- Title: [[MIT 2.830J Control of Manufacturing Processes - Lec 4]] - Type: #source/video - Author: - Reference: - Published at: - Reviewed at: [[2021-10-24]] - Links: https://www.youtube.com/watch?v=R4lUaI7VsK4&list=PL7CF97E01FDE7C51A&index=4 --- [[_Media/MIT Control of Manufacturing Processes S08/lecture4.pdf | Lecture 4 Notes]] [web](https://ocw.mit.edu/courses/mechanical-engineering/2-830j-control-of-manufacturing-processes-sma-6303-spring-2008/lecture-videos/lecture4.pdf) # Probability models of process control e.g measuring diameter on a CNC lathe ![[_Media/MIT Control of Manufacturing Processes S08/CNC lathe diameter.png]] - Looks like there might be drift - something either systematic of periodical - maybe the worker is over-correcting - might be only 1 measurement per run or multiple measurements per run at different points on the same part - induced temperature - induced heat in the part - Shift change mean shift - operator measurement technique difference - Different input material batches - how far the material is in the chuck Measured observations from experiments will always have **variation** that we can attribute to $ randomness + deterministic (systematic) change $ Recall the process model with output $Y$ and process parameters $\alpha$ can be thought of as a function of those process parameters $Y = \Phi(\alpha)$. Some of those process parameters can be treated control inputs $u$ giving $Y = \Phi(\alpha, u)$. Then recall, the variation equation is just first order variation in observable outputs $Y$. $ \Delta Y = \frac{\partial Y}{\partial X}\Delta\alpha + \frac{\partial Y}{\partial X}\Delta u $ Observations are often not enough, you need information from design to characterize the variation (e.g. variation within a part might be by design) There can also be inherent variation in the measurement tool and technique. Guage analysis seeks to characterize measurement error by looking at repeatability and reproducibility of the measurement system. ## How to describe randomness We can start moving from the data towards inferring probability distributions and using intermediate techniques like histograms. This course focuses mostly on parametric probability distributions but there are plenty of non-parametric and computational statistical methods that can infer distributions from the empirical data. Recall that since you can since, for continuous variables you need to ask about the probability of a range of values $Pr(X<x)$ it's useful to to sum (or integrate) the probability density to get the cumulative distribution function (CDF) and therefore the probability density function is just the derivative $pdf(x) = dP/dx$