How to Build a Flying Car,II: Control Theory

How to Build a Flying Car: an occasional series of posts that examine various points of interest.

Here you are in your flying car. You are at 10,000 feet and need to go somewhere. You have 50 thruster fans pointed down through the floor of your car and 10 more pointed back. What you need now is some way to tell all of them how much to thrust. You don't have 50 hands, and even if you did, you don't have the brainpower to use them all properly to avoid the fate of the VertiJet's test pilots.

You need control theory.

Control theory has been a substantial part of engineering since Jame Clerk Maxwell published On Governors in the Proceedings of the Royal Society in 1868. This was the day of the steam engine and the Watt governor, descended from the spinning-ball governors used in windmills, which kept the engines running at an even speed.

The Watt governor is fairly simple to understand; the speed of the engine makes the balls try to stand out by centrifugal force, and thus to pull up on a yoke that cuts off the steam supply. There is a happy medium where the supply and speed match just right and so that is the speed the engine will try to run. Setting the balls and the steam-valve right was something of an art, it was an attempt to reduce this to a science that was Maxwell's aim.

Maxwell's analysis sets the stage for another trend in control theory: it contains no less than 73 mathematical formulae and equations. And I can assure you that it is easier to read than anything published on the subject in the 20th century.

What I would like to do in the next few posts is try to give an overview of a small part of control theory, nowhere near what an engineering student would learn in the course of a degree, but enough to  give you an intuitive grasp of what is going on. Note that even this little part is usually covered using differential equations, Laplace transforms, and their maps in the complex plane; but I will strive to avoid that here.

So, here is the problem. Here is your flying car, and I have set the fans so that the thrust is exactly equal to the weight of the car. It ought to hang there motionlessly. But there is also a disturbing force, which here I have labelled "wind" and modeled with a randomly varying force. 

The wind is the wiggly orange and green lines at about 0, from two separate runs. The blue and red lines, respectively, represent the altitude of the car. As you can see, the car doesn't stay put. Whenever it gets a couple of seconds of sustained gusts, it starts moving, and its momentum is enough to keep it going even though the wind returns to an average of zero.  

So what's a flying car to do? A first attempt is to use a controller of a kind we're all familiar with: it's like the thermostat in your home. It's either on or off (or in a more sophisticated version, turns on the heat if it's too cold, and the air conditioner if it's too hot.) But full force in either case. For that reason, a controller of this kind is often referred to as a "bang-bang" controller. For heating and cooling, this actually works rather well. But for our flying car, we have a little problem:

 The scale of the wind is the same here as before. We have drawn in orange the signal for control either pushing up or pushing down. But the blue signal, the car's altitude, is engaging in sickening leaps and plunges, which are getting worse as time goes on!

What's happening, of course, is that the corrective force acting all the time the car is above its intended altitude, is pushing it so hard that by the time it gets back to zero, it's going down so fast it can't stop. For heating and cooling, the system doesn't have that kind of momentum; for flying cars, it does. And that's enough to make the control problem quite a bit more complex.

Something like this happens in airplanes under human control, as well: when it does, it's called PIO for Pilot-Induced-Oscillation. A human pilot, of course, doesn't operate by slamming the controls back and forth to the stops; we try for a proportional response to what we observe. But PIO can still occur; the base reason for it is very different from "bang-bang syndrome." 

Enlightenment lies ahead in our next episode!