When you lift your hand to catch a ball it is very unlikely that your brain actually formulates or solves the equations of motion in working out when and where to place your hand to catch the ball. It’s much more likely that you use, unconsciously, something that’s much more akin to the geometrical methods underlying the

**Geometric Process Control (GPC)**technology described in this article.

Geometric Process Control (GPC) combines the three key plant applications of Process Control, Production Control (i.e. Achievement of Business Objectives such as producing in-spec product, maximising Yield or Recovery, minimising waste and many other Key Performance Indicators) and Alarm Management.

It improves all three applications which were previously quite separate as they lacked a unifying mathematical basis. They came together only in the different brains of individual process operators hence inconsistently. Substantial economic and safety improvements result.

The mathematical basis for the breakthrough is the use of n-dimensional geometry together with Inselberg's coordinate transformation that makes it possible to see a multi-variable graph containing perhaps several hundred variables (such as temperatures, pressures, flows and product qualities) and thousands of different observations in a single picture. Let’s say that again. One graph that can show all the contents of a spreadsheet of hundreds of columns and thousands of rows in a single picture. Today, and for the last several thousand years, the world has been restricted to graphs that could show at most half-a-dozen variables. This has affected and limited understanding of multi-variable processes by forcing over-simplification. Behaviour could then only be represented within the limitations imposed by the very few variables that could be shown in a simple graph.

We use the multi-variable graph to choose a set of ‘Best Practice’ operating points, the values of all the variables, which gave us the good results we wanted and exclude the points that didn’t. We obtain the data to decide on these points from analysing past process operations. Then we construct a multi-dimensional solid object from those points and having done that we can now test any other point against that object by saying that geometrically we need to stay inside the object so that we achieve the same ‘Best Practice’ objective used in choosing the good points. A new-format Operator Display is part of the method.

We have used GPC on test cases from oil refineries right the way down to bakeries so another benefit is that the methods are very simple to apply. It doesn’t need advanced mathematics, algebra or equations. We cut out the first two steps of traditional mathematical modelling which were (1) describe your problem with equations (2) solve them. We go directly from observation to the model in a few minutes. And this means we can use this technique in small plants where they don’t have control engineers or advanced mathematics available to them. We have broadened the application of this advanced technology to a much wider range of industry than was possible before.

Economic Benefits arising from the Unification are, perhaps unsurprisingly, substantial and an outline is given at the end of actual achievement at one of the Field-Trial sites. The value of our first-ever method to calculate values for process alarm limits was recognised by the European Process Safety Centres 2003 Award for the biggest single contribution to improving process safety.

### Setting-Up a Process

Process industries have coped with the problems by finding operational ranges of process variables within which they have been able to make in-specification product at acceptable yield or recovery levels. The finding of these ranges is known as ‘Setting-Up’ a process and the results are documented as Operating Procedures. Operations, Production Planning and Process Control all then aim for process operation at or close to a Target operating point (an operating point is the instantaneous set of values of all the variables) which is commonly in the centre of these ranges (‘mean-centred operation’ is a phrase that will be familiar to some).Geometry Unifies Process Control, Production Control and Alarm ManagementViewed geometrically, the set of ranges define a multi-dimensional rectangular Box. It is rectangular because simple ranges on individual variables imply independence of variables so that for the simple two-variable example. The range AD on variable V1 and DC on variable V2 imply that we can operate anywhere inside the rectangle ABCD and get good product.

But the variables are not independent so it will be found that good product is actually obtained only when operating in the sub-ranges EH and HG. The two rectangles ABCD and EFGH are obviously connected by the shaded oval shape shown along one of the diagonals of ABCD.

The oval shape is a ‘contour’ within which good product (by some criteria separate from V1 and V2) can be made. ABCD is the enclosing rectangle and EFGH is one of many possible enclosed rectangles that have been found by the process operators by trial-and-error to produce best results. Different operators may have different working Boxes.

This geometrical view is easy to understand in the two dimensions needed to consider a two-variable problem or even in the three dimensions needed to understand a three-variable problem. It gets difficult in a real process when one may have to consider two or three hundred variables because we have not previously been able to view two or three-hundred dimensional objects. Obviously though the same principles apply to the enclosing Box that we seek as the result of Process Set-Up, to the operators preferred operating Box that must be one of the many hypercube equivalents of EFGH, and to the multi-dimensional sausage (intentionally avoiding a noun with mathematical implications) that represents the ‘contour’ within which good product is made.

So whats the problem? Doesn’t everyone have their Box and why not find the shape of the Sausage, put that in the Operating Procedures and operate there instead? There are several problems in practice.

Geometry Unifies Process Control, Production Control and Alarm ManagementThe first is that most Boxes resulting from Process Set-Up are inconsistent largely because there has been no way to see or otherwise verify their consistency. By consistency we mean that all the ranges taken together exactly define a rectangular box without the overlap or underlap that you would get if you tried to assemble a three-dimensional box from sheets of cardboard of assorted sizes as shown.

Relatively few plants think in these terms and so have not checked their Operating Procedure Box for consistency. Many plants don’t even operate wholly inside their Box because they haven’t understood the concept and the fact that results outside the Box are always bad whereas results inside the Box are often good. Control algorithms haven’t understood this either; perhaps because they are Operating Point-focussed so don’t provide assistance in keeping the process inside its Box.

A very few plants have diligently, using designed experiments or even one variable at a time, found a Box mostly interior to the sausage in which the proportion of good results is even higher but have not appreciated that this partly-interior Box is one of many and that therefore an even better one may be available to them. A practical limitation is that it may be beyond the capability of the control system (whether manual or automatic) to maintain operation inside the small ranges that can result on some variables from the use of wholly or partially-interior Boxes.

We thus begin to see that accepting the Box which encloses the sausage where we always make good product as an operating space begins to provide a way to bring together the achievement of Business Objectives and the Operating Point-focused topic of process or equipment control. In most plants it will also give an immediate increase in the proportion of good results.

We can also see that the sausage or Best Operating Zone would be an even better place to operate but have lacked the necessary ability to describe a multi-dimensional object quantitatively such that we could document it in operating procedures and communicate it to others. In consequence other mathematical methods such as Partial Least Squares (PLS), Dimensionless Numbers and Principal Components (PCA) have attempted to describe the sausage (or at least its major features) with equations and then to make it visible by introducing new variables (e.g. Reynolds Number) that reduce the dimensionality of the problem by replacing several source variables with one composite variable with the ultimate aim of shrinking the problem to just two new compound variables that can be plotted in an x-y graph. While Reynolds Number itself has become a familiar variable whose significance is understood by most engineers, the compound variables introduced by the other methods tend to have no known meaning to process engineers or to process operators. The 2-dimensional “You Are Here” plots produced are less useful than might be hoped because the space in which they are drawn does not intuitively relate to plant operations.

### Parallel Dimensions

Geometry Unifies Process Control, Production Control and Alarm ManagementRecognising that the key problem is visualisation we use a new type of graph in which the axes are drawn vertically and parallel to each other, this allowing many axes, and a related set of variable values (a multi-dimensional Point) are represented by a polygonal line connecting the value s of each variable plotted on its own axis. For an example see the illustration where all the process values of variables P1 through P14 at approximately 0900 on the 14th of the month have been associated with their resulting lab qualities (q4 through q8) and shown as one polygonal line.The graph is more useful when many points are shown and in this one 996 data points are present. The quality specifications of top-grade product are shown as red triangles delineating the allowable ranges of the specifications and all polygonal lines passing through all five ranges have been coloured yellow.

Notice that the five ranges on the Quality variables are themselves a Box and that they are inconsistent in allowing far more range than is required on q4, q5, q7 and q8. There is economic opportunity in improving specifications and/or eliminating unnecessary laboratory analyses.

Geometry Unifies Process Control, Production Control and Alarm ManagementThe enclosing Consistent Box on the process variables is easily constructed using the yellow ranges on each of the process variables and is shown. The Box encloses all of the good product points and the additional points that lie inside the Box but outside the sausage. The extent of these can be seen on the quality variable axes. The good product points were 12% of the total whereas the Box encloses 39% of the total points. The Yield of good product would rise to 12/39 or approximately 30% if the process was always operated to stay inside the Box. This represents an overnight Yield increase of 250% and is probably achievable without capital investment. Yield would increase by 250% by operating only inside this Box.

There are many other Boxes that are of interest. For instance, the Box on the Manipuable variables can give us information about whether our Manipuable variables are currently capable of controlling to stay within the Box on the process variables or whether and to what extent we should be justifying process control improvements.

### The Best Operating Zone

The Box is defined by the values of the variables on their axes whereas the BOZ (the sausage) is defined by the set of yellow selected points. Separating these points gives the set of points that define the Best Operating Zone. We use these to document and model a multi-dimensional solid object and then have as our operating objective the geometric objective of remaining at all times as an interior point of this object thus ensuring that we achieve the objective (top grade product) by which the points were selected. The beauty of this type of modelling is that it is wholly data-driven. The mathematics are independent of the type of process so are the same for modelling a bakery or an oil refinery and can thus be wholly concealed from the user so avoiding the need for a bakery or other smaller plant to employ mathematically-specialised engineering staff.Geometry Unifies Process Control, Production Control and Alarm ManagementThe resulting BOZ model is shown and is in fact the Operators Display. It includes only variables already known to the operator.

The dark blue dots are the instantaneous values of the process variables read from the real-time control system, the red envelope is the chosen BOZ view and the green envelope represents the set of usable ranges on all the process and quality variables that are a result of the need to remain as an interior point of the BOZ. Quality variables are shown in turquoise. Their values are not available in real-time so their ranges are predicted. Actual values are shown here as turquoise dots in this after-the-event replay to allow the engineer to compare the predicted ranges with the actually achieved values.

Obviously as the process values alter from one time-step to the next so the green ranges change and the shape of the green envelope alters. We have to stay inside this green envelope at all times and this is the defined role of process control. Operators rapidly appreciate that the green envelope shows them the usable range of each process variable relative to their operating objective and to the current operating point of the process. This is information that they have never had before but which they immediately understand. The display shows, for instance, two variables on the 7th and 14th axes with values just below the mid-line of their red (fixed) ranges. They would previously have been taken to be good values but in fact they are almost on the low limit of their green ranges so are in fact values that the operator should be watching closely.

### New Definitions for Process Alarms and a New Role for Process Control

Process alarms fit into this geometric display too. Process Control, whether it is wholly manual or fully automatic, has the role of keeping us inside the green envelope at all times and in a properly built model is capable of achieving this. This if an individual variable should stray outside its green range it is the first indication of a problem in that process control has lost the capability to control. It is therefore the earliest time at which we are entitled to raise a High or Low alarm. If the variable value should go outside the red range it is a violation of the proven capability of the process and confirmation of change sufficient to alter the process capability. This is where we should raise a HighHigh or LowLow alarm. The HighHigh/LowLow alarm limits thus stay fixed in value, as they do today, whereas the High/Low alarm limits move about as the process moves so allowing them to interact with each other and very substantially reducing the number of false alarms compared to the previous fixed values, which were in fact always wrong.### Operating Advice for the Operator without a Rules Base

Geometry Unifies Process Control, Production Control and Alarm ManagementAlarms are shown to the operator on the display. Further use is then made of geometry to generate advice to the operator that will help him clear the alarm. Each process variable is identified as Manipulable or non-Manipulable (A flow is Manipulable whereas a temperature isn’t in a manual control system. In an automated control system Setpoints, Targets and even Limits or Constraints of the control scheme may be considered Manipulable.) The Advisory Algorithm attempts to find a set of moves on the Manipulable variables only that will change the shape of the green envelope to clear the alarm condition.Here we see two process alarms and a resulting quality prediction outside the in-specification range. The two process alarms on non-Manipulable variables can be cleared by making moves on three Manipulable variables from the present black dots to the new blue dots. This will cause the envelope shape to change from the present green position to the new blue position which changes process operation sufficiently to clear the two alarms on the non-Manipulable variables and also predicts a quality value which is once more within specification. It is important to note that this advice is generated by a same-for-everyone algorithm. A Rules Base is not required thus reducing the cost by one or two orders of magnitude.

### Affordable RealTime Optimisation for All Plants

The same algorithm is used when no alarms are present to find moves that will maximise and minimise variables previously identified for optimisation this delivering RealTime Optimisation (RTO) without the expense of developing a first-principles model.Those in the pharmaceutical industry will recognise this as the solution to the PAT requirement. Others will recognise it as the solution to the multi-variable product quality control problem which has eluded them for so long.