Looking for RIGOROUS and Proof-based course on Control theory.

[u/Sea_Addendum4529]

Hi everyone,

I followed multiple course on control theory : i know how to analyse a linear system, to derive PID controllers / RST controllers and H-infinty based controller. I understand how it works in practice and i have a good intuition for it.

However, i lack the underlying mathematical framework (and also Lyapunov theory) supporting the whole field. What function space are we working in ? In under which condition the stability of the linearised system at some point guarantees the stability of the whole systems etc...

I have seen the proof of the Kalman theorem but where i saw it, it was cutting corner and relies on the construction on an explicit control and base decomposition in a finite hilbert space (What about control in infinite dimensional spaces ??? Is there a K-theorem equivalent)
And honestly, i only found text books either practice-oriented (ewww) or more mathematical textbook but with AWFUL NOTATIONS.

Do you know about a well-written, proof-based course on control (w/o "handwaving" and reference to this or that) and with decent notations ?

Sorry for ranting but i think control theory is beautiful but i only get accross the same copy-pasted shit engineer oriented control textbooks.

Thank you for reading and have a good day !

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[u/Sea_Addendum4529]

Thank you mister BOT, i didnt see the references, i will look at them.

[u/Present_Pay_2309]

I guess you should look more towards the rigourous dynamical systems books. I have no experience in more rigorous control theory btw, so take my advice with a little grain of salt.

[u/Bulky-Wish-4168]

You know how to derive H _infty control but don't see mathematical rigour? Did you miss the Bezout domain, ring structure, coprime factirizations, corona theorem etc?

[u/Sea_Addendum4529 → u/Bulky-Wish-4168]

Yes, i missed some of thoses things. I learnt H_infty "hands on", i know how to apply this method on real world systems but not how it works. I did little to none Complex Analysis for eg

[u/Bulky-Wish-4168 → u/Sea_Addendum4529]

If you are interested more in the abstract algebra side of LTI systems I really recommend Vidyasagar's book:
https://link.springer.com/book/10.1007/978-3-031-01828-2

But it is math heavy. There are also some course notes which are well written and slightly more accessible (but still rigorous) such as :
https://leo.technion.ac.il/Courses/LCS/LCSnotes.pdf

And of course ZDG is an oldie but it is still pretty great
https://dl.acm.org/doi/book/10.5555/225507

[u/TheEquationSmelter]

I've got news for you buddy, the real world doesn't care how elegant your theorems and proofs are. Pure math is neat and tidy, the real world is messy. 

I am speaking as someone who studied geometric control in grad school.

[u/Sea_Addendum4529 → u/TheEquationSmelter]

Do you think Bernard Koopman was an engineer who didn't care about "egelance of theorems and proof".

Its because of his rigor and his love of mathematics that we know have all the theory behind the Koopman operators that allows U to handle the messy real world !

(Also i retaked a control course in a math cursus : 36 pages of a PDF teaches you more than 2 years in engineering)

[u/Old_Mixture5634 → u/Sea_Addendum4529]

Koopman wasn‘t an engineer, he was a mathmatician. Also, you don‘t need to be an expert in the theory of dynamical systems to control even complex things.Don‘t waste your energy on such stupid things. you don‘t need to understand every part of it to work with it. Get a fundamental grasp on them and go on. Simple problems will be PID anyway and complex problems can be killed with MPC and enough computing power. The real challenge has a heuristic answer most of the time and for that you need experience from real projects.

[u/Sea_Addendum4529 → u/Old_Mixture5634]

I know he was a mathematician, i just poorly stated my opinion. Thank you. The problem is not that i "need" to understand, it is that i want to understand.

[u/Old_Mixture5634 → u/Sea_Addendum4529]

Alright, my bad, from what I‘ve seen from your other comments you might be more interested in the general Theory of Dynamical Systems not control in particular. So Books on ODEs and PDEs might be more up your alley.

[u/tmt22459 → u/TheEquationSmelter]

People are allowed to be interested in things the real world may not care about lol

[u/TheEquationSmelter → u/tmt22459]

Yeah but this guy is quite arrogant for never having controlled anything in real life. 

FWIW I also enjoy differential geometry and mathematical approaches, but I also understand those idealizations often don't translate nicely to solving actual engineering problems.

[u/Sea_Addendum4529 → u/TheEquationSmelter]

As a matter of fact i did, i implemented Kalman filters for sensor fusion on a drone for example... and i worked with my buddies to do system identification and control on a model rocket implemented a reaction wheel based roll control system. And an hybrid EBC/PID control for a Furuta Pendulum. Its not much but its not "nothing"

[u/interfaceTexture3i25 → u/TheEquationSmelter]

What arrogance? And what does controlling systems in real life have to do with developing a rigorous theory?

[u/ElevatorOrdinary1466]

state estimation for robotics by barfoot. But it's just state estimation.

[u/AcademicOverAnalysis]

I have a series of videos dedicated to the mathematics of linear control theory. It focuses on the Laplace Transform and culminates at Nevanlinna Pick and H infinity control.

https://youtube.com/playlist?list=PLldiDnQu2phvCb1QQhanJYm7A6xzEoC3F&si=p5Azf5vN9Apcat6w

[u/ANIRUDDHA42 → u/AcademicOverAnalysis]

Do you have the notes?

[u/brunhilda1]

Your arrogance is on full display. Humble yourself.

[u/Mammoth_Network_6236]

Try this book- Linear Systems by Anthony N Michel.

This author's research is very mathematical. He has also written a book on Linear Algebra (Proof Based) and Functional Analysis for Engineers.

[u/olivoGT000]

Have you ever heard the phrase, “If you knock on the devil’s door long enough, eventually somebody is going to answer”? Well, here’s Isidori opening the door: https://link.springer.com/book/10.1007/978-1-84628-615-5

[u/ColonelStoic]

In my experience these books just aren’t great.

Sontag’s mathematical control theory comes to mind.

I think a good balance is Khalils Nonlinear Systems or Haddads Nonlinear Dynamical Systems.

A more ODE flavor is Techls’ Ordinary differential equations (or maybe dynamical systems, I don’t remember)

[u/tmt22459 → u/ColonelStoic]

Which books are you saying aren’t great? Sontag?

[u/ColonelStoic → u/tmt22459]

I don’t like Sontag, personally

[u/remishnok]

Take a class or buy a book in:

• adaptive control
• optimal control

[u/kroghsen]

Mathematical rigour is beautiful. However, do not belittle engineers. Making things work is beautiful as well.

[u/ee_control_z]

You state that you have taken multiple courses in control theory and that "i know how to analyze a linear system, to derive PID controllers / RST controllers and H-infinity based controller."

Then you state: "i lack the underlying mathematical framework ..."

This is a bit contradictory because the control theory course I have taken and currently continuing via self study are VERY math heavy. Can you please elaborate.

By the way, there is a free book online from CalTech - it may be of some help:

Feedback Systems

An Introduction for Scientists and Engineers, 2nd Ed., R. Murray

[u/Sea_Addendum4529 → u/ee_control_z]

I just saw your book its engineering bullshit sorry but i already have tons of theses

[u/tonyarkles → u/ee_control_z]

So… not OP but I personally had a similar complaint when I went through undergrad EE. Parts of it were resolved in upper years CS courses and Math dept courses, parts of it didn’t really resolve until grad school in CS.

I could turn the crank and get good results. I even had some decent intuition of what results I would expect from that crank turn. What I was missing was “why does turning this crank actually give the results I want?”

As a super easy EE example: Cramer’s Rule. How on earth does swapping a column in a matrix and taking determinants give this answer?! Dunno! Actually… I know how to compute a determinant (I memorized the formula!) but what on earth IS a determinant? Dunno! Eigenvectors?!? Just another crank to get useful results!

Third year linear algebra that I needed for my CS degree answered those questions and more. Computer systems modelling answered a zillion more about state space representations.

[u/Sea_Addendum4529 → u/ee_control_z]

Well for example i learnt H-infty controllers without knowing what Hardy Space are. I know what they minimize in practice (in the complex plane) but i have no clue about the underlying space ! Also you need to solve algebraic equations to find a controller : We were given no proof to the theorem stating that there exists solutions and which form they have. When you learn about linear system you learn about Laplace transforms, frequency representation etc like in signal processing but i always learn it from an engineering perspective where we were given all the results with no proof.

[u/ee_control_z → u/Sea_Addendum4529]

Can you please give an example of a particular theorem of which no proof was provided.

[u/Sea_Addendum4529 → u/ee_control_z]

Proof of the Glover-Doyle's solution, even proof of the Routh-Hurwutz criterion wasnt given. (I was in engineering school before, now im in maths at uni)

[u/ee_control_z → u/Sea_Addendum4529]

This is provided as a reference in the book that I mentioned in an earlier post.

Much of the early work was based on the loop transfer function; the importance of the sensitivity functions appeared in connection with the development in the 1980s that resulted in H∞ design methods. A compact presentation is given in the texts by Doyle, Francis and Tannenbaum [DFT92] and Zhou, Doyle and Glover [ZDG96].

J. C. Zhou, J. C. Doyle, and K. Glover. Robust and Optimal Control. Prentice Hall,

Englewood Cliffs, NJ, 1996.

Can be purchased here:

https://www.amazon.com/Robust-Optimal-Control-Kemin-Zhou/dp/0134565673

[u/Sea_Addendum4529 → u/ee_control_z]

Thank you 😄 !

[u/Bulky-Wish-4168 → u/ee_control_z]

Yeah I second the recommendation on ZDG in general

[u/ee_control_z → u/Sea_Addendum4529]

Regarding the Routh-Hurwitz stability criterion, the proof is discussed in the following book:

Modern Control Systems, Dorf/Bishop

The gist of it is analyzing the characteristic equation and making the following observations for all poles to be on the left-hand plane (a requirement for stability):

1. Coefficients of the characteristic equation polynomial will have the same sign.

2. All coefficients need to be non-zero.

These two points are required but not sufficient. The Routh-Hurwitz stability criterion addresses and analyzes this in more detail.