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.

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?)

I think this is what you want kzfaq.info/get/bejne/np2efZmLsZqwlmQ.html