someone very unfair gave a thumb down! what could be bad here? Wrong? Expensive? unclear? This is a very neat work by an expert sharing for free his research online!
My mind was blown when you showed that the obsv and ctrb matrices multiplied into the Hankel matrix 🤯 Trying to figure out how this relates to HAVOK 🤔 BPOD relies on a linear model. So I'm guessing you can't reverse this procedure with nonlinear measurements to get some analogous ctrb and obsv matrices for nonlinear systems: I.e. decompose a Hankel matrix of some set of nonlinear measurements into a direct and adjoint impulse response of a nonlinear system. This is confusing, yet very fascinating.
I got a bit confused when you Prof. Brunton talked about C = [B AB ...] as matrix of measurements. This is the controllability matrix, isn't it? Except of this I have to say:"Well explained lecture on BPOD" I am not a fan of youtube lectures. But in your case I am happy to make an exception. Greetings from Germany
Um...isn't this just Hankel SVD from HAVOK, PCT, all that, in a different guise? I mean, this literally looks identical. What's the connection here? Is the adjoint system related to the Perron-Frobenius operator?
Good question. Most of these methods are related somehow. But the history and timeline is important. The ERA (eigensystem realization algorithm) and SSA (singular spectrum analysis) are probably the first of these methods that perform SVD and regression on a Hankel matrix (1980s). Balanced truncation came out around the same time. But Balanced POD was much later, in the early 2000s. DMD came out later, around 2010, and performing DMD on a Hankel matrix (also called "delay DMD") was introduced by Jonathan Tu et al in 2014 in a JCD paper (arxiv.org/abs/1312.0041). My wife Bing was one of the first to use this idea on data from the field of neuroscience. We first introduced a connection between Hankel DMD and the Koopman operator in our HAVOK paper (2017, www.nature.com/articles/s41467-017-00030-8). Then Arbabi and Mezic followed up with a theory paper to strengthen the connection. PCT is the most recent in this line, and uses the full system state instead of a scalar measurement for the embedding. Most of these methods are very close in practice to ERA, but a lot of thought has gone into how the algorithm works for high dimensional data (BPOD and DMD) or nonlinear systems (HAVOK/Koopman).
@@Eigensteve The systems here are linear, right? whereas HAVOK/PCT handles nonlinear/fully chaotic behavior? Is there a connection between the adjoint system in this video and the Perron-Frobenius operator? Is it known why HAVOK seems to produce the almost-invariants sets of that operator in your Lorenz example (and does this happen in any other systems?)
Thank you for this amazing lecture series, professor! I had a couple questions from this lecture: - I did not understand how the adjoint system was obtained from the direct system and what its physical meaning was (it doesn't seem to be a simple adjoint because the direct system depends on A and B while the adjoint system depends on A and C). Also, I understand what giving an impulse input u means but I did not understand what it means for y since that is something we measure and not something we control. Are the two x's in the direct and adjoint systems the same state variable or different? - Regarding the Hankel matrix being symmetric - for SISO systems the structure of the matrix guarantees this. But in general, for MIMO systems, for the matrix to be symmetric we also require the component block matrices CB, CAB etc to also be symmetric. But we know that for any arbitrary system, A, B, C matrices are completely independent of each other and this condition is not guaranteed (or am I wrong about this?).
The adjoint system will probably require a lecture on its own. But regarding the Hankel matrix, you are absolutely right, this will only be block symmetric in the MIMO case.
someone very unfair gave a thumb down!
what could be bad here? Wrong? Expensive? unclear?
This is a very neat work by an expert sharing for free his research online!
My mind was blown when you showed that the obsv and ctrb matrices multiplied into the Hankel matrix 🤯 Trying to figure out how this relates to HAVOK 🤔 BPOD relies on a linear model. So I'm guessing you can't reverse this procedure with nonlinear measurements to get some analogous ctrb and obsv matrices for nonlinear systems: I.e. decompose a Hankel matrix of some set of nonlinear measurements into a direct and adjoint impulse response of a nonlinear system. This is confusing, yet very fascinating.
I got a bit confused when you Prof. Brunton talked about C = [B AB ...] as matrix of measurements. This is the controllability matrix, isn't it? Except of this I have to say:"Well explained lecture on BPOD" I am not a fan of youtube lectures. But in your case I am happy to make an exception. Greetings from Germany
Thank you!!
Where does the adjoint equation come from? Why is it correct? Any sources where I can find explanation? TIA
Um...isn't this just Hankel SVD from HAVOK, PCT, all that, in a different guise? I mean, this literally looks identical. What's the connection here?
Is the adjoint system related to the Perron-Frobenius operator?
Good question. Most of these methods are related somehow. But the history and timeline is important. The ERA (eigensystem realization algorithm) and SSA (singular spectrum analysis) are probably the first of these methods that perform SVD and regression on a Hankel matrix (1980s). Balanced truncation came out around the same time. But Balanced POD was much later, in the early 2000s. DMD came out later, around 2010, and performing DMD on a Hankel matrix (also called "delay DMD") was introduced by Jonathan Tu et al in 2014 in a JCD paper (arxiv.org/abs/1312.0041). My wife Bing was one of the first to use this idea on data from the field of neuroscience. We first introduced a connection between Hankel DMD and the Koopman operator in our HAVOK paper (2017, www.nature.com/articles/s41467-017-00030-8). Then Arbabi and Mezic followed up with a theory paper to strengthen the connection. PCT is the most recent in this line, and uses the full system state instead of a scalar measurement for the embedding.
Most of these methods are very close in practice to ERA, but a lot of thought has gone into how the algorithm works for high dimensional data (BPOD and DMD) or nonlinear systems (HAVOK/Koopman).
@@Eigensteve The systems here are linear, right? whereas HAVOK/PCT handles nonlinear/fully chaotic behavior? Is there a connection between the adjoint system in this video and the Perron-Frobenius operator? Is it known why HAVOK seems to produce the almost-invariants sets of that operator in your Lorenz example (and does this happen in any other systems?)
Thank you for this amazing lecture series, professor! I had a couple questions from this lecture:
- I did not understand how the adjoint system was obtained from the direct system and what its physical meaning was (it doesn't seem to be a simple adjoint because the direct system depends on A and B while the adjoint system depends on A and C). Also, I understand what giving an impulse input u means but I did not understand what it means for y since that is something we measure and not something we control. Are the two x's in the direct and adjoint systems the same state variable or different?
- Regarding the Hankel matrix being symmetric - for SISO systems the structure of the matrix guarantees this. But in general, for MIMO systems, for the matrix to be symmetric we also require the component block matrices CB, CAB etc to also be symmetric. But we know that for any arbitrary system, A, B, C matrices are completely independent of each other and this condition is not guaranteed (or am I wrong about this?).
The adjoint system will probably require a lecture on its own. But regarding the Hankel matrix, you are absolutely right, this will only be block symmetric in the MIMO case.
Sir can you share the link where you have explained c2d transformation?
I think this is what you want ruclips.net/video/h7nJ6ZL4Lf0/видео.html