Beyond Control: Maths of a Control System

Control systems are all around us, and understanding them is going to make you much better at hardware design. In the last article — Beyond Control: The Basics of Control Systems — we looked at an overview of what a control systems are in general with the example: “everything in between water and time is a control system”. We also observed control systems in nature, where I described my keen ability to fill a glass of water without catastrophic results. That discussion involved the basic concept of a block diagram (without maths) and we expanded that a bit to see what our satellite dish example would look like (still without maths).

I promised some big ugly maths in this article, and we’ll get to that in a bit, never you fear. First let’s have a look at how some basic elements: resistors, inductors, and capacitors are defined in the time domain. Don’t let these first few definitions turn you off. No matter how you feel about calculus, you don’t necessarily need to fully understand each equation. What’s more important is how the equations themselves combine to solve the circuit. Also important is that I will do everything possible to get out of doing difficult math. So stick with me through the article and you’ll learn that agony-saving trick for yourself!

A quick recap on transfer functions before we get going might be beneficial. A control system is used to define electromechanical behavior. For example: our satellite dish (from the previous article) at some point will need to be moved from one position to another position and as control engineers it is our job to determine just how this action will take place. I’m not talking about setting the mood for the big emotional robotic rotation, more like: not damaging the equipment or any people that might be nearby when moving the dish. For many reasons the dish would need to be moved with extreme care and in a very precise manner. The control system is the mathematical definition of that movement. Often the maths of the definition are nasty differential equations, (remember I’m avoiding any math that can be avoided, right?) so, instead of using differential equations to define the system, the transfer function will define the system with algebra, relating the output of the system to the input.

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Beyond Control: The Basics of Control Systems

Control systems are exactly what you think they are: systems designed to control something. Perhaps a better way to put it is systems design control the behavior of something. The term “control systems” does an excellent job of being vague and most of us (originally) don’t think too much about it until its brought to our attention or we crash a robotic armature into itself and investigate how that horrifying event was allowed to happened. Usually during this investigation our internal dialog has a loop running that goes something like: “why the hell will the system allow me to manipulate it in a self-destructive way!?!”

What I found was my own ignorance, I hadn’t implemented a proper control system. One could make a case claiming that I hadn’t taken ANY control system into account whatsoever. I jumped in too deep, too fast (sound familiar?) and paid the price of crashing a rotating arm into another part of the system. Luckily, a friend stepped in and repaired the arm for me and metaphorically pointed to a large neon sign on the wall and said “you can’t ignore this”. He walked over to pull the chain dangling beneath the sign, the high voltage energized the gas in the tubes blinding me with the now unavoidably obvious words: Control Systems.

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