When did you last use calculus on the job? My guess is never. So why do all the control theory textbooks bombard us with equations and expect us to understand how that relates to our real world processes? In these days of computer aided design and analysis tools – the need to solve a differential equation has been all but eliminated. This means that most engineers have losttouch with the concept of calculus, and how it applies to the real world. This is a shame because in many engineering disciplines and particularly in process control – the ability to visualise a problem mathematically is what really separates the real pros from the rest of the crowd. In fact even the humble PID contains those scary sounding calculus terms Integral and Derivative. This sectionaims to give you a feel for what these terms really mean. This will give you a real edge when understanding what’s going on in your controller.
Go into the control room of a process plant and ask the operator: “What’s the derivative of reactor 4’s pressure?”
And the response will typically be: “Bugger off smart arse!”
However go in and ask: “What’s the rate of change of reactor4’s pressure?”
And the operator will examine the pressure trend and say something like: “About 5 PSI every 10 minutes” He’s just performed calculus on the pressure trend! (don’t tell him though or he’ll want a pay rise) So derivative is just a mathematical term meaning rate-of-change. That’s all there is to it. Testing your understanding
Suppose you have a box of electronics that calculatesthe derivative of its input signal. Its output is connected to an analogue meter which reads zero when vertical, negative to the left and positive to the right. Look at the diagram below and draw on it where the meter would be pointing for each of the 4 input signals.
Scroll Down for the answer…
Were you right? If not – remember that the absolute value of the input signal does not matter,all that matters is whether it is changing through time, and if so in which direction:
• • • •
If the input signal is not changing the output will be zero. If the input signal is increasing linearly, the output will be positive and stationary. If the input signal is decreasing linearly, the output will be negative and stationary. If the input signal is getting steeper with time, then the meterwill be positive and moving to the right.
And so on. If you understand these concepts, then you know everything you need to about differential calculus in order to understand the PID algorithm.
Integrals without the Math
Is it any wonder that so many people run scared from the concept of integrals and integration, when this is a typical definition?
What the!?!? If you understood thatyou are a smarter person than me. Here’s a plain English definition: The integral of a signal is the sum of all the instantaneous values that the signal has been, from whenever you started counting until you stop counting. So if you are to plot your signal on a trend and your signal is sampled every second, and let’s say you are measuring temperature. If you were to superimpose the integral of thesignal over the first 5 seconds – it would look like this:
The green line is your temperature, the red circles are where your control system has sampled the temperature and the blue area is the integral of the temperature signal. It is the sum of the 5 temperature values over the time period that you are interested in. In numerical terms it is the sum of the areas of each of the bluerectangles: (13 x 1)+(14x1)+(13x1)+(12x1)+(11x1) = 63 °C s The curious units (degrees Celsius x seconds) are because we have to multiply a temperature by a time – but the units aren’t important. As you can probably remember from school –the integral turns out to be the area under the curve. When we have real world systems, we actually get an approximation to the area under the curve, which as you can see...