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Batch process monitoring and improvement

7.7. Batch process monitoring and improvement

A batch makes its product in a fixed time and is then emptied, so what it leaves behind is not a row of numbers but a table: every recorded tag, at every sample of the batch. Stack the tables of many batches and the data set is a three-way array of batches by tags by time.

Every model of the trajectories in this section reaches that array the same way. The table of one batch is flattened into a single long row, so the array becomes an ordinary matrix with one row per batch and one column per (tag, time) pair, and the tools of the preceding sections apply to it unchanged. This is batchwise unfolding, set out for PCA by Nomikos and MacGregor (1994) and carried to PLS in their multi-way partial least squares paper the following year. A component is then a weight on every (tag, time) cell, which says both which tags matter and when in the batch they matter. Westerhuis, Kourti and MacGregor (1999) compare it with the other ways of arranging the same array.

What such a model can be asked is fixed by what it is given. That single choice, and not the choice of method, decides what each of the three case studies below can and cannot say.

Three worked case studies accompany this page. Each is built on a dataset that can be downloaded and on models that can be reproduced with the code shown on the page:

7.7.1. What each model can be asked

Each study gives its model one more thing than the study before it, and each addition buys an answer the previous model could not give.

What each case study’s model is given, and what that buys.

Study

What the model is given

What the model answers

What it cannot see

DuPont reactor

Ten trajectories

Which batches differ from the rest, in which tags, and at which samples

Anything whose cause leaves no trace in those ten tags

SBR reactor

Six trajectories and five quality attributes

What the quality will be, from part of a batch, and whether a running batch is departing

A quality attribute the trajectories do not drive

FMC dryer

Eleven charge-chemistry measurements, nine operating conditions, eleven trajectories and eight quality attributes

Which of the three input blocks a quality problem came from

A cause in none of the three input blocks

Read the last column. A model is silent about whatever it was not given, and that silence is not visible from inside the model. Four of the DuPont batches made poor product and sit among the normal batches on both statistics the ten trajectories support. The first case study ends on that point: the condition of a batch has to be observable through the measurements before any model of them can report on it.

Adding a block is therefore a measurement decision before it is a modelling one. It also costs something: in the third case study the chemistry block keeps under a quarter of its own variation over the two components of the multiblock model, against half over the two components of a PLS on that block alone, because the shared direction is chosen for what predicts quality across all three blocks.

7.7.2. The two numbers every one of these models reports

Whatever the model was given, a batch is judged on two distances, and they carry the same meaning in all three studies.

  • Hotelling’s \(T^2\) is how far the batch lies along the model’s components, compared with the reference batches. A large value is a batch doing more of what the reference batches did.

  • The squared prediction error, SPE, is how far it lies off them. A large value is a batch doing something the reference batches never did.

Which of the two is large says what kind of departure it is, and that changes what can be done next. A departure along the components can be extrapolated: in the second case study the model forecasts the remainder of such a batch in the right direction, and more closely the later the forecast is made. A departure off the components can be flagged, and the tags and samples carrying the residual can be named, but not forecast, because nothing in the reference set describes where it is going.

Nomikos and MacGregor (1995) give the charts and the limits for both statistics, and Nomikos (1996) the contribution plots that name the tags behind a large value. How well such a chart detects a fault of a given size is assessed by Ramaker and co-workers (2006).

7.7.3. Two decisions taken before the model

The first is alignment. Unfolding assumes that sample \(k\) means the same stage of the batch in every row, so the batches have to be put on one time axis before any of this. Batches that run to different lengths are aligned within each phase against a maturity variable, a measured quantity that rises steadily through that phase, as the third case study does with the collector level and the dryer temperature. González-Martínez, Ferrer and Westerhuis (2011) set out dynamic time warping, which aligns the raw trajectories instead, and does so for a batch that is still running.

Alignment discards the durations, so a process where the duration itself carries information keeps it as a column, which is what the eleventh trajectory of the batch dryer does.

The second decision is which batches count as normal. A monitoring model reports how far a batch is from its reference set, so that set defines what “unusual” means, and the rule Nomikos and MacGregor give for it is that it carries only batches with acceptable operation and acceptable product. A model built to understand what happened rather than to judge a new batch is under no such constraint, and Kosanovich, Dahl and Piovoso (1996) use one that way. The first case study keeps its poor batches in the model on purpose, to show what the trajectories do and do not separate; the second fits its on-line model on the normal batches alone, because it is judging a running batch against them. Where normal operation is itself moving, as in a startup or a grade transition, Duchesne, Kourti and MacGregor (2002) show how the reference set is built.

7.7.4. Beyond monitoring: changing the batch

A model that says which trajectory shape goes with good product can also be read backwards, to ask what shape to run. Duchesne and MacGregor (2000) optimize the trajectories themselves this way, and Flores-Cerrillo and MacGregor (2004) adjust a running batch to hit a quality target, a mid-course correction, while their batch-to-batch paper carries information from one batch to the next, for a process that repeats the same recipe. The second case study watches a running batch without adjusting it.

Estimating the scores of a batch observed so far is a missing-data problem. García-Muñoz, Kourti and MacGregor (2004) carry that estimate through a running batch and set the limits it is judged against, which is what the on-line section of the second case study builds. García-Muñoz and co-workers (2003) are the source of the batch dryer, and Wold and co-workers (2009) model the same dryer with landmark features in place of the raw trajectories.

Two theses carry the derivations and the fuller case studies behind these papers: Nomikos (1995) for the charts and their limits, and García-Muñoz (2004) for the batch dryer and the score estimate of a running batch.

The columns need not be process tags alone. Gurden, Westerhuis and Smilde (2002) take a spectrum at every sample, so each wavelength is a trajectory, and Kourti, Nomikos and MacGregor (1995) group the columns into blocks, which is what the third case study builds on.

7.7.5. References

7.7.5.1. Foundational multiway PCA / PLS for batches

7.7.5.2. Control, optimization and product quality

7.7.5.3. Performance, alignment, and spectroscopy

7.7.5.4. Missing data and the scores of a batch so far

A batch that is still running is a row whose later cells are not yet measured. Nomikos and MacGregor (1995, Technometrics, listed above) compared three ways to fill those cells, and one of them treats them as missing data; the case studies take that route, estimating the scores from the observed cells alone. The estimators themselves, single-component projection, projection to the model plane and trimmed score regression, are set out in:

7.7.5.5. Theses (McMaster University)

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