Reddit mentions of Nonlinear Systems (3rd Edition)

I would recommend Nonlinear Systems by Khalil

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https://www.amazon.com/Nonlinear-Systems-3rd-Hassan-Khalil/dp/0130673897/ref=sr_1_1?keywords=hassan+khalil&qid=1550277980&s=books&sr=1-1

For Non-Linear systems I love Khaleel. Though I dont remember if he has much on Sliding Mode.

I learned Kalman Filters from [Thrun] (http://www.amazon.com/Probabilistic-Robotics-Intelligent-Autonomous-Agents/dp/0262201623/ref=sr_1_1?s=books&ie=UTF8&qid=1348256381&sr=1-1&keywords=probabilistic+robotics). Though he really approaches it from a software perspective and not necessarily controls (they are merging now!). He also has all different flavors of filters, which is awesome.

Of course! There are a lot of okay entry-level control theory books, but the really good books are a bit more advanced. The /r/controltheory wiki here has some good book suggestions (in particular the WikiBooks book on control theory), but I'd really recommend watching Steve Brunton's Control Theory Bootcamp on youtube to get a good overview of intro grad level control. Brian Douglass (also on youtube) has also a bunch of great videos on control theory, if you are interested in diving deeper into specific topics.

I used Chen's "Linear Systems: Theory and Design" as my intro book, but it's not exactly the most riveting. My favourite book now is Ian Postlethwaite and Sigurd Skogestad's "Multivariable Feedback Control: Analysis and Design" (apparently control theorists really like colons in their titles).

Now none of these books will use anything beyond advanced linear algebra and functional analysis, so for the nonlinear control that uses the fancier differential geometry, I'd recommend Bullo and Lewis and "Nonlinear Systems" by Khalil. Note that Khalil has another book called "Nonlinear Control", which is just Nonlinear Systems but cut in half. Don't get that one.

Control theory also intersects with optimization (they share the same arXiv classification), so for optimization I'd recommend Convex Optimization by Boyd and Vanderberghe. It's really a fantastic book. Calculus of variations is also essential for studying optimal control.

For your second question, I guess it depends if you want academic or industry positions. I can happily say that right now the job market for control theory is super hot in both. Aerospace and car companies are hiring controls people to do autonomous car stuff and spacecraft GNC (think the spaceX rocket landing), and a few of the car companies even opened up industrial labs where academics can do research and publish papers. It's pretty good. I'm graduating this year, and I managed to line up a few tenure track job interviews. I think like 40 R1-level places were hiring controls people, mostly for autonomous systems work.

That being said, you should definitely study something you are interested in. I have the fortunate problem of being interested in literally everything, so I kind of picked research topics that were hot for academic jobs. I wouldn't focus so much on choosing between "pure" and "applied", because the line is very blurred sometimes, and I think control theory definitely fills a large span of what people consider "pure" and "applied". So I think you are right in that you can study some very pure math topics, and then use those to do controls work. For example, my mathematical interest from undergrad was graph theory, and now all my controls papers that I write are using neat things like spectral and algebraic graph theory. Other things like spacecraft controls uses stuff like Clifford algebras to do the quaternion computations rigorously.

One control-theory-esque thing that is very hot in math departments right now is optimal mass transport. The math department at my university interviewed two faculty candidates doing OMT work. If you are interested, I'd recommend the books by Cedric Villani. The connection to control theory was done by Brenier and Benamou.

When you learn about your graduate admissions, if you want I can take a look at the faculty and see who does more theoretical control theory stuff and make recommendations. Its completely normal to be indecisive, especially if you are an undergrad about to start grad school. Definitely explore a bit, both on the math and the controls side, and feel free to message me if you have more questions. Good luck!

Cant compare to other books, but this is what we used for nonlinear.

https://www.amazon.com/Nonlinear-Systems-3rd-Hassan-Khalil/dp/0130673897

The majority is on nonlinear systems in general, but the last 2 chapters teach control techniques. (book is available online as pdf if you want to take a look at it first)