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A little background here. I'm an undergrad in the final year. I have decided academia as my career path. My grades are not high but my research caliber is good and I have ongoing projects that are promising. My field of interest is control systems.

My question: I had a course in basic control systems. I did well, I can understand the basics. For my literature survey for my ongoing project, I went through the current trends in control systems and it's completely greek and latin to me. It's disheartening but I really want to try.

I cannot understand how one goes from MIMO, SISO systems to sliding mode control, fuzzy logic control and all the other advance stuff that seems to be the talk of the town.

How do I cross this? Are there any standard books, literature bridging this gap or at least elaborating on the prerequisites to understand the current state of art in control system math.

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First, get a decent grasp of linear algebra. Most control systems techniques use linear algebra extensively.

Second, don't worry much at first about not understanding much of the mathematics. The truth is that most controls researchers also don't understand the underlying mathematics. At least not the real, hard math. They might understand what L2 minimization is, but it's rare to see any controls folks actually get into the functional analysis of their problem. (As a mathematician working in controls, it's often frustrating.)

Ultimately, I don't think there's a single book that bridges these concepts. Your best bet is to work on numerical applications of controls systems, and try to read progressively more advanced books, and also to work out examples.

If you look hard enough, you can find some nice canned examples of some canonical controls problems -- a 5-state aircraft model, for example -- implemented in MATLAB/Simulink.

The best thing you can do to educate yourself is to try to implement the techniques against a model you can understand. In controls, learning is often accomplished by doing. Truthfully, the field is not as mathematically rigorous as it should be. But that's OK, because it's very useful (and sometimes very easy) to develop good, working solutions to real problems without requiring the mathematical rigor. Once you can conceptually handle some of the more advanced methods, you can then work on understanding the mathematical rigor.

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    $\begingroup$ I would second Arkamises comments. I think for a degree of intellectual satisfaction, you need a fairly rigorous mathematical background (I would add ODEs, smooth manifolds, matrix computations and more linear algebra). However, such a background would give intellectual strength, but not necessarily a solid grounding in control systems. I would focus on developing real intuition (read physics, mechanics, etc) first, rather than mathematics first. I would strongly recommend reading Gunter Stein's IEEE Control Systems magazine article "Respect the Unstable". $\endgroup$ – copper.hat Oct 28 '13 at 18:43
  • $\begingroup$ Detailed answer, thank you. @copper.hat, I'll read that, thank you very much. Also, my major is mechanical engineering and I can totally understand the mechanics part of it. Almost all systems I have come across now. After all, I did spend 4 years doing exclusively this. $\endgroup$ – SPRajagopal Oct 28 '13 at 19:23
  • $\begingroup$ @SPRajagopal There's an old aphorism that says, "an engineer is someone for whom every Taylor series truncates exactly after the linear term." The truth is that engineering mathematics is very procedural, and most of it elides most of the interesting actual mathematics going on. This is the road your curriculum has brought you down, for better or for worse. It may help, or at least provide comfort, to know that most research engineering mathematics pulls from theoretical mathematics only that which is absolutely and minimally necessary to solve the problem. (cont'd)... $\endgroup$ – Emily Oct 28 '13 at 20:23
  • $\begingroup$ (cont'd) So if some of the concepts in things like fuzzy logic control (which is just a radial basis function network with different membership functions), adaptive control, etc. seems like they're pulled out of nowhere... they probably are. If the folks developing these concepts don't know what a Radon measure is, then it's OK if you don't either. $\endgroup$ – Emily Oct 28 '13 at 20:26

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