.E. Hardt. -4 -3 -2 -1 5/1/08 Manufacturing What if the Disturbance is not NIDI? di NIDI Sequence a z-a Filter (Correlator) cdi CD(Z )(z − a) = aD( z) cdi +1 = a(di − cdi ) unknown 5/1/08 expect some correlation, expect therefore ability to counteract some of the disturbances disturbances known Lecture 20 © D.E. Hardt. 38 Manufacturing What if the Disturbance is What not NIDI? not Proportional Control Simulation Kc n Disturbance σ2Ctc /σ2o 1 0.89 0.77 0.69 1.39 39 0.7 z+0.7 NIDI Sequence + Sum Kc Gain Filter (Correlator) 1/z Process + + Sum1 0 0.1 y Ouput 0.25 0.5 0.9 Lecture 20 © D.E. Hardt. 5/1/08 Manufacturing Gain - Variance Reduction Gain 2 σ CtC σ2 2.0 Increasing Correlation 1.0 0 5/1/08 0.5 Lecture 20 © D.E. Hardt. 1.0 Loop Gain 1.5 40 Manufacturing Conclusion - CtC with Correlated Conclusion CtC (Dependent) Random Disturbance (Dependent) • • • Mean error will be zero using “I” control Mean Variance will decrease with loop gain Best Loop Gain is still KcKp =1 Best 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 Lecture 20 © D.E. Hardt. 5/1/08 41 Manufacturing Conclusions from Conclusions Cycle to Cycle Control Theory Cycle • Feedback Control of NIDI Disturbance Feedback will Increase Variance will – Variance Increases with Gain • BUT: If Disturbance is NID but not I; BUT: NID but We CAN Decrease Variance We – Higher Gains -> Lower Variance Higher – Design Problem: Low Error and Low Design Variance Variance Lecture 20 © D.E. Hardt. 5/1/08 42 Manufacturing How to Tell if Disturbance is Independent – Look at the Autocorrelation Look – Effect of Filter on Autocorrelation • Correlation of output data • 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 Could Effects Effects – Step Mean Changes from Cycle Step Disruption 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 Manufacturing Process Model for Bending Local Response Curve (Mat'l I) 60 50 40 Angle Angle 30 20 10 0 1.15 0.025” steel y = 150.85x - 156 2 R = 0.9992 1.2 1.25 1.3 Punch depth Lecture 20 © D.E. Hardt. 1.35 1.4 5/1/08 49 Manufacturing Results for Kc=0.7;Δμ=0 Open-loop Closed-loop 36.5 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 Manufacturing I-Control Δμ≠0 Control Δμ 36 35 34 33 32 31 30 29 28 27 26 0 Material Shift 5 10 Run number 15 20 2 σ CtC = 1.01 2 σ Tar t Kc=0.5 Angle ss= 34.95=0.14% Lecture 20 © D.E. Hardt. 5/1/08 51 Manufacturing Minimum Expected Loss Minimum Integral-Controller Integral Calculated Expected Loss vs. Gain Performance Index versus Feedback Gain ( σ^2 = 0.0618, S = 7.2581, N = Verification Experimental 20) 16 Performance Index, V 14 12 10 8 6 4 2 0 0 0.2 0.4 0.6 0.8 1 1.2 Fe e dbac k Ga in, Kc Lecture 20 © D.E. Hardt. 5/1/08 Kc 0.5 0.98 1.5 V 4.47 3.11 4.89 1.4 1.6 1 .8 2 52 Manufacturing Disturbance Response for Disturbance “Optimal” Integral Control Gain 39 37 Target 35 33 31 29 27 25 0 Material Shift 5 10 15 20 25 Angle K=1 0.025” steel to 0.02” steel Run number Lecture 20 © D.E. Hardt. 5/1/08 53 Manufacturing Injection Molding: Injection Process Model Process Levels 5 sec 0.5 in/sec 20 sec 6 in/sec ˆ Y = 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) 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 2.92E-04 SS 49.6 0 0 0 0 49.6 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 0.831 Fcrit 4.35 4.35 4.35 4.35 p-value 0 0.004 0.255 0.373 ˆ Y = β 0 + β 2 ⋅ X2 Lecture 20 © D.E. Hardt. 5/1/08 Final Model 54 Manufacturing P-Control Injection Molding 1.45 1.445 1.44 Open Loop Closed Loop, Gain = 0.5 Diameter 1.435 1.43 1.425 1.42 1.415 1.41 0 Experiment Open-loop 20 Closed-loop Mean ( Hot measurement) Variance Variance Ratio Hot Cold 1.437 9.97E-06 40 60 Run 1.439 2.34E-06 Lecture 20 © D.E. Hardt. 80 0.234 100 5/1/08 55 Manufacturing Output Autocorrelation 0.8 0.6 0.4 0.5 0.8 0.7 0.6 0.2 0 -0.2 -0.4 -0.6 -0.8 0.4 0.3 0.2 0.1 0 -0.1 -0.2 0 5 10 15 20 0 5 10 15 20 Bending Injection Molding Matlab function XCORR Lecture 20 © D.E. Hardt. 5/1/08 56 Manufacturing P-control: Moving Target Closed Loop Target = 1.436 Closed Loop Target = 1.44 Open Loop 1.455 1.45 1.445 Diameter 1.44 1.435 1.43 1.425 1.42 1.415 1.41 Experiment Mean ( Hot measurement) Variance Variance Ratio 0 Closed-loop, target = 1.436Ó 20 40 Closed-loop, target = 1.440Ó Open-loop 1.438 60 80 6.79E-06 100 Run 1.440 4.54E-06 1.437 1.44E-05 0.471 120 0.315 - 140 Lecture 20 © D.E. Hardt. 5/1/08 57 Manufacturing Injection Molding: Integral Control 1.4 5 1 .4 4 5 1.4 4 1 .4 3 5 1.4 3 1 .4 2 5 1.4 2 1 .4 1 5 1.4 1 Diameter Experiment 0Closed-loop, 10 target = Mean (Hot measurement) 1.438 Run 20 Variance 3.94E-06 30 Variance Ratio 0.395 58 40 1.436 Open-loop, from first 1.437 9.97E-06 experiment Lecture 20 © D.E. Hardt, all rights reserved 5/1/08 Manufacturing Conclusion • Model Predictions and Experiment are Model in Good Agreement in – Delay - Gain Process Model Delay – Normal - Additive Disturbance Normal – Effect of Correlated vs. Uncorrelated Effect (NIDI) Disturbances (NIDI) Lecture 20 © D.E. Hardt. 5/1/08 59 Manufacturing Conclusion • 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 Can Correlation Sure Helps!!!! • Can be Extended to Multivariable Case – PhD by Adam Rzepniewski (5/5/05) • Developed Theory and demonstrated on 100X100 Developed problem (discrete die sheet forming) problem Lecture 20 © D.E. Hardt. 5/1/08 60 ...
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This note was uploaded on 09/24/2010 for the course MECHE 2.830J taught by Professor Davidhardt during the Spring '08 term at MIT.

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