I work with atomic force microscopes (AFM’s). At their heart is a PID control system that keeps the probe doing the right thing, and giving us ultra high resolution images of a variety of things (topography, elasticity, etc). But I realized that I knew little about actual control systems. I was not an engineering student in university, so I wasn’t exposed to the concepts.
Now I am diving in. Fortunately, there is a ton of great information on the web about the theory and practice of control systems, so that is cool. However, it is clear that my physics degree, while heavily laden with mathematics, was deficient in discrete mathematics. In physics, in general, you look for closed form solutions. Thus you are integrating and differentiating continuous functions to get to a solution. Mostly, you don’t care about numbers and numerical values. That is something that a calculator can give you with the equation you generate.
However, engineers deal with concrete things. Forces, lever ratios, stasis, and most importantly actual real world values for their problems. As such, they use a lot of mathematical modeling, and computers to simulate results. Hence, the use of discrete mathematics instead of continuum mathematics.
Of course, I took numerical analysis, and numerical methods for physics, and have hacked around enough in Matlab to know some of the basics, but I got that information empirically, not from first principals.
Now I am filling in that gap.