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Fractals, Chaos, and Control Systems on Rails


Facebook's Control Theory group, you know you are a Control theorist when

Posted by Harry Seldon on February 26, 2009

You will want to join the Control Theory group on Facebook if, or you know you are a Control theorist when:

  • For you, Vibration Control is not only a Sex toy.

  • Finding the good frequency is not only the problem of your girlfriend.

  • “Slower, Faster, now it is good” reminds you your work.

  • You see everything as a control system. (which is called being controlphrenic)

  • You have already thought “laws are way too complex, a PID would do the job”.

  • You have already said to your lawyer “Don’t tell me about laws, I am designing them all day”.

  • Every 10 words you say is either feedback, stable, unstable, transient, control or state.

  • Kalman is some god.

  • Navigation is for you another word for Kalman filter.

  • You know the difference between guidance, navigation and controls.

  • For you, airplanes, rockets, cars, finance, politics, trains, teams should be controlled with some good sensors, good actuators and good controls.

  • “Everything is under control” means something to you.

  • Socrates should have said “control yourself” instead of “know yourself”.

  • Every Pilot should be an “Automatic Pilot”.

  • But actually, you do not enter an automatic vehicle ,”you know too much” ;-)

  • ‘Chaos’ does have not the same meaning as for anybody else.

  • Control theory can save the world

Feel free to complete the list in the comments!

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Good news from the combat against the crisis

Posted by Harry Seldon on February 22, 2009

Julia island
In my 2009 wishes, I wished that people better understand the world. For this purpose, I suggested they read The (Mis)behaviour of Markets: A Fractal View of Risk, Ruin and Reward by Benoit Mandelbrot.
Actually, for one year and a half we have been inside this growing up crisis I have been appalled by the small number of articles in the media about Mandelbrot’s work on Finance and more generally by the small number of articles about chaos theory and economy. Indeed more than ten years ago, Mandelbrot explained why the finance world was walking on its head. To make it short, it is because financial theory fundamental principles are wrong and over-optimistic. These principles state that the hazard involved in the market theory is regular enough to be smoothed by the big numbers law. This hazard is “benign”. Basically, with the standard theory, crisis can’t happen. So yes, you must be dreaming, there is no crisis, unless you listen to Mandelbrot that tells you that the hazard is actually “savage” and potentially leading to a collapse of the market: a crash.

So where is the good news ?

It looks like main stream media are slowly understanding Mandelbrot’s work and they are speaking about it. I found yesterday an excellent article in one of the main French financial newspapers about Mandelbrot’s work on finance. The second good news, contained in that article, is that Mandelbrot’s book are being republished. A third good news, is that the author of the previous article, Philippe Herlin is very proactive and has just created a Facebook group called “Finance & Mandelbrot”. I joined immediately ! An instant interest of this group is that Philippe Herlin gives a list of news articles about the subject (given below). Please notice the link called How Fractals Can Explain What’s Wrong with Wall Street which is dated on February 1999. Yep, it is not a typo for 2009 !
Last but not least, there is a significant political news. Mandelbrot showed that the financial system was fractal, exhibiting the same behaviour at all scales. Actually that was the start of his long and brilliant career. Later on he showed the same behaviour for climate, geology, ecology, or biology. Actually about any natural system is fractal and even chaotical. However, there is a good way to smooth the behaviour of a system, it is called control. Our world needs more control (distributed control whenever possible). That is to say, more rules, more ways to enforce them and more ways to measure their efficiency. So the good news is today European leaders backed sweeping new regulations for financial markets. Questions remain, will the rules be good and will they be enforced?

By the way, if you want to understand a little better the whole world, if you want to hear about economy, climate, geology, ecology, or biology in less than 200 pages, just read these two amazing books. Your vision of the world will change and for almost free because these books are quite old (but have never been so fashionnable).

Nostalgia: I was offered these books 13 years ago. I read them then. They are definitely among the best gifts I have ever received. They stand in my bookshelf, one arm away from where I am writing these lines. Believe me these books are far easier to read than the Bible so you can give them to your children. I read them again 2 years ago after completing my studies in Dynamical Systems and Controls. I understood plenty of things I had missed in the first reading. So you can also offer these books to adults. They will learn quite a few things about the nature of our world.


In English

In French

If you know other good links about fractal, chaos theory and economy, please leave them in the comments.
Mandelbrot set

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Control Systems 101: a Flight Control System example

Posted by Harry Seldon on January 03, 2009

Flight control system
The about page of this blog says this blog is about Linux, Ruby on Rails, Controls and the Thinkosphere project. It is now time to speak about controls.
First things first. What is a control system (CS) ? I am glad you ask because it is not an easy question. A way to answer is to give various examples which I will do in this post and next ones. Right now, I would say that a control system is a set of methods, processes and machines or humans that aims at shifting the state of a system from an initial state to a new determined state while keeping the behaviour of the system inside some bounds. OK, the definition is not very sexy. However, you notice immediately that many systems (if not all) will fall into this definition. That is the point of this blog. I want to show you that control systems are about everywhere. Everything and everybody is either a control system or a part of a control system.

What does a control system consist in ?
A CS is made of actuators, sensors, controllers. Another way to say it is: a CS is made of muscles, perceptors, brains. In order to enter the details of how a control system works I will take the example of an Aircraft Flight Control System (FCS) because it is what I know best as I am a Flight Mechanics and Controls Engineer, and a private pilot.

What is the aim of the FCS ?
Basically, nothing else than conducting the flight to move safely the airplane from a Point A to a point B minimizing the cost (Fuel consumption), minimizing the time and maximizing the passenger comfort… Yep quite a lot of things to do ! CS Design is not something very easy.

Actuators act on the system. For a mechanical system actuators apply forces and torques on the system.
For our FCS, actuators are:

  • Engines, they generate the thrust
  • Ailerons, they allow tilting the aircraft towards the left or right (along the roll axis, that is one of wings is higher than the other one) and thus they allow the aircraft to change its heading)
  • Elevator, it allows tilting the aircraft towards the up or down (along the pitch axis, that is the nose will be higher than the queue or the opposite)
  • Rudder, it allows tilting the aircraft (along the yaw axis, towards the left or the right in the plane of the aircraft)
  • Spoilers, slats and flaps to better act on the lift

Actuators receive their orders from the controller.

Some other things that fall into the actuators category: action mean, levers, muscles.

The sensors observe the state of the system. They measure critical parameters. For our FCS, sensors are:

  • Gyroscopes, they measure the angular velocities of the aircraft (roll, pitch, yaw rates)
  • Accelerometers, they measure linear accelerations.
  • Angle of attack probe, it gives the angle between the velocity vector and the pitch attitude angle.
  • Pitot tube measures the velocity with respect to the air.
  • Barometer allows estimating the altitude through the air pressure.
  • GPS to give the aircraft position

The controllers gets information about the system from the sensors.

Some other things that fall into the sensor category: feedback, detectors, observers, measures, references.

The controller gets the high level orders from the pilot (where do you want to go, heading, altitude, etc ?). Then, using the control laws it converts the order to low level electric orders sent to the actuators. Next, it checks that the aircraft does what it told it to do. The small discrepancy between the order and the measure is analyzed by the controller which sends an update to the actuators and the control loop goes on. The frequency of the whole process is typically 8 Hz.
The controller is made of computers and softwares. Softwares themselves are made of quite a lot of different parts, the core part being the implementation of control theory algorithms.
For sake of simplicity I am including the estimators, these algorithms that use several data measurements to filter the measurements or compute not measured data, inside the controller.

Some other things that fall into the controller category: guidance, navigation, control,stability, GNC, brain, think, computer, estimators, software, automatics, algorithms, methods, processes, regulator, commander, boss, manager.

In controls, we often speak about loop and feedback. The scheme at the top of this article is the representation of the order in which the parameters are sent and obtained. The pilot enters the parameters into the flight computer. The computer sends the first orders to the actuators. Actuators act on the aircraft. Sensors measure the state of the system and feed them back to the computer. Then, the computer verifies if everything behaves accordingly with what it expects, otherwise (for instance if there were external perturbations or modelling errors) it sends new orders. And the cycle goes on.
Finally, you should basically understand what a control system, controllers, actuators and sensors are. In future posts, I will explain the differences between Guidance, Navigation, and Control. I will give you more example of CS and what their controllers, actuators and sensors are. Some controlled (or not!) systems I have in mind are Economy, Climate, Politics, Society, Company, etc.
In the mean time, you can play the following game. Whatever you see, ask yourself if it is more a controller, a sensor or an actuator and for which CS. Or just do that for the following things: justice, police, lawyer, economist, trader, SEC, president, soldier, medical doctor, clothes, volcano, and ThinkoSphere.
Playing this game, you will quickly understand this motto I often have in mind: “It’s all about controls”!

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