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I'd suggest getting a book on the subject. I've got this one in hand atm. It's relatively cheap, but your library may have a copy of it on hand if you're still in school. Chapter 3 (out of 6 chapters) is basically all about the subject you're trying to tackle.

I can fill in gaps here and there, but I doubt I'd do the subject much justice trying to paint the whole picture.


In broad strokes: A functional( J[ x(t), u(t), t_f ] in your case) is a function of a function. That is, it takes a function defined on an interval as input, and outputs a number. You are trying to minimize this number by varying the function u(t); i.e. computing a control history. The calculus of variations is about learning to take derivatives of functionals with respect to functions(as opposed to functions with respect to coordinates or parameters), and then developing an equivalent to the first and second derivative tests from calculus so as to find extrema.




I found an answer to a similar problem to yours, but it's not quite the same. I really suggest you dig deeper into the subject before anyone like me throws an equation at you.

Yes, I'm using a sort of knock-off Arduino Uno. Wikipedia says the ATmega238 (the MCU on the board) can operate at 20 MHz, which I think should be plenty.

I found an interesting fact when playing around with the code. At first I had the code run every 3 - 5 milliseconds. I Then took that constraint away to have it run every loop and it didn't change a whole bunch. I then had a line print to the serial monitor cause I was timing how fast the loop was, and it seemed that the serial printing added just enough delay to make the oscillation very very close to being stable. The oscillation was there but it didn't grow to be unstable, it was just constant. So, somehow adjusting how fast the MCU is computing the input based on the state helps with the oscillations. That leads me to believe I'm definitely having a noise issue in how I'm calculating the derivates. I think the Kalman filter suggested above will be a good thing to implement and hopefully settle that oscillation

There's a good introductory textbook on nonlinear dynamical systems and applications to biological systems called Modeling Life that I help teach a class for. It's aimed to be easily digestible for college freshmen so it has some introductory calculus in there but there's also some really nice connections between dynamical systems and real life systems that's outside of what's normally taught in college differential equations courses.

This book has been a great help in Kalman filters.

If you like Python; this other book on Kalman and Bayesian filters in Python is a must.

My professor is willing to have a directed study course for me. Thanks again for the awesome suggestion. This is the book she is recommending. Have you heard of it?
http://www.amazon.com/Applied-Estimation-Analytic-Sciences-Corporation/dp/0262570483

I would definitely advise you to use the book from Borelli, Morari and Bemporad called Predictive Control. It is pushing the best book available now and intended includes a lot of things on hybrid MPC.

It is available here.

Robust and Adaptive Control: with Aerospace Applications by Lavretsky and Wise helped me get my foot in the door for advanced control of aerospace systems.

http://www.amazon.com/Robust-Adaptive-Control-Applications-Processing/dp/1447143957

This book sounds pretty good based on the description, and it was published in 2014.
https://www.amazon.com/dp/1107424151/ref=cm_sw_r_cp_awdb_t1_SuRBCbBTMH1AP

This is a great text that covers adaptive control using neural and fuzzy approximators:

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https://www.amazon.com/Adaptive-Control-Estimation-Nonlinear-Systems/dp/0471415464/ref=sr_1_4?keywords=kevin+passino&qid=1550449828&s=books&sr=1-4-spell

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One of the authors also has code from the book on his website:

http://www2.ece.ohio-state.edu/~passino/SACENSbooksoftware/file_index.html

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I am currently taking a Optimization & Control course, using this textbook: https://www.amazon.com/Numerical-Optimization-Operations-Financial-Engineering/dp/0387303030

I haven't spent much time with it yet, but I have had several people tell me it's a highly recommended book.

Possibly https://www.amazon.com/Living-Control-Systems-III-Fact/dp/0964712180 and Behavior:Control of Perception from the same author. I'm coming from a psychology background, and these were quite useful.

Check out Applied Control Theory for Embedded Systems:

https://www.amazon.com/Applied-Control-Embedded-Systems-Technology/dp/0750678399/

The author is active on reddit, especially in r/embedded and r/ECE.

I’m also in the process of writing some articles and guides on this very topic that will eventually morph into a book. I’ll update this post in a little while when I have something to show.

I like [this one] (https://www.amazon.com/Feedback-Control-Dynamic-Systems-7th/dp/0133496597).

Essentials of Robust Control, Kemin Zhou


amazon link

Flight Dynamics Principles. Michael Cook.

http://www.amazon.com/Flight-Dynamics-Principles-Third-Edition/dp/0080982425

Please don't delude yourself that knowing which buttons to press in Matlab is a substitute for actually knowing the material.

I suggest you buy this book. Then you'll have a chance of understanding what Matlab is doing under the hood, and you can just do what needs to be done. And no, the book doesn't mention Matlab, to my knowledge -- it just explains how the math actually works.

If everyone just got this book, their lives would be simple. Most of what you need to know to tune a PID is in here.


http://www.amazon.com/Modern-Control-Engineering-5th-Edition/dp/0136156738

I used Ogata for my control subjects at uni. I thought it was a decent introductory text- although I never pursued the subject beyon what I did at school. There's a pdf floating around on the internet somewhere.

Start with classical control

• ODE, Laplace theory
  
• 1^st and 2^nd order systems
  
• Poles and zeros, feedback, stability (Root Locus, etc)
  
• System type and order, steady state errors
  
• PID
  
  That is represents a basic set of topics to properly understand linear systems. However, since we know that systems are not linear and implemented using computers (discrete-time control), this is just the beginning. It would probably take 6 months to learn the linear stuff given a reasonable foundation of mathematical knowledge. Depending on your preference and leaning when it comes to theoretical vs practical knowledge, I would suggest looking at Nise - Control Systems Engineering, which is more inline with the American, theoretical style or Dutton - The Art of Control Engineering, which more in keeping with the UK style of application.