I upvoted this post and your comment, that is equivalent to upvoting the post twice.
Re:Functions are Vectors
by sheepscreek on Sunday July 30, 2023 @09:46AM
(#36931098)
My first thought was about its striking conceptual similarity to Fourier transforms. Truly fascinating. Going to explore this a bit more. Thanks for sharing.
I am so thankful for Steve Brunton's content. Had I been able to access his content a decade ago during my M.Sc, I would have probably pursued a PhD out of the passion and quality he brings to this domain - instead I just felt done with academia, strugglign to find grants, reading yet again terse books all alone, and just moved on.
Solid YouTube educators are creating immense future opportunities and we'll all benefit from that. Control theory connects the dots between all sorts of things and can be a joy for those of us who love seeing patterns and structures everywhere (iirc Steve has a recent video about control theory for social models also).
Another fantastic application that the website doesn't mention is the composition / Koopman operator. In control theory (e.g. autonomous drones, cars, robot arms, etc.), most real-world systems are described by nonlinear dynamics which are very difficult to work with (e.g. safety/stability guarantees, optimizing over forward horizons using NMPC, state estimation, etc.) The Koopman operator however gives a globally relevant linear approximation of non-linear systems. In other words, you can treat a nonlinear system as a linear system with fairly high accuracy. This greatly simplifies control and estimation from a computational perspective. You can also learn these linearizations from data. Steve Brunton has some good materials on Koopman theory [2][3], and there are some great applications to control of systems such as soft robots [4].
[1]: https://arxiv.org/abs/1904.02539
[2]: https://youtube.com/playlist?list=PLMrJAkhIeNNSVXUvppZTYNHKQ...
[3]: https://arxiv.org/abs/2102.12086
[4]: https://arxiv.org/abs/1902.02827