Hi Steve! I think this video is the most interpretable and wonderful lecture that illustrates the entire concept of model reduction and balanced truncation and thank you for your dedicated efforts. I have a question about the balanced truncation method. Is this way still can be an applicable way when dealing with nonlinear systems? Thank you so much!!
Hi Steve, start from 11:45, I am still confused about why we can pop out the inv(T). You said we assume T is some eigenvectors or orthogonal... But how this assumption help us eliminate the exponential?
Really good question. In the definition of the matrix exponential, e^{B} = I + B + (1/2)B^2 + (1/3!)B^3 + ..., if we plug in invT * A * T, we get e^{invT * A * T} = I + invT*A*T + (1/2)(invT*A*T*invT*A*T) + (1/3!)*(invT*A*T*invT*A*T*invT*A*T) + ... you can write I=invT*T and then we get e^{invT * A * T} = invT*T + invT*A*T + (1/2)(invT*A*T*invT*A*T) + (1/3!)*(invT*A*T*invT*A*T*invT*A*T) + ... = invT* ( I + A + (1/2)*A^2 + (1/3!)*A^3 + ... ) * T = invT * (e^A) * T* See time 16:00 in the following video: ruclips.net/video/nyqJJdhReiA/видео.html
Almost forget this. Thanks for pointing out! As I watch the following video, another question pop out. At 2:50, you assume ~wo and ~wc are diagonal. But we don't have such assumption for original wo and wc. I can see this makes calculating T and scale it easier. Can we can also do that without this assumption ?
Yeah, here, we are assuming that we have performed the balancing transformation that makes these two Gramians equal and diagonal. It is not obvious that this is possible, but it is. I show how to derive this transform in ruclips.net/video/IlrnTjwujBk/видео.html
Hi! Very nice video, but I think there is a small mistake towards the end. hat(A)*= T* A* T-* (then T* and T-* cancel out correctly in the integral) :)
Sweet baby jesus!!! THIS LECTURE SERIES IS F*CKING AWESOME!!!!!!
Hi Steve! I think this video is the most interpretable and wonderful lecture that illustrates the entire concept of model reduction and balanced truncation and thank you for your dedicated efforts. I have a question about the balanced truncation method. Is this way still can be an applicable way when dealing with nonlinear systems? Thank you so much!!
Hi Steve, start from 11:45, I am still confused about why we can pop out the inv(T). You said we assume T is some eigenvectors or orthogonal... But how this assumption help us eliminate the exponential?
Really good question. In the definition of the matrix exponential, e^{B} = I + B + (1/2)B^2 + (1/3!)B^3 + ..., if we plug in invT * A * T, we get
e^{invT * A * T} = I + invT*A*T + (1/2)(invT*A*T*invT*A*T) + (1/3!)*(invT*A*T*invT*A*T*invT*A*T) + ...
you can write I=invT*T and then we get
e^{invT * A * T} = invT*T + invT*A*T + (1/2)(invT*A*T*invT*A*T) + (1/3!)*(invT*A*T*invT*A*T*invT*A*T) + ...
= invT* ( I + A + (1/2)*A^2 + (1/3!)*A^3 + ... ) * T
= invT * (e^A) * T*
See time 16:00 in the following video: ruclips.net/video/nyqJJdhReiA/видео.html
Almost forget this. Thanks for pointing out! As I watch the following video, another question pop out. At 2:50, you assume ~wo and ~wc are diagonal. But we don't have such assumption for original wo and wc. I can see this makes calculating T and scale it easier. Can we can also do that without this assumption ?
Yeah, here, we are assuming that we have performed the balancing transformation that makes these two Gramians equal and diagonal. It is not obvious that this is possible, but it is. I show how to derive this transform in ruclips.net/video/IlrnTjwujBk/видео.html
Wow! That was so unexpected. Amazing!
@@Eigensteve Thank you for the explanation and @shengjian thank you for the question !
awesome videos, thanks a lot for sharing!
Hi! Very nice video, but I think there is a small mistake towards the end.
hat(A)*= T* A* T-* (then T* and T-* cancel out correctly in the integral) :)
I've just seen you corrected that in the next video of the series. Keep up the good work!