2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)

Transcription

1 MIT OpenCourseWare J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303) Spring 2008 For information about citing these materials or our Terms of Use, visit:

2 Control of Manufacturing Processes Subject Spring 2007 Lecture #20 Cycle To Cycle Control: The Case for using Feedback and SPC May 1, 2008

3 The General Process Control Manufacturing Problem Desired Product CONTROLLER EQUIPMENT MATERIAL Product Equipment loop Material loop Process output loop Control of Equipment: Forces, Velocities Temperatures,,.. Control of Material Strains Stresses Temperatures, Pressures,.. Control of Product: Geometry and Properties 2

4 Output Feedback Control ΔY = Y α Δα + Y u Δu Manipulate Y u Δu = Y α Δα Actively Such that Compensate for Disturbances 3

5 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( increase Robustness ) Measure Sensitivities via Designed Experiments Adjust free parameters to minimize Measure output and manipulate inputs Feedback control of Output(s) 4

6 The Generic Feedback Regulator Problem D 1 (s) D 2 (s) R(s) - G c (s) G p (s) Y(s) H(s) Minimize the Effect of the D s Minimize Effect of Changes in G p Follow R exactly 5

7 Effect of Feedback on Manufacturing Random Disturbances Feedback Minimizes Mean Shift (Steady- State Component) Feedback Can Reduce Dynamic Disturbances ?

8 Typical Disturbances Equipment Control External Forces Resisting Motion Environment Changes (e.g Temperature) Power Supply Changes Material Control Constitutive Property Changes Hardness Thickness Composition... 7

9 The Dynamics of Manufacturing Disturbances Slowly Varying Quantities Cyclic Infrequent Stepwise Random 8

10 Example: Material Property Manufacturing 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 9

11 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 10

12 What is Cycle to Cycle? Measure and Adjust Once per Cycle d r - Controller Process y Execute the Loop Once Per Cycle Discrete Intervals rather then Continuous 11

13 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 pp 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

14 Cycle to Cycle Feedback Manufacturing Objectives How to Reduce E(L(x)) & Increase C pk with Feedback? Bring Output Closer to Target Minimize Mean or Steady - State Error Decrease Variance of Output Reject Time Varying Disturbances 13

15 A Model for Cycle to Cycle Feedback Control Simplest In-Process Dynamics: d(t) u(t)) y(t) 4τ p Cycle Time T c > 4τ p d(t) = disturbances seen at the output (e.g. a Gaussian noise) τ p = Equivalent Process Time Constant 14

16 Discrete Product Output Measurement u(t)) d(t) y(t)? y i Continuous variable y(t) to sequential variable y i i = time interval or cycle number 15

17 The Sampler u(t) d(t) y(t) T S y i y(t) y i T S i 16

18 A Cycle to Cycle Process Model y(t) y 2 y 3 y 4 y i y i With a Long Sample Time, The Process has no Apparent Dynamics, i.e. a Very Small Time Constant 17

19 A Cycle to Cycle Process Model a discrete input sequence at interval T c u i K p y i a discrete output sequence at time intervals T c -or - u i Equipment Material y i α e α m 18

20 Cycle to Cycle Output Control Process Uncertainty Desired Part - Controller Process Part Output Sampling 19

21 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 20

22 Delays z n = n - step time advance operator e.g. z 1 * y i = y i+1 y i-2 y i-1 y i y i+1 y i+2 z 2 * y i = y i+2 and z 1 * y i = y i 1 i 21

23 A Pure Delay Process Model u i-1 K p y i y i = K p u i 1 u y z -1 K p 22

24 Modeling Randomness Recall the Output of a Real Process Width 2 In) u(t) d(t) y(t) T S y i Random even with inputs held constant 23

25 Output Disturbance Model u(t) d(t) y(t) T S y i Model: d(t) is a continuous random variable that we sample every cycle (T c ) 24

26 Or In Cycle to Cycle Control Terms d r - G c (z) G p (z) y Where: d(t) is a sequence of random numbers governed by a stationary normal distribution function 25

27 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 26

28 The Gaussian Process... i+2 i+1 i 27

29 Constant (Mean Value) Disturbance Rejection- P control d i ~NIDI(μ,σ 2 ) r - K c u i K p /z y i y i = d i + K p u i 1 y i = d i + K p K c (r y i 1 ) u i 1 = K c (r y i 1 ) if d i = μ (a constant), we can look at steady - state behavior: y i y i 1 y = d i 1 + K p K c + r K p K c 1 + K p K c 28

30 And For Example Thus if we want to eliminate the constant (mean) component of the disturbance y d i = K p K c = K Higher loop gain K improves rejection but only K = eliminates mean shifts 29

31 Error: Try an Integrator u i = K c i e i j =1 u i+1 = u i + K c e i+1 running sum of all errors recursive form (e i = r y i ) zu = U + K c ze G c (z) = K c z z 1 = u e d i r - K c z z 1 u i K p z y i 30

32 Constant Disturbance - Integral Control r - K c z z 1 u i K p z D y i Y (z) = z 1 z 1+ K c K p D (Assume r=0) or y i+1 + (1 K c K p )y i = d i+1 d i Again at steady state y i+1 = y i = y And since D is a constant y (2 K c K p ) = 0 Zero error regardless of loop gain 31

33 1.0 Effect of Loop Gain K on Time Response: I-Control Step Response 1.0 Step Response Amplitude TextEnd K=0.5 TextEnd K= Time (samples) Time (samples) 1.0 Step Response 1.0 Step Response Amplitude TextEnd K=0.9 TextEnd K= Time (samples) Time (samples) 32

34 Effect of Loop Gain K = K c K p Step Response 1.0 Amplitude TextEnd K= Time (samples) Best performance at Loop Gain K= 1.0 Stability Limits on Loop Gain 0<K<2 33

35 What about random component of d? d i is defined as a NIDI sequence Therefore: Each successive value of the sequence is probably different Knowing the prior values: d i-1, d i-2, d i-3, will not help in predicting the next value e.g. d i a 1 d i 1 + a 2 d i 2 + a 3 d i

36 Thus This implies that with our cycle to cycle process model under proportional control: r e - K c u K p /z x d i y The output of the plant x i will at best represent the error from the previous value of d i-1 x i = K c K p d i 1 will not cancel d i 35

37 Variance Change with Loop Manufacturing Gain σ 2 CtC σ Loop Gain 36

38 Conclusion - CtC with Un-Correlated (Independent) Random Disturbance 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

39 What if the Disturbance is not NIDI? d i a z-a cd i NIDI Sequence Filter (Correlator) CD(Z)(z a) = ad(z) cd i +1 = a(d i cd i ) expect some correlation, therefore ability to counteract some of the disturbances unknown known 38

40 What if the Disturbance is not NIDI? Proportional Control Simulation K c σ 2 Ctc /σ2 o NIDI Sequence 0.7 z+0.7 Filter (Correlator) n Disturbance Sum Kc Gain 1/z Process + + Sum1 y Ouput

41 Gain - Variance Reduction 2 σ CtC σ Increasing Correlation 1.0 Loop Gain

42 Conclusion - CtC with Correlated (Dependent) Random Disturbance Mean error will be zero using I control Variance will decrease with loop gain Best Loop Gain is still K c K p =

43 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 42

44 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 43

45 Is Disturbance is Independent? Correlation of output data Look at the Autocorrelation Effect of Filter on Autocorrelation Φ xx (τ) = Reaction of Process to Feedback If variance decreases then data must have some dependence x(t)x(t τ )dt 44

46 But Does It Really Work? Let s s Look at Bending and Injection Molding 45

47 Experimental Data Cycle to Cycle Feedback Control of Manufacturing Processes by George Tsz-Sin Siu SM Thesis Massachusetts Institute of Technology February

48 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 47

49 Process Model for Bending u i K p y i y i = K p u i 1 Y(z) = K p z K p =? 48

50 Process Model for Bending 60 Local Response Curve (Mat'l I) 50 Angle Angle steel y = x R 2 = Punch depth 49

51 Results for K c =0.7;Δμ= Open-loop Closed-loop 35.5 Target Run number 2 σ CtC σ 2 =1.67 Theoretical =

52 I-Control Δμ 0 Tar t σ CtC = 1.01 σ 2 K c =0.5 Angle ss = 34.95=0.14% Material Run number Shift 51

53 Minimum Expected Loss Manufacturing Integral-Controller Performance Index, V Calculated Performance Expected Index Loss versus vs. Gain Feedback Gain (σ^2 = , Experimental S = , N = Verification 20) Kc Fee dbac k Ga in, Kc V

54 Disturbance Response for Optimal Integral Control Gain Target 35 Angle K= steel to 0.02 steel Mat erial Shift Run number 53

55 Injection Molding: Manufacturing Process Model ˆ Y = β 0 + β 2 X 2 + β 3 X 3 + β 23 X 2 X 3 Initial Model Process inputs Levels X2 = Hold time (seconds) 5 sec 20 sec X3 = Injection speed (in/sec) 0.5 in/sec 6 in/sec Effect beta SS DOF MS F Fcrit p-value E X2 (Hold time) -1.04E E X3 (Injection speed) -3.75E E X2X3 ANOVA on model terms (Hold time*injection speed) 2.92E E Error E-06 Total Y ˆ = β + β X Final Model 54

56 P-Control Injection Molding Open Loop Closed Loop, Gain = 0.5 Diameter Hot Cold Experiment Mean ( Hot measurement) Variance Variance Ratio 0 Open-loop E Run Closed-loop E

57 Output Autocorrelation Bending Injection Molding Matlab function XCORR 56

58 P-control: Moving Target Closed Loop Target = Closed Loop Target = 1.44 Open Loop Diameter Experiment Mean ( Hot measurement) Variance Variance Ratio 0 Closed-loop, 20 target = 1.436Ó E Closed-loop, target = 1.440Ó Run 4.54E Open-loop E-05-57

59 Injection Molding: Manufacturing Integral Control Diameter Experiment Mean (Hot measurement) Variance Variance Ratio Run 3.94E Closed-loop, target = Open-loop, from first experiment E-06 - Lecture 20 D.E. Hardt, all rights reserved 58

60 Conclusion Model Predictions and Experiment are in Good Agreement Delay - Gain Process Model Normal - Additive Disturbance Effect of Correlated vs. Uncorrelated (NIDI) Disturbances 59

61 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) I 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) 60