The Secret to Success in Control Theory

 It is possible to get an intuitive understanding of control theory from the vocabulary alone, but to understand how it works, or to use it, you must get into the mathematics of it.

Frankly the mathematics of control systems is, on the surface, very challenging. The equations are often ghastly differential equations, meaning that they are relations between quantities arising from Newtonian Calculus. The quantities are typically rates of change, and rates of change of rates of change, and so on. They typically depend on each other through complex functional relationships which are classified as nonlinear (implying they are very difficult to solve except in special cases).

Then, as with everything in engineering, after one learns this fact about the difficulty of the mathematics that describe the systems involved, one learns that for a great many such systems, including most if not all of the ones that that one is interested in, it is possible, through what seems like a magical trick*,  to simplify these ghastly equations into mere algebraic ones, which can often be solved using the techniques one learned by eighth grade.. Moreover, in conjunction with these simple-to-solve equations, one can often resort to graphical diagrams to infer most of what one needs to know to design a good system.

This kind of drastic simplification of the mathematics makes the design and solution possible for a wide variety of types of real world systems. It is the reason one can design many of the wonderful conveiences that have given us the modern world: transistors, solid.state amplifier circuits, computer chips, jet airplanes, modern eco-friendly car engines, etc.

Even with this simplification, there are still numbers involved, and that is enough to scare a lot of people away. Others are not scared away but have mathematical aptitude and are curious to learn more.  Those people, if they carry through on this curiosity, can be great students, educators, and technicians,. They can provide valuable opinion in other fields where actual knowledge of the engineering and science is required to translated to general public (instead of just relying an intuitive understanding).

As many have pointed out, to be very good at engineering or physics, and to make it your true profession, you need to have a burning desire to understanding, even to see, the underlying mathematical workings of the real world in action. You have to want to pursue the beauty of that mathematical understanding of the world.

It doesn't mean you have to "think in numbers". Any Hollywood image you have of that kind of thing is wrong, because it was given to you by people who don't understand the mathematical curiosity I'm talking about.  

Mathematics isn't just numbers and equations with symbols. Of course can be geometry too, but that is also a limited case. The mere connectivity of things one to another is a branch of mathematics. Whether surfaces are punctured with holes or fold back on themselves is mathematics. 

The grouping of things together, and the exclusion of things one from another is considered among the most fundamental areas of mathematical thought, as fundamental as counting. Together they have yielded many of the great advancing in computing over the last half century, that have created the Millennial world.

To see the world in mathematical beauty is to look at various parts of it, that are in front of your understanding and consciousness, and to know at once that many different branches of mathematics may describe various parts of what is going on, and that underneath it all, beyond one's ability to see it all at once, is a giant mathematical machine, predictable or otherwise, at work. To investigate even just a tiny portion of that reality in mathematical terms, and to truly see the mathematics at work, can be a full-time, lifetime pursuit.

*The Laplace transform. 

The Laplace transform is named after mathematician and astronomer Pierre-Simon Laplace (1749-1827), who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions in Essai philosophique sur les probabilités (1814), and the integral form of the Laplace transform evolved naturally as a result.