Wow. I am an applied mathematician works in mathematical ecology (population dynamics) and epidemiology. I also teach nonlinear dynamics to students. Its really nice how you teach asymtotical stability in linear dynamical system. I hope you will do more videos on dynamical systems and other areas of mathematics.
There are no more videos for 11 months! Hope you'll be back soon! I feel really lack of such content like this, especially on so highly advanced topics here on youtube!!! I guess the demand for these videos will only grow in time, as more programmers over the world will get into more complex tasks
You are absolutely correct, and that would be an alternative (and often, easier) way of checking stability. The Lyapunov approach has a few attractive properties tough. For example, and this is subjective of course, It is more intuitive (it doesn't require knowledge of eigenvalues, and how they affect stability). More fundamentally, Lyapunov's approach (i) generalizes to nonlinear system, and (ii) can be used to not only decide stability of some known dynamical system, but can also to optimize over stable dynamical systems.
@@VisuallyExplained If I may add, it is also beneficial to *have* the Lyapunov function since, in some cases, it can beinterpreted as a cost-to-go / value function in control theory. With local / greedy optimization of this function, one may derive a stabilizing controller for an unstable system. So it is useful to synthesize controllers, not just to analyze stability.
Cool video! Something I’m confused on is that you said there is no Lyapunov function if the SDP is infeasible. But all that about the SDP was under the assumption that the Lyapunov function is a quadratic. So couldn’t the SDP be infeasible, but there exists a Lyapunov function which doesn’t necessarily take a quadratic form? Then wouldn’t the system be asymptomatically stable due to the iff?
Great question. As it turns out, a linear system is stable iff it has a quadratic lyapunov function. (This is of course not true for nonlinear systems)
Thanks for making this video! I have a question about your comments at 4:25. Could you explain more or point me to a reference that explains more about this method for changing the strict inequality to an SPD one? At first glance, it seems that an approximation is applied, but you said there is no loss of generality
@2:25 is it possible to explain why the trajectory can't settle into a "loop" on the paraboloid, i.e. doesn't converge to 0, but converges to an ellipse?
If the goal is to determine the asymptotic stability of x_{k+1} = A x_{k}, can you not compute the eigenvalues of A (which is a simpler task)? Applying semi-definite programming here seems unnecessary?
For these animations: Did you overlay seperate blender and manim animations in post, or did you integrate manim into the internal blender python script and render it all via blender?
Thanks for the comment! I will do a video about my workflow at some point in the future. For now, you can look up the Blender "Donut's videos" on youtube, that how I started ;)
I LOVE the content, but as someone who is new to data analytics, it's challenging to keep up with the pace. Any chance you could speak more slowly so for those of us who all this terminology is new, we can have a better shot at keeping up with you. Thanks!!
Still waiting for the max cut video, the unreasonable effectiveness of semidefinite programming. P/S some researcher in wireless communication also use this :))
Wow. I am an applied mathematician works in mathematical ecology (population dynamics) and epidemiology. I also teach nonlinear dynamics to students. Its really nice how you teach asymtotical stability in linear dynamical system. I hope you will do more videos on dynamical systems and other areas of mathematics.
Can you make videos on Lebesgue meausure and integration ? Thanks
There are no more videos for 11 months! Hope you'll be back soon! I feel really lack of such content like this, especially on so highly advanced topics here on youtube!!! I guess the demand for these videos will only grow in time, as more programmers over the world will get into more complex tasks
Spectacular animations
Nice job! Great video!
What do you use for the drawings?
Thanks! I use Blender3D and the python library "manim" to produce the animations, and then adobe premiere/after effect to stitch them together.
@@VisuallyExplained OMG that's definitely a lot of efforts behind this beauty , that's inspiring !
@@VisuallyExplained have you ever thought about outlining the workflow? :)
If A is linear, can't you just check if the largest eigenvalue of A has a magnitude less than one? Then A^∞ = 0
You are absolutely correct, and that would be an alternative (and often, easier) way of checking stability. The Lyapunov approach has a few attractive properties tough. For example, and this is subjective of course, It is more intuitive (it doesn't require knowledge of eigenvalues, and how they affect stability). More fundamentally, Lyapunov's approach (i) generalizes to nonlinear system, and (ii) can be used to not only decide stability of some known dynamical system, but can also to optimize over stable dynamical systems.
@@VisuallyExplained If I may add, it is also beneficial to *have* the Lyapunov function since, in some cases, it can beinterpreted as a cost-to-go / value function in control theory. With local / greedy optimization of this function, one may derive a stabilizing controller for an unstable system. So it is useful to synthesize controllers, not just to analyze stability.
So grateful this video is up
Cool video! Something I’m confused on is that you said there is no Lyapunov function if the SDP is infeasible. But all that about the SDP was under the assumption that the Lyapunov function is a quadratic. So couldn’t the SDP be infeasible, but there exists a Lyapunov function which doesn’t necessarily take a quadratic form? Then wouldn’t the system be asymptomatically stable due to the iff?
Great question. As it turns out, a linear system is stable iff it has a quadratic lyapunov function. (This is of course not true for nonlinear systems)
Great presentation
Thank you for supporting the channel!!
Thanks for making this video!
I have a question about your comments at 4:25. Could you explain more or point me to a reference that explains more about this method for changing the strict inequality to an SPD one? At first glance, it seems that an approximation is applied, but you said there is no loss of generality
This is awesome!
These videos are great, thanks for the upload
Loved it!! Looking forward to the 4th video!
Yay! Thank you!
@2:25 is it possible to explain why the trajectory can't settle into a "loop" on the paraboloid, i.e. doesn't converge to 0, but converges to an ellipse?
If the goal is to determine the asymptotic stability of x_{k+1} = A x_{k}, can you not compute the eigenvalues of A (which is a simpler task)? Applying semi-definite programming here seems unnecessary?
Underrated video!
For these animations: Did you overlay seperate blender and manim animations in post, or did you integrate manim into the internal blender python script and render it all via blender?
IMHO, at 00:57 sec in, do you mean to say "..related to the PREvious state"? Ie., the state u_(t+1) in related to the previous u_t (by a function)?
Superb
Wow! Could you tell how you do such great visualisations on blender?
Thanks for the comment! I will do a video about my workflow at some point in the future. For now, you can look up the Blender "Donut's videos" on youtube, that how I started ;)
I LOVE the content, but as someone who is new to data analytics, it's challenging to keep up with the pace. Any chance you could speak more slowly so for those of us who all this terminology is new, we can have a better shot at keeping up with you. Thanks!!
Thanks for the nice comment! I will keep your feedback in mind for the next videos
Still waiting for the max cut video, the unreasonable effectiveness of semidefinite programming. P/S some researcher in wireless communication also use this :))
Coming soon...
Can you make a video on positively invariant sets? Thanks for your work.
en.wikipedia.org/wiki/Positively_invariant_set