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Sort of a tangential question (?). Does the eigenvalues of a system correlate to the lyapunov function somehow? Because I notice that negative eigenvalues means that the system is stable, a positive eigenvalue means that a system's exponents diverge, and a 0 eigenvalue means that the system is constant somehow and doesn't diverge to infinity or converge to 0. I feel like I'm drawing lines but I can't seem to generalize it.

The energy level correlation to the lyapunov function is beautiful and so intuitive

What video do I watch before this one?

Thank you! This was awesome

Prof, are your lecture notes posted online?like to study it besides watching your lecture video, Thanks

Thank you for explaining so clearly

Thanks! Really good explanation 👌

jesuse is god! ☺️ puzzle master

Hello, Richard! I believe you're implicitly assuming that $Im(A) \subset Im(B)$. Otherwise substituting $u$ like this wouldn't really work. Consider for instance A = [[0, -1], [1, 0]], B = [[0], [1]]; clearly the system is controllable, yet your argument doesn't work here.

You are the best, thank you very much

thank you so much for showing a proof thats accessible

Does a similar relationship hold between the magnitude and phase of transferfunction of solely the controller that is not set in a negative feedback loop?

holy cow are you writing mirrored? youre insanely talented

Very awesome video. I really wasn't able to understand it until I saw the video. Thank you very much.

can we think of the terminology of short and fat / tall and thin matrices in away associated with the over-determined system and under-determined system in relation with the number of unknowns and the number of equations

🌷🌷🌷 the best)

Thank u))

How do you approach to a general solution for stability from a stand point of a LYAPUNOV functions for a class of nth order nonlinearar differential equations?

You explained in 10 minutes what my university couldn't explain in 60.

had that figured out before I had even heard of lyapunov. Isn't this obvious?

sir give more information related to rechable set estimation using ellipsoid estimation ,polytopes,zonotopes

Amazing

Thank you Professor

It would be really great if these could be rerecorded without the heavy breathing noises.

when modelling the state space for a given system does y have to be the signal that is completely measured or can we choose a y where some compoenents of y aren't measurable ?

Hello Prof. Richard, Thank you very much for your useful lectures. I have a question, please. At time 23:20, you said that the b^2 will cancel each other. Can you kindly explain this, please? there might be a math error here

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dude you are life saver thanks

love the transition from "imagine we have a linear system" to state vector nihilism almost immediately 😂

The theorem says that M is the largest invariant set in E, but what you described is positive invariance. Invariance is for all t, not just for nonnegative t.

I am outsider of the topic, I have a homework, and I understand nothing about it

Nyquist plot should start at -1 and and in 0.

Omg ur the Son of the Red Dwarf computer 👏🏽

Hi Richard, I am Joel from Argentina and I am currently studying A&C in Germany.. Your videos are really useful, I really appreciate your dedication and effort to help students ! I will recommend your RUclips channel with my classmates, wish you a incredible and successful future

Great video, thanks for this explanation.

Exceptionally clear. Thank you! Just one bit that confused me: you indicated the argument that F(s0) makes with the Re axis as negative, and the argument of F(s1) as positive. Both are BELOW the Re axis, though - is this not inconsistent?

Thank for you dedication, every other content about this is so confusing. your more graphical explanation at least made realize how is suposed to use this method

I noticed that this fact is trivial if the matrix is diagonalizable.

This is beautiful.. Did you make it in manim or blender..?

Thank you very much!!!!!🥰

Thanks for making these videos! They help reinforce my school lectures

Thanks for making understand

Hello proffesor, i like your explanation, but I already studied Lyapunov stability from Slotine book(Applied nonlinear control) and it talks that the theorem is valide around origen(e.g. x*=0). For you, can x* be any equilibrium point define in Omega?. Peace!!

Wonderfully explained! Thank you. I was wondering though, how would a state-space model's controllability be determined when the underlying dynamics are non-linear and/or time variant? The matrices A and B could still be computed then, but they'd change over time.. how would this affect the process of determining controllability?

Thank you for the explanation

that's informative.

Where are these so called lecture notes? Could a random like me access them somehow?

Thanks! Great explanation!

Dear sir, Your way of teaching this subject is very helpful. Thanks a lot. Just wanted you to note that the B1 matrix would be [k ;0] instead of [1;0].

thank you very much for this beautiful video, is the v (you were trying to solve ) represent the input?