FWIW, whenever I teach undergrad control, I always aim to emphasize one key fact: complex exponentials of the form exp(st), where s is complex, are eigenfunctions of linear time-invariant systems. The reason why you need s to be complex (as opposed to purely imaginary, which would only give you sinusoids) is that the corresponding set of signals (sinusoids with exponentially decaying or exponentially growing envelopes) is sufficiently rich to allow for things like system ID and for analyzing both transient and steady-state behavior.
Loving this series. Has me side-eyeing the spine of my Ogata textbooks from across the room.
Note sure if this fits in with your gameplan yet, but I'd be interested to hear how/if robust control fits into the modern optimization-industrial complex.
Prof. Recht -- killer blog, love literally everything you post. Looks like you have some conversion problems at some point in the Mathjax/Latex > Markdown > HTML > live site pipeline, though -- I see raw Latex in the post. Or at least, that's my guess; my blog (the less interesting social science grad student version of this: (https://griffinjmbur.github.io//counting-blogpost/) suffers from similar problems. I write in Mathjax and only one kind of in-text Latex escape character works for getting it to a GH blog with Jekyll (the clunky \\(x+y\\) style), which I did not realize until I began publishing my notes, so I've made a lot of manual conversions from $x+y$ to that (unfortunately ChatGPT is strangely bad at automating this), and I have lots of stuff like "\(\frac{P}{1+PC}\)", which I see in much of your post, in there as a result.
Just a friendly heads-up. You are literally my favorite blogger about anything remotely serious.
Interesting, I never thought about Laplace transforms for sound. I have used them in EEG signal source decoding, and we tend to call it the Laplacian transform. Math is everywhere!
FWIW, whenever I teach undergrad control, I always aim to emphasize one key fact: complex exponentials of the form exp(st), where s is complex, are eigenfunctions of linear time-invariant systems. The reason why you need s to be complex (as opposed to purely imaginary, which would only give you sinusoids) is that the corresponding set of signals (sinusoids with exponentially decaying or exponentially growing envelopes) is sufficiently rich to allow for things like system ID and for analyzing both transient and steady-state behavior.
Loving this series. Has me side-eyeing the spine of my Ogata textbooks from across the room.
Note sure if this fits in with your gameplan yet, but I'd be interested to hear how/if robust control fits into the modern optimization-industrial complex.
Prof. Recht -- killer blog, love literally everything you post. Looks like you have some conversion problems at some point in the Mathjax/Latex > Markdown > HTML > live site pipeline, though -- I see raw Latex in the post. Or at least, that's my guess; my blog (the less interesting social science grad student version of this: (https://griffinjmbur.github.io//counting-blogpost/) suffers from similar problems. I write in Mathjax and only one kind of in-text Latex escape character works for getting it to a GH blog with Jekyll (the clunky \\(x+y\\) style), which I did not realize until I began publishing my notes, so I've made a lot of manual conversions from $x+y$ to that (unfortunately ChatGPT is strangely bad at automating this), and I have lots of stuff like "\(\frac{P}{1+PC}\)", which I see in much of your post, in there as a result.
Just a friendly heads-up. You are literally my favorite blogger about anything remotely serious.
robotics are powerful
A whole new world when we replace Laplace with z-transform: z = exp(st).
> The tricky complex analysis sticks when you hear a concept in a song you like.
Missing an opportunity to include audio examples!
Interesting, I never thought about Laplace transforms for sound. I have used them in EEG signal source decoding, and we tend to call it the Laplacian transform. Math is everywhere!