controls for operations personnel

Lesson 1

This lesson will describe some basic definitions used when talking about controls.  The focus is on how a control affects operations.  The definitions provided here are: control, average value, variability

I. Control

The dictionary (www.thefreedictionary.com) has several definitions for control – two of which apply here: 

“An intelligence agent who supervises or instructs another agent”

In our case, the intelligence agent is what we normally call a control.  It can be software in a Distributed Control System (DCS), a standalone controller, a Programmable Logic Controller (PLC), a hardware based drive controller, or any number of other possibilities.  The “other agent” is typically a valve, motor, or some other piece of process equipment.  Sometimes the other agent is itself another control.  In other words, a control supervises/instructs another control.  Where one control directly controls another, it is often referred to a cascaded control.  Cascaded controls have special maintenance procedures.  When one or more controls control one or more controls, these are often called supervisory controls.

“To adjust to a requirement; regulate”

The requirement part of the control is perhaps the most important from an operator’s standpoint.  The requirements can be considered as two large groups:

On/off control is a type of control where the system is in one of only two possible states – on or off.  This might be a motor, a pump, or even a valve.  In the case of the valve it is either completely closed or completely open.  For example, the quench valve of a ClO2 generator should be 100% open during a puff or 100% closed during normal operation.  There are no circumstances where the valve is run at 50% open.  It is easily determined if an On/Off control is operating properly – is the system on/open/off/closed when it should be.  Another example of on/off control is the household furnace.  It provides full output when running.  When the thermostat determines the house temperature is below a certain threshold, it turns the furnace on, when the temperature rises above another threshold, the furnace is turned off.  The actual temperature in the house regularly varies between the lower and upper threshold.

Continuous control is a bit more difficult.  Here, the requirement is to maintain some value over time.  For example, in the on/off example, the household temperature thresholds may be set at 71.5 and 72.5 degrees to maintain a temperature of about 72 degrees.  In continuous control, we would want to maintain 72 degrees.  Industrial examples of continuous control include maintaining a set: level in a tank, pressure in a system or rate.  The big question for operators, control engineers and E&I maintenance personnel is: “What does maintain a value really mean?”  Two important concepts here are: average value over time and variability over time.

II. Average

The average value of a set of readings can be found by adding up all the readings and then dividing by the number of readings.   If the readings of a randomly varying process were graphed, the average value would be the line that has half the readings above it and half below.  Figure 1 below shows a graph of a regularly varying process and its average value. Figure 2 shows the same data for a randomly varying process.  Although the process values are dramatically different, the average value is the same.



Figure 1: Graph of average value of a regularly variable process


Figure 2: Graph of average value of a randomly variable process

 

It is much easier to visualize the average value in Figure 1. Randomly varying processes, as shown in Figure 2, usually need some mathematical analysis to determine the average.  Unfortunately, randomly varying processes are typically much more common than regularly variable processes.  Fortunately, the variability itself is often of more interest than the average value, so operators do not usually calculate the average value of a process variable.  It should also be noted that if a regular cycle (or a regular cycle plus some random variability) is seen, then there usually is an identifiable cause.

III. Variability

If we look at Figure 1, we see the values range from some minimum to some maximum.  Consider the household temperature example given previously.  This looks a little like what we might expect from on/off control.  We know that there will be variability in the on/off control, but, in truth, there will be variability in the continuous control as well.  This course will discuss sources of variability.  If Figure 1 is an example of variability in on/off control, then Figure 2 is an example of variability in continuous control.  This section will discuss how variability is measured.  This is probably the most important concept in the entire course.


One way to describe variability is: "a measurement of deviation from the average".  Two commonly used measures are the range and the standard deviation.  The range is simply the distance between the maximum value and the minimum value over some period of time.  Mathematically, It is (the maximum value – the minimum value). 

In Figure 1, the maximum value is 6 and the minimum is 4 so, the range is 2.  Range is an easy way to determine the performance of a loop.  Assume that a customer has a specification on some variable as 10 ± 2.  If during production, a randomly varying loop maintains an average of 10.1 with a range of 3.2, then we can guess that the maximum value seen was 11.7 and the minimum was 8.5.  Therefore, all product met this spec.  If on the other hand, the range was 6, then we know that some product was out of spec because the minimum would now be 7.1 and the maximum would be 13.1 whereas the spec requires values to be between 8 and 12.  Given that we don't know the distribution of the values, we do not know how much was out of spec.

The other measure of variability is the standard deviation.  This is the number that helps us to understand the distribution of values within a range. Standard deviation (or s)  is a little more complex to calculate than the range.  Because it is a mathematical calculation, it is described using a formula that uses symbols that represent certain calculations and numbers. Don't get scared away by this. After you see, the formula, we will describe in detail what it really means and how to use it.

The formula for calculating a standard deviation is as follows:


standard dev formula
Equation 1: Standard Deviation


In this equation,

s (or s for sigma) is the standard deviation (the number you are trying to calculate)

...... is the square root sign. Once you have calculated everything else, you will take the square root of the resulting number to find the standard deviation. Finding a square root means finding the number that when multiplied by itself will give you the number under the square root sign. For example, say that you did all the other calculations and the number under the square root sign came out to 4. The square root of 4 is 2 (because 2 times itself (or 2 x 2) = 4), so the standard deviation would be 2.

.... - the first symbol under the square root sign is called a summation sign. It means to add everything together once you have done all the other calculations. More on that in a minute.

y is the measured value for a single sample

(or y-bar) is the average value of all of the samples

 is the measured value for a single sample minus the average for all the samples. This is calculated for each and every sample taken.

is the sum of all of the above terms.  This sum is then squared - that is, multiplied by itself.

n is the number of samples (that is, how many measurements you took).

(n-1) is the number of samples minus 1. There is a mathematical reason for using this that we won't go into here but you take the sum of all of the (y - y-bar) values squared and then divide by (n-1) to find the number that goes under the square root sign.

Operators rarely need to calculate the value as it is often provided by a computer program – but they should know how to calculate it. An easy way to understand what is happening with this calculation is to use an example.

Example 1

[put simple example here - whole numbers or no more than one decimal]

Example 2

Process variables are rarely measured in such simple numbers, before moving on, let's look at a more process-like example. Given the following table of numbers:


4.951104

5.0504

5.218094

5.377307

5.048452

4.661929

4.892677

4.693085

5.231365

5.119092

5.031102


Table 1: Random process data

 

We add up all the numbers and find that the sum is 55.27460636.  As there are 11 values the average is 55.27460636/11 = 5.024964214.  We now build a new table with three columns, column 1 is the process value, column 2 is the process value – the average value and column 3 is the value of column 2 squared.

Value (y)

4.951103513

-0.05867

0.003442

5.0503999

0.040628

0.001651

5.218093587

0.208322

0.043398

5.377307288

0.367536

0.135083

5.048451525

0.03868

0.001496

4.661929398

-0.34784

0.120994

4.892676721

-0.11709

0.013711

4.693085303

-0.31669

0.10029

5.231365263

0.221594

0.049104

5.119092121

0.109321

0.011951

5.031101735

0.02133

0.000455


Table 2: Support table for Standard Deviation Calculation

 

We now calculate the total of the third column as 0.479035622.  Our standard deviation formula then becomes

 

So what does standard deviation mean?  Remember that standard deviation is a way to measure variability. In general, for a normal process operating in control, about 67% of measured values will be within 1 standard deviation of the average (either above or below).  In our example the average is 5.025 and the standard deviation is 0.151 so we should find about 7 samples between 4.873 and 5.176.  Looking at the table we see that 6 values (4.951103513, 5.0503999, 5.048451525, 4.892676721, 5.119092121, 5.031101735) do indeed lie between the one standard deviation limit. 

Statistics is never exact, especially on small samples, so this result is what we would expect.  More commonly we use the 2 standard deviation limit.  In our example that would be between the average ± (2 * 0.151) or between 4.722 and 5.328.  95% of samples should fall within the 2 standard deviation limits.  It can be seen that in our example there are 8 samples that do, so again the data follows the rule. If you look closely, you may notice that about half the sample values are within 2 standard deviations above the average and half are within 2 standard deviations below the average.

But what happens if our data does not follow the statistics rule?  The statistics work for what is called a Gaussian distribution (or a bell curve).  This simply describes the random variation often seen in nature.  Most processes should follow the Gaussian distribution.  However, if they do not, then more or fewer values will fall outside the prescribed limits.  More likely what will be seen is that the right number falls outside the limits, but most values are either above or below the limit.  This indicates that the process is not random, that is, there is some specific circumstance.  For example, consider the case where the right number of points are outside the 2 standard deviation limit, but all of them fall on the plus side of the limit.  This means that the process “operates differently” above the average than it does below the average. 

Consider a case where consistency is being controlled by a controller and a dilution valve.  Under normal conditions the dilution valve operates in the range of 40%-60% open to maintain a consistency of 3%.  A standard deviation s (also called sigma) is calculated and 95% fall inside the 2s range.  However more than half of the points are above the 3% consistency limit.  A possible explanation could be that the dilution valve is unable to open over 50% due to some mechanical binding.  When incoming consistency is randomly high - although the controller may be correctly asking for a >50% valve opening - the valve only opens to 50% and sticks, resulting in higher than desired consistency.  When the incoming consistency drops sufficiently that less than 50% dilution valve opening is required, the valve resumes its normal control.

Because the calculation of standard deviation assumes that the base data is Gaussian, it tends to drive the answer that is expected.  In other words, even if the data is not truly Gaussian, about 95% of the samples will fall within the 2s range.  More importantly, the value of s does not really tell us anything by itself.  What is of more interest in terms of evaluating control performance is the value of s over time.  If it is getting larger, then there is likely a degradation of the process or control.

The next lesson will discuss how evaluating s (or standard deviation) over time is used to determine process and control system performance.

Exercises

Complete exercise 1 of the COP101 Exercises

Next Lesson Variability and Statistical Process Control

Notes.
1) Statistics for Experimenters, George Box, William Hunter, J. Hunter. 1978 John Wiley and Sons. p40

controls for operations personnel c 2007 managed reliability llc