Linear Systems [Control Bootcamp]

Linear systems of ordinary differential equations are analyzed using eigenvalues and eigenvectors. This will be the mathematical foundation of this bootcamp on linear control theory.
Chapters available at: databookuw.com/...
These lectures follow Chapter 8 from:
"Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control" by Brunton and Kutz
Amazon: www.amazon.com...
Book Website: databookuw.com
Brunton Website: eigensteve.com
This video was produced at the University of Washington

[@matteovaiente214]

As a mathematical epidemiologist interested in learning and applying control theory to disease dynamical systems, I've found no better resource to provide an accessible, yet sufficiently rigorous, introduction to the subject. Thank you for your efforts and for providing an accompanying, high-quality textbook.

[@Eigensteve]

Many thanks!

[@zhaosinicholas921]

thx professor its absolutely the most profound series I ve ever watched on modern control theory

[@Eigensteve]

Thanks!

[@zhaosinicholas921]

​@@Eigensteve Very cool! I am now watching the third videos of this series and planning to watch them all. Thes videos provide me some great perspectives that I never really look it in that way. I also have a question about this field and I hope you could anwser me :). My major is measurement and control. I m now in my junior year. And so many students around me turn to the AI field when they finish their study of undergraduate. They told me that control theory is a dying subject that theres no more big problem to be solve and hardly can we do the innovations. They also said that PID solve 95 percents of the problems. Eventually, they conclude that AI is now popular and there are more chances, but both of these two field is largely about mannipulating the matrix. So why not turn to the AI, a rising field? I think they might be not totally right because I found that sometime its either hard to understand the control theory or can we mannipulate the theory smoothly. However, the grad students in my university also told me that its all about the theories and simulations and they can seldom make the stuffs that are practical. They said that its the atmosphere of control theory study in China now. So what do you think of this, professor? I hope you counld anwser my question in your spare time maybe. But anyway, I really appreciate all your videos. Thanks again and wish you all the best!

[@Eigensteve]

@@zhaosinicholas921 , this is a fair question, as it impacts long-term career choices. My feeling is that control theory is always going to be important, as there are still many very important unsolved challenges in this field. Machine learning is of course also a great field that is rapidly growing. The intersection of the two, or more generally ML with any field of hard engineering, is particularly exciting for me. But whether or not any one field is a "fad" (and ML and control are definitely not fads), building a solid background in linear algebra, optimization, and statistics will never go out of style. So ML and control will build your "math muscles", which will be useful for the rest of your life.

[@zhaosinicholas921]

@@Eigensteve Thanks very much for your time and consideration, professor!

[@19CH09]

I am so glad to finally see a control video that pronounce correctly the Greek letters!!!! YES!! It's "xeee" not "kasaii" [ksi]!! BRAVOO!

[@Eigensteve]

Awesome -- nice to have the feedback, since I was going into that pronunciation blind :)

[@junninghuang4343]

I love the second lecture about linear systems. It explain that why we introduce eigenvectors in linear systems: It's a kind of coordnation transformation, from the x space to its eigenvector space, and then back to the x space. The introduce of eigenvectors will decouple all the components of x^{dot} and x. That's really cool! BTW, I find a video of 3Blue1Brown related to the same topic, titled "change of basis".

[@ThatLucasGuy93]

You are really phenomenal at understanding how to organize a high level overview of a topic like this. I badly needed the review materials compiled in one place - lifesaver. I feel like everything is motivated now and clicking into place. Can't thank you enough, keep churning these out, I hope to follow you into an extremely wide range of mathematics knowledge.

[@moacirfonseca6625]

THANK YOU SO MUCH! You are helping me a lot in my Optimal Control classes. These are the best classes I've seen on KZbin.

[@tommyfan6911]

Thank you so much. I am currently applying for a control system development job, and your boot camp saved my time. Time is Life, so you saved my life!

[@toastrecon]

Man. I wish I'd had these during engineering school.

[@everythingaccount9619]

Could'nt agree more

[@owen7185]

I'm really lucky, I'm in 2nd year now, this man is awesome his videos help so much

[@sogandmaga]

Thank you very much professor! Your explanaiton is excellent and in each video there are a couple of eureka moments that makes the audience understand the significance of each step.

[@Eigensteve]

Thanks so much! Glad you like the videos!

[@ahmedkamiss7643]

At long last I finished ME564 and 565 and started on this. My goal is to watch each and every lecture on your channel ( yes, I take notes :). Thank You !

[@Eigensteve]

Wonderful!

[@owoeyebabatope2425]

Wow! Prof. I'm a big fan. Thank you. This provides a holistic and insightful view of the control model useful for practical control. Thank you once more. I look forward to your lecture on data driven science and engineering.

[@Eigensteve]

Many thanks!

[@ahmedrista164]

thank you, professor, I'm so happy to learn about linear systems and improve my knowledge for the better , I wish you all the best

[@Eigensteve]

You are very welcome

[@gastonparra2971]

Hi, I am new to this topic and enjoy seeing these classes. They are so interesting.

[@alegian7934]

Im Greek . The way you pronounce ξ is very accurate!

[@tiddlywinks497]

I swear every time I need a math explanation you have one on the topic, thanks so much

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@climbscience4813]

This is a really good explanation of eigenvalues and eigenvectors. Thank you!

[@shreyas9312]

Haha 2 lectures in and i am sitting here thinking - man, i cannot continue watching this until i figure out how was the video captured and processed haha. Great content so far Steve. Looking forward to finishing the series in the coming few days as a refresher

[@michaelmarkoulides7068]

Wow I’ve never seen an eigenvector coordinate approach to linear systems , computationally makes life so much easier the derivation is a a lot but the end result is so elegant

[@Eigensteve]

Thanks for watching!

[@rogiervdw]

Excellent treatment, wow what a way to give insight & intuition. Absolute world class teaching, thank you!

[@abhaykela]

Thank you professor for this amazing series of lectures, you made controls easy for me.

[@Eigensteve]

Glad you like them!

[@trueroughly1691]

so at 4:20 we can define e^At because we can find the equivalent summation as RHS? which is pretty much a generalization of e^ct in R?

[@kevinairis317]

Dr. Brunton, you are my hero. That's all I have to say.

[@kevinairis317]

I am a grad student at Texas A&M now, but I really wish I took controls with you during my undergrad at UW. Keep up the great work!

[@Cesarneuro]

Yo, My hero. Your classes are pretty awesome. filling up all the gaps I was missing.

[@Drone.Robotics]

Thank you Sir....I have seen the whole playlist and it cleared a lot of my concepts about control theory. Your videos are just great and your way of teaching complex things in simple manner is appreciable. Thanks Again.

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@abderezakabidi4510]

Thank you professor for this lecture, you explain very well, so could you please add us a playlist about a nonlinear systems.

[@musicarroll]

Nice to hear an engineer who actually understands mathematics (cause he studied math before he studied engineering).

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@animeshsinghal3405]

Beginners should go through 3Blue1Brown's playlist on linear algebra before watching this lesson: kzbin.info/aero/PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab And when you come back, everything that Prof. Steve teaches will make so much sense solidifying your understanding of linear algebra.

[@lindsaysmall7471]

Basic question sorry, why is the relationship AT = TD? I thought it should be AT = DT to be more the like the initial cze relationship. Thanks so much for the video series, just a fantastic resource.

[@Eigensteve]

Good question! This is one of the fun things about matrices. Although the equation for a single eigenvector is A*x = lambda*x, when we stack the eigenvectors into a matrix T, we get AT = TD. You can write this out explicitly and convince yourself that this is how to write it.

[@junthx3138]

kzbin.info/www/bejne/eqvUdmqqn5yYf68 This video solves your question. Watch video 40&41 in the playlist.

[@danielglazar6811]

@@Eigensteve Thank you for your content! I'm actually getting AT = DT as well. In TD, we're taking a linear combination of nth component of eigenvectors (nth row of T) with constant scaling factor λ_n. In DT, we're scaling the nth eigenvector ξ_n with λ_n. Am I missing something?

[@danielglazar6811]

Derp, nevermind. AT = TD is correct.

[@Backstabbio]

This guy is magic at writing backwards.

[@linliu3665]

How do you deal with the case when matrix A does not have n linearly independent eigen vectors? In other words, what if T is not inversible?

[@neobermeo]

Thank you for these impressing lecture on control systems. Could you activate the automatically generated english subtitles for this lecture and the lecture on controllability? Thanks again, professor Brunton

[@mohamadebrahim8890]

Perfect, thanks so much for this playlist.

[@pouriashafee5414]

Really good lecture. just I don't know why it doesn't have subtitles. Sometimes if I don't understand something, I just copy the text into ChatGPT and get more information. It would be better with subtitles

[@eddieseabrook8614]

This is a fantastic youtube channel, I'm just sad I didn't discover it sooner! So many fascinating topics tied together One question, is it possible for a linear control system to be described by a matrix A which is not diagonalizable?

[@user-hk3ej4hk7m]

Are the equations using z analogous to applying variable separation to the system of ODEs expressed in terms of A and x? I'm doing a course on classic control at uni and I'm trying to warp my head around how these concepts map to using transfer functions and block diagrams.

[@user-hk3ej4hk7m]

Btw, thanks for providing such high quality content! Your courses on fourier, laplace and frequency space really helped me understand my "Signals and Systems" courses at uni.

[@evanparshall1323]

Such a beautiful explanation

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@MelonLuminus2845]

many thanks from Vietnam!

[@abbddos]

This is pure gold...

[@Rowing-li6jt]

where does this correspond to in the textbook?

[@calebbreazeale6548]

Is this the same as a LDU factorization?

[@Bedmonds91]

Does the z form in this video have anything to do with the z-transform?

[@SkielCast]

I was mind blown to see the relation of SVD inside all of this, so this approach basically replaces Laplace transform with eigen decomposition?

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@maneki9neko]

Brilliant. I love this.

[@Eigensteve]

Thanks!

[@manojsethia2606]

Why is your stuff not inverted as you right it on front side; Great series BTW........

[@Eigensteve]

Thanks!

[@alexmustang8177]

They are using camera in mirror image mode

[@nigelmaccuver9122]

😀 Nicely explained as always but what is this topic applies to??

[@kanshudo1]

I love your fantastic videos. One small question: I believe you state that expressing our system in terms of eigenvectors makes the system dynamics become diagonal. Would it be incorrect to say that it makes them "orthogonal?"

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@santoshmaharana4020]

How you got this much knowledge? How you got to know you want to learn this?

[@michaelpalin8953]

Wait a minute... he is left-handed!

[@sansha2687]

5:25

[@medochi8427]

You are awesome ❤❤

[@ajj7794]

I Understood all of this lesson except how to go from vector x_dot = Ax therefore x(t)=e^At I try to take laplace and see if i get the same solution by saying |x1_dot| = |A1 A3| * |x1| thus |x1_dot| =A1x1(t)+A3x2(t) thus sX1(s) = A1X1(s) + A3X2(s) this is where i get stuck |x2_dot| |A2 A4| |x2|

[@jumpo121]

do you still need the way how to get to it?

[@ajj7794]

@@jumpo121 thank you for your offer, it might help other students following this playlist so pls do explain it. would you know of any good youtube play list for digital control by chance? many thanks for your help

[@jumpo121]

@@ajj7794 i would like to explain it to you this way: What we have is a model in statespace, right : x_dot = A*x (1) . Now, Imagine, that we have a normal function (not in statespace), called: ydot = E*y (2), where E is a constant. (1) and (2) seem to be almost equal ( you can see the A in (1), like the E in (2) in ) right :) ? Now we transform (2) to ydot - E*y = 0 (2*). what we get is homogenous diffential equation and to solve this equation i will use the exponential approach. so y = e^(lamda)t ydot = lamda*e^(lamda)t. if we put everything in now into (2*) we will get: lamda*e^(lamda)t -E*e^(lamda)t = 0; we divide with e^(lamda)t and we gonna get, lamda-E=0 ,right? so the result is that, lamda = E. NOW, i will put lamda= E to our exponential approach. y=e^(E)t. and this my friend explains how we get to x(t)=e^At . (i did it not in statespace form, cause it is easier to understand it without matrices etc.)

[@ajj7794]

@@jumpo121 thanks, I did not solve it that way and would have never thought of that approach, thank you for enlightening me. The approach I took was to assume that the matrixes where constants and taking the Laplace transform. Many thanks for the nice solution

[@mbonuchinedum6225]

Thanks steve !!!

[@emergency.jergens]

I like this video a lot

[@menelbenaissa9134]

Just magistral

[@senorPFox]

beautiful.

[@vishruthn3875]

How is he so conveniently writing from the other end?

[@IceTurf]

Control Bootcamp - video 2

[@kubigaming3606]

are u writing everything backwards so we re able to see that i a proper form? can't stop thinking about that XD

[@ahsanyusob915]

he might have just written it normally at first on a glass wall, then mirror the video. Idk. Unique. I really like the concept.

[@manfredbogner9799]

very good

[@balajip5030]

Thanks.

[@murphp151]

Sooooo good.

[@maziarghorbani]

Thanks for the tutorial. Just one criticism: the pen’s sound is very unpleasant

[@gustavbrochmann]

Please get some none squeaky pens

[@lorrewu8156]

bravo!!!

[@mktsp2]

Good video but lots of squeaking!

[@QuicksilverSG]

Ugh, this is like 95% linear algebra, 5% control systems.

[@mauriciocarazzodec.209]

👏

[@kormybeats]

🐐🐐🐐

[@caleb7799]

Maybe take some allergy medicine to stop some of that drainage.

[@eevibessite]

kzbin.info/www/bejne/ZnLLm2uJgaman8k

[@jarrettmosco9624]

2:25