There is a missing factor of 1/n! in the Taylor series. Luckily here it is of no consequence as it only affects the higher order terms that are dropped. Great series of lectures!
I’ve never thought about DEs in this manner with the fixed points, etc. interesting. To me, what’s even more interesting are BVP on irregular domains. Like how the solution to the Helmholtz equation on a rectangle is the 2D fourier series, but, if you go to a rectangle with one quadrant missing, the eigenfunctions are nearly impossible to to represent in a “clean” fashion.
Differential equations are equivalent to vector fields and so studying vector fields provides different perspective. Specifically closed integral paths are precisely periodic solutions of the differential equation. When you use irregular domains you are excluding these integral paths as solutions. The boundaries are then not "natural" in the sense that they interrupt the natural flow of these integral paths forcing more complex solutions.
I'm pretty rusty on this, hence why I'm watching these to try and refresh my memory (10 years out of uni). I always liked to think about the local stability by imagining the state space as an n dimensional space with gravity. If you choose a point and drop a marble, you can watch which direction it rolls. If it falls into a low point and stops, it's stable. If it rolls away forever it's unstable. There are also points where the marble can roll away and then get stopped somewhere else. If you want to develop a controller, you have to figure out what force vectors you need to apply to keep the marble fixed in the point that you dropped it. In real life, the state spaces can be massive, so you can just choose a small sample that you can stay within, so that allows you to approximate it linearly.
Yes, x0 is a fixed point of a differential equation if and only if x(t)=x0 for all t is a solution of the differential equation. Of course in our differential equation x'=f(x), x0 is a fixed point if and only if f(x0)=0.
Dear sir, I hope you will answer my question. If we linearize a non-linear system near the equilibrium point, then we are limited only a very small region of our whole system. My question is what if I want to solve or operate at any other location except the equilibrium point? And since the linear version of non-linear system explains a very small region, I think this is not so meaningful if we are interested in our whole non-linear system. In that situation, how do we explain or solve the system?
After linearizing the system, its operation can tested by simulation or analytic methods, so you can assess how far from the equilibrium point you can go without losing control characteristics. If one point is not sufficient to properly control the system, you can choose other points to cover the entire range of operation of your control system and use a gain scheduling approach to change the model. Try to search for "What Is Gain Scheduling? | Control Systems in Practice" from Matlab channel, it is a good starting point.
@@darkside3ng Thank you so much sir for giving me the instruction. I was looking for this answer in different books also, but didn’t get a satisfactory solution.
Hello sir, according to my knowledge, the linearity around the equilibrium point of a nonlinear system is only true within a small range (in vicinity) around this equilibrium point. Could you please help me with a method to quantify the vicinity around any equilibrium point of a system?
On the illustration drawn near the beginning of the video we see two fixed points, and it seems like our dynamical system flows from one fixed point into the other. Is this always the case? can we have multiple fixed points but the phase portrait only flows around their own fixed points and never crossing into each other?
I feel like taking the Taylor series in powers of delta x could use a bit more elaboration, quite a jump from simple Taylor series expansion. Especially confusing by the overuse of variables that are variations of x in this lecture… x, x bar, delta x :)
For a vector function, you take the iteration fo the jacobian with is the tensor product between the gradient vector and the function vector, for terms of O(x^2), you are dealing with tensor of rank bigger than 2.
Please try to keep ur vedios a bit shorter like I feel interested to watch the series and I see these long vedios and I dnt thnk I have the time to watch every vedios and catch up to the current vedio
![Linearizing Around a Fixed Point [Control Bootcamp]](http://i.ytimg.com/vi/1YMTkELi3tE/mqdefault.jpg)
There is a missing factor of 1/n! in the Taylor series. Luckily here it is of no consequence as it only affects the higher order terms that are dropped. Great series of lectures!
What a wonderful explaination! Thanks for saving my life.
This is so interesting and helps me so much with my research. Thank you very much, Dr. Brunton. Keep'em coming.
Video looks so elegant with the colors and background! Great lecture, beautiful work!
I’ve never thought about DEs in this manner with the fixed points, etc. interesting.
To me, what’s even more interesting are BVP on irregular domains. Like how the solution to the Helmholtz equation on a rectangle is the 2D fourier series, but, if you go to a rectangle with one quadrant missing, the eigenfunctions are nearly impossible to to represent in a “clean” fashion.
Differential equations are equivalent to vector fields and so studying vector fields provides different perspective. Specifically closed integral paths are precisely periodic solutions of the differential equation. When you use irregular domains you are excluding these integral paths as solutions. The boundaries are then not "natural" in the sense that they interrupt the natural flow of these integral paths forcing more complex solutions.
i wish i could remember everything from my ODE class! i sort of turned towards more computational and statistical work, but pdes are beautiful math
you make me love math
thanks for your lectures
❤❤❤
Very useful. Thank you professor!
I'm pretty rusty on this, hence why I'm watching these to try and refresh my memory (10 years out of uni).
I always liked to think about the local stability by imagining the state space as an n dimensional space with gravity. If you choose a point and drop a marble, you can watch which direction it rolls. If it falls into a low point and stops, it's stable. If it rolls away forever it's unstable. There are also points where the marble can roll away and then get stopped somewhere else.
If you want to develop a controller, you have to figure out what force vectors you need to apply to keep the marble fixed in the point that you dropped it.
In real life, the state spaces can be massive, so you can just choose a small sample that you can stay within, so that allows you to approximate it linearly.
Thanks prof. Big fan.
Excellent explanation. greetings from Peru.
Great work!
ti's getting interesting thanks
I love you! Thank you for this video!
Great video. Thank you
Many thanks!
Yes, x0 is a fixed point of a differential equation if and only if x(t)=x0 for all t is a solution of the differential equation.
Of course in our differential equation x'=f(x), x0 is a fixed point if and only if f(x0)=0.
Dear sir,
I hope you will answer my question. If we linearize a non-linear system near the equilibrium point, then we are limited only a very small region of our whole system. My question is what if I want to solve or operate at any other location except the equilibrium point? And since the linear version of non-linear system explains a very small region, I think this is not so meaningful if we are interested in our whole non-linear system. In that situation, how do we explain or solve the system?
Hello sir, did you find any resources on this topic?
@@zaynbashtash Not yet
Control Theory 🎃 See NASA
After linearizing the system, its operation can tested by simulation or analytic methods, so you can assess how far from the equilibrium point you can go without losing control characteristics. If one point is not sufficient to properly control the system, you can choose other points to cover the entire range of operation of your control system and use a gain scheduling approach to change the model.
Try to search for "What Is Gain Scheduling? | Control Systems in Practice" from Matlab channel, it is a good starting point.
@@darkside3ng Thank you so much sir for giving me the instruction. I was looking for this answer in different books also, but didn’t get a satisfactory solution.
Hello sir, according to my knowledge, the linearity around the equilibrium point of a nonlinear system is only true within a small range (in vicinity) around this equilibrium point. Could you please help me with a method to quantify the vicinity around any equilibrium point of a system?
On the illustration drawn near the beginning of the video we see two fixed points, and it seems like our dynamical system flows from one fixed point into the other. Is this always the case? can we have multiple fixed points but the phase portrait only flows around their own fixed points and never crossing into each other?
great Sir
Plz make videos on how to draw this graph in mathematica or matlab
I feel like taking the Taylor series in powers of delta x could use a bit more elaboration, quite a jump from simple Taylor series expansion. Especially confusing by the overuse of variables that are variations of x in this lecture… x, x bar, delta x :)
For a vector function, you take the iteration fo the jacobian with is the tensor product between the gradient vector and the function vector, for terms of O(x^2), you are dealing with tensor of rank bigger than 2.
in what book can i find this theory ? i can not find it
What if there are no fixed points?! Does it mean linearization is not an option? Then what should we do?
Perfect
Classrooms Are Using dx For dz All The Time And Ignoring dt/dt In Maxwell's Equations
Greedy Instructor Perspective
Nice
thank youuu
How does this man write backwards
The image is mirrored. In one of his lectures he mentiones he is left-handed
@@marekw4353 that makes so much sense, thanks!
What's going on in here
if only i can summon a girlfriend like you summon your delta x
Please try to keep ur vedios a bit shorter like I feel interested to watch the series and I see these long vedios and I dnt thnk I have the time to watch every vedios and catch up to the current vedio
Bro he is giving education for free. Don't be entitled
Feel intersted? What a low frustration tolerance.