[Math] Mathematics for Guidance, navigation and control

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I'm finishing my math degree this week and have been looking for some subject to practice and study on my own while I'm doing some work as a programmer. I'm interested in getting my master's later but for now I'll dig through a couple books to stay busy.

Can anyone recommend the areas of math used in designing guidance, navigation, and control systems (like what may be used at places like Nasa, spacex, etc)?

I have a strong background (2 or more semesters each) in

  • real analysis (through basic measure theory and Lebesgue integration)
  • complex analysis
  • linear algebra
  • (partial) differential equations
  • numerical analysis

I have a semester in group theory, and another in rings/fields (culminating with Galois theory) as well, but I expect it won't be as much help.

(I know this is kind of 'soft', so if this question is better suited elsewhere, or needs more specification, let me know.)

Side note – I know there's a field of math called "Control Theory" that's got a basis in classical and functional analysis, and even some topology. I guess the question I'm really asking is if such rigor as is provided in these topics typically would be useful at all if I were to get a job in this field one day, or if I should just learn how Kalman filters are implemented and how to program embedded C code.

Best Answer

Here are some of the books I've enjoyed. Note that I'm from an engineering background, and I do robotics and specialise in optimal-control and systems with non-linear dynamics.

  1. Nonlinear Dynamics and Chaos, Strogatz. This is very approachable and he actually also has lectures online if you prefer that format. It's not specific to the applications you have in mind, but it does cover a very important part (nonlinear dynamics), which is where a lot of research is pushing at the moment.
  2. Dynamic Programming & Optimal Control, Berstekas. From my point of view, this text is quite rigorous. Very thorough if you want to get into optimal-control, and stuff like trajectory-optimization.
  3. The Variational Principles of Mechanics, Lanczos. A bit old, but a classic. More on the theoretical than practical side.
  4. Optimal Control & Estimation, Stengel.

Also, here are a couple of MOOCS that imho are excellent.

  1. Underactuated Robotics, this deals with optimal-control for non-linear systems, and focuses quite heavily on robotics applications. However it does a lot of stuff like trajectory optimization which is really relevant to orbit planning etc. and just overall is super fun and well taught! Downside is the next session will probably be next fall. You can watch all the vids and see the exercises anyway, but you'll need your own matlab license to do them.
  2. Discrete Optimization This one does optimization from a more CS point of view, and is also really fun. I'd say not as close, but it still does stuff like travelling salesman (essentially shortest path problem), and touches on a lot of algorithms that will be useful.

Note that the second course in particular is very hands-on, and not so much about the math. The first course is imho a very good balance of both. Not a lot of proofs, but still a lot of insight into the math behind everything.

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