My college major was control theory/system and I wish you were my teacher back then. I never fully understood what it was about but had to follow all the dry materials. Now I am working with dynamical systems as a grad student and almost finished my dissertation on Boolean network control and applying it in psychological systems (children's emotion regulation dynamics). I want to have a better foundation in the classic control theory and your classes are really helpful! Thank you for taking the time to record them and putting them here

@AlexanderWebster_

3 жыл бұрын

Was this a masters? I feel like I rarely see Control Systems Engineering as a major, as it is largely covered in Mech. Engineering.

@testxy5555

3 жыл бұрын

@@AlexanderWebster_ I did my undergrad in China, control is a major, but maybe nowadays more and more pivoting toward AI. The Boolean network control is my PhD dissertation work in Quantitative Psychology. There have been a few applications of control in psychology by Peter Molenaar, mostly in the intervention settings. There is a scarcity of applying control in psych, I think mainly because human subject research does not have the luxury to collect such intensive time-series, but now we have mobile sensors and researchers are adopting experience sampling study design more often, so there might be an increase of need for control theory in psych.

@molomono9481

3 жыл бұрын

@@AlexanderWebster_ Control Systems Engineering is only partially covered in other fields of engineering. But on its own is a field in applied mathematics. Even the math behind AI for the most part is a subset of the field of control theory. (also other fields, but when looking at the algorithms i think this is fair to say)

@AdvancedGT

Жыл бұрын

I completely understand. I also took a major with significant focus on control theory but never understood it. The reason for stability is so intuitively and clearly explained here. It's really amazing. 20 min youtube video gave me more than 3 1.5h lectures

My control teachers were textbook smart, yet they had no empathy with students' needs. They would go hubris on students for getting problems wrong or confusion on math concepts. This is the should-be of a control lecture. Cheers!!

Thank you so much for putting this online. Your explanation has allowed me to get my head around how it all works. I am a home hobbyist that builds all sorts of robots and am currently building a self balancing bike. All the previous control systems I have built were more built using the trail and error method till I could get them ti work. This series of videos will forever change my methods.

Soy de Perú, estudiante de ingeniería mecatrónica y este bootcamp, me ha parecido una de las mejores explicaciones para entender control. I'm from Peru, student of engeneering and I really like this bootcamp, the way that he teachs is very easier and interesting. I really love this bootcamp!

@Eigensteve

9 ай бұрын

Thanks for watching!

I'm brushing up on some material for my aerospace masters and I have to say the way you explain some of these concepts in such an intuitive manner is beautiful. It's really changed the way I view control systems as a topic in my major.

It's is simply the best control course I've ever seen with very good explanation. I learned these concepts in university but never understood of its significant. Thank you & please keep updating your video gradually, I always find interesting to watch & learn them.

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.

Wow! For the first time the fog has cleared on this. These concepts are very well presented! Thank you!

Thank you, sir! I haven't seen such a clear explanation of eigenvalues in terms of stability. I read books, and papers, and couldn't understand the actual cause behind them. You are the best with great knowledge. I can see from your explanation. You are just putting things in a very organized way to be understandable.

Thank you! Your example with rotating liamda was perfect and clear!

This is so amazingly helpful oh my god thank you. Another UC Berkeley grad student recommended your channel. This is 110/10. Thank you for existing

HERO!!!! thx prof ! Thank you for distilling the most important content, presenting the most intuitive point of view and real thinking prototype! This makes mediocre teachers who directly read the textbook ashamed!

Amazing explanation. Such a bummer these lectures were not around when I was in college.

Thank you Steve, your explanation is really clear and understandable. I am studying evolutionary game theory, but for a long time, I am confused about why they judge stability on real part of the eigenvalue of the pay-off matrix, I didn't expect I found the answer here. I enjoy your video a lot. Many thanks for your work.

Its really understandable. First when i got to my lectures in control theory, since i had really poor background, it felt more like i took chinese or arabic language course. Im having exam in one month, and when i pass, it will be because i managed to learn all objectives given in the course, which i would never be able to comprehend in such extend if not of Control Bootcamp. Furthermore i haven't found any course with such structure, which is perfect for people who want to get familiar with control theory. Big THANKS!

So beautifully explained! Thank you 🙏

amazing depth and communication of knowledge here

great video series , very well explained

Please encourage Nathan to put his lecture in Fourier Analysis in this format. I’m sure he is already thinking about it. :)

@Eigensteve

3 жыл бұрын

Will do. I think he has a lot of cool stuff in the works. Here is a playlist on Fourier Analysis I put together for our book (kzread.info/head/PLMrJAkhIeNNT_Xh3Oy0Y4LTj0Oxo8GqsC)

Thank you so much for this!

Woow! Thank Prof Brunton

This solved one of the greatest mysteries in control theory for me: Why discrete case is related to unit circle while continuous is on the imaginary axis! It's because the discretization of x'=A*x is x'=e^Adt*x! Thanks so much

You are incredible, thank you so much. A chemical engineering student here from RPI.

@Eigensteve

3 жыл бұрын

You are very welcome!

Thank you sir for such a nice video........

Thank you very much.

Thank you very much for the amazing videos and for the discussion in the comments! I would just like to point out a correction when you start talking about discrete time domain and relate the value of state X after N intervals to the eigen value of Atilde. You have mentioned that Atilde^N corresponds to lamba^N though I think you mean lamba_tilde^N. When I worked out the math, Atilde^N corresponded to lambda*N. Do correct me if I am wrong in my understanding.

Nice , thanks

Can you say something more about k? Is it just a positive integer? Is there possibility of having negative delta x? Perhaps the question doesn’t make much sense because I think about time as some positive quantity but I can imagine cases where you don’t know the initial condition and would like to evolve the system (back in time) for some particular purpose.

Thank you. I was just thinking during the lecture, how could you write inverse this fast! Oh, it seems I have to watch it again!😊😅

Amazing

Hello Steven, Thanks for the lecture. What happens in the case of Eigenvalues with R=1 (for the discrete case)? I mean when we are on the edge between stable and unstable behavior.

@Eigensteve

3 жыл бұрын

Great question. These eigenvalues would correspond to persistent behavior. If they have an imaginary part, then they will come in complex conjugate pairs, corresponding to sine/cosine pairs at a given frequency. If lambda=+1, then nothing changes along this eigenvector direction (multiplication by one doesn't change anything), and if lambda=-1, then the amplitude flips back and forth from -1 to +1 and so forth for every iteration.

Perfect

Steve, my question is, not all eigenvectors are orthogonal. so, is it right to say that any arbitrary X can be uniquely transformed into its components along the eigenvectors?

Dr Brunton... how do you remember the expression AT=TD. What is the one line mathematical motivation that naturally helps you to remember it?

thanks

Sorry, I am not to much into discrete calculations. But under which thought can you call the derivative of x_k x_k+1? I would have tried a differential fraction like (x_k+1-x_k)/delta t

Unique and excellent lectures on control theory at a higher level. Really appreicate it. One thing that i didn't quite get was the cancellation of the imaginary parts of eigenvalues when they add up to form the real coordinate A matrix? , I think because of different eigenvectors they would add up with some leftover term behind?? isn't it?

@Eigensteve

4 жыл бұрын

This is one of those really interesting properties of real-valued matrices, that even if they have complex eigenvalues and eigenvectors, they have to recombine to return the original real matrix.

@adeeljamal6846

4 жыл бұрын

@@Eigensteve ,thanks a bunch for the explanation. I will check this out.

We can use eigenvalues to compare degree of stability in two systems??

nice videos! a question - at the 11:28 mark I see A^tilde = exp(A*deltaT) at the top left. when I try to derive it though I get A^tilde = T * exp(A*deltaT) * T^-1🤔. Intuitively, from a physics point of view A^tilde should have dimensions - they have physical contants no?

@Eigensteve

4 жыл бұрын

Good question. This is a bit confusing. dt=deltaT is different than T. Atilde is just the discrete time version of continuous-time A, so we take e^{A*dt}. We could expand this in eigenvector coordinates as: Atilde = e^{A*dt} = T * e^{D*dt} * T^-1. And good intuition to keep track of dimensions. Always good practice! Oddly, the matrix "A" has dimensions of "1/seconds" (or "1/time") because it maps "x" to "xdot". the matrix "Atilde" has no dimensions, because it maps x at time k to x at time k+1. So that is another way to check that Atilde = e^{A*dt} makes sense. A has units of "1/s", dt has units of "s", and so if we multiple and exponentiate, Atilde is dimensionless, which is what we expect.

@michaelghorgan

2 ай бұрын

​@@Eigensteve if Atilde = T*e^{D*dt}*T^-1, then doesn't that mean that Atilde^N = T*e{N*D*dt}*T^-1, in which case the eigenvalues are scaled as N*lambda and not lambda^N?

@michaelghorgan

2 ай бұрын

Oh I figured out what I was misunderstanding. The discrete-time eigenvalues are different from the continuous-time ones. lambda_discrete = exp(lambda_continuous * dt), so when Re(lambda_continuous) > 0, lambda_discrete > 1.

What would happen if R = 1 in DT and lambda = 0 in CT?

1:34 If Dr. Brunton or any of his students have any book or URL references to deeper learning material on the "whole classes on ODEs and phase portraits and saddles and sinks and centers", the rest of us would be so grateful if they could post a link to such material here so that we can follow up on these connections. Thank you, and thank you for this course!

@roboticsresources9680

3 жыл бұрын

You can refer Prof. Steven Strogatz's book 'Non-linear Dynamics and Chaos', his lectures on KZread of the same name and the book ' Differential equations and Dynamical Systems' by Perko

@the1111code

Жыл бұрын

3blue1brown channel on YT

Hello Dr. Brunton - quick question: it seems that you use the variable "z" throughout the video to represent system dynamics in the complex plane. The way you described it seemed like a logical flow from continuous time to discrete time. This got me wondering, where/when in this logical flow does this variable switch from the s-plane to the z-plane? In other words, when you moved into discrete time our variable normally would change from s to z, but there was never a change of VARIABLE in your wonderful lecture. Could you help me reconcile this? My advisor wasn't much help. Thanks in advance!!

@adamgronewold7364

3 жыл бұрын

A follow up - I am not sure I follow lambda^n=(R^n)*e(i*n*theta). Where does that come from?? x_k=T*(e^(D*deltat))^k*T^-1*x_0 gives us T*(e^(k*D*deltat))*T^-1*x_0 not T*(e^(D^k*deltat^k))*T^-1*x_0 meaning we will have no lambda^n terms only n*lambda terms.

The eigenvalues are the poles of the system and they must be in the left hand side of the complex plane for the system to be stable.

What is meant about how T is a mixture of the different frequency/time dynamics of e^Dt at around 2:40?

@rajat4640rajat

3 жыл бұрын

You can think T^-1 as the projection matrix for projecting the actual vector X on the eigenvectors Z and T doing the the inverse effect. So, exp(Dt)zo becomes your solution along the independent orthogonal directions in the Z space while T combines these solutions to present you the solution in the X space. Steve may comment.

8:33 Not really. The system dynamics doesn’t only depend on the poles. As correctly shown in the video, the poles determine the Neper frequencies of exponentials, angular frequencies of sines and cosines, and exponents of power functions. But the zeros (along with the poles) determine the amplitude of each term of the signal. Thus the signal depends on both its poles and zeros.

Sir, Can You make a course on Fractional order Control system? It'll be very helpful

nice video, one question because I am confused: the D matrix (so the eigenvalues) of the contious time system and the eigenvalues of the discrete time system are not the same are they? otherwise if x_(k+1) = A_tilde*x_k = e^(A*delta_t)* x_k so for k+1 = N: x_N = Te^(D*N)T^(-1)x_0; where for N --> inf.: x_(N-->inf.) --> inf. if Re(lambda)>0 which is the same as for time contious system BUT if x_(k+1) = A_tilde*x_k = T D_tilde T^(-1)* x_k so for k+1 =N: x_N = [T D_tilde T^(-1)]^N*x_0 = T (D_tilde)^N T^(-1)* x_0; where for N --> inf.: x_(N-->inf.) --> inf. if Re(lambda) > 1 which is what you show for discrete systems. So my point is, the D is not the same for continuous and discrete-time systems is it? I hope whatever I wrote is comprehensible.

@michaelghorgan

2 ай бұрын

Yes I think that's right, D_tilde = exp(D*delta_t), so when Re(lambda) > 0, the lambdas in D_tilde are greater than 1

When you’re writing in the transparent board? Are you *mirror writing*? I imagine you’re doing so but I’m not quite sure.

@alphaomikron2856

3 жыл бұрын

He's writing on a mirror

@Eigensteve

3 жыл бұрын

Yep, although writing backward would be more fun! :)

Thanks alot for this nice video. I want know the condition that makes the following discrete system stable: x(k+1)=Ax(k)+B Where A and B are a known square matrices an x' in R^n. Thanks in advance sir.

7:58 Wow, this is so beautiful! 🤯 B*u comes from the feedback controller, right?

@rajdivyeshmehta868

3 жыл бұрын

Yes it comes from the controller.

It seems that there is a contradiction! we can investigate the same system in continuous time or discrete time. The dynamic behaviour should not be different, but we have different stability condition! I appreciate it if you could make it clear

if two poles of the system have -ve real parts and one is equalled to zero, what would the stability be?

Can he do a video on how to write backwards like a god?

this lecture is uwu