I would recommend Nonlinear Systems by Khalil
​
So, basic dynamical systems theory (see chapters 10-11 of Khalil's book) goes through how to model systems that are timescale separated. The most introductory example of this is a light bulb. The current is a 60Hz, which is much faster than the dynamics of the tungsten filament. The way you simulate these systems is to freeze the state of the slow system, let the fast system converge, then take a small step in the slow system.
For climate interacting with economics, the climate system is clearly the slow system (as per the IPCC; I'm a dynamicist, not a climate scientist). So, when we model the resulting behavior, we need to hold climate constant and then see how the fast economic system converges before taking a small step in the slow system. Everyone subtly does this exactly backwards. They hold the fast economic system constant and take steps of varying size in the slow climate system. "Imagine if, suddenly, X was underwater. It'd be really disruptive to the economic system, because we'd have to move Y."
Since my degrees are in aerospace, I like to use the example of if we used this kind of reasoning on an aircraft. Fuel usage (and thus weight and weight distribution) is a slow system, whereas the orientation of the aircraft is clearly much faster. If we squinted our eyes and held the fast system constant, then made a big jump in the slow system ("Imagine if, suddenly, half of our fuel disappeared"), then yeah, there would be a huge kick to the orientation of the vehicle. I've actually done this in a flight sim, where you can just magically change the weight characteristics instantaneously. Suddenly, everyone considering taking a flight should need to be alarmist about the coming airplane fuel disaster. When in reality, the plane's orientation easily changes over time to account for the very slow fuel usage (and much more rapidly due to other factors). (We literally have students do this calculation in class.) Clearly, our reasoning has gone horribly wrong.
> One reply here mentioned that bias changes as a function of current meaning the state of the system can be dependant on what came before it. Wouldn't this be considered hysteresis?
I am not 100% sure whether to call it hysteresis. It's likely that it's another non-linearity. There are more non-linearities than just hysteresis ;) Non-linear modeling is truely a field on it's own and it's really hard to explain in a reddit comment. Besides that, I really don't understand a lot of the material myself.
What I would do is try to find the time domain transfer function of your system and then try to transform it to a state-space or laplace domain (laplace only works after linearisation). I don't know about how to do it for non-linear systems (I am still in grad school, but I get my non-linear courses next year).
If you really want to learn about non-linear systems, my classmates recommend this book: https://www.amazon.com/Nonlinear-Systems-3rd-Hassan-Khalil/dp/0130673897
Be prepared to do some insane math though.
> I don't understand the connection to PWM.
Frankly I don't understand that either. It could also be that the person from the image in your original post is shifting the bias voltage up/down. That way the the ratio of positive/negative voltage (also some form of duty cycle) changes.
If you push the average voltage up, you get closer and closer to the saturation point on the positive half and further away from the negative half. This leads to asymmetric distortion (distortion only on the top half and no distortion on the bottom side). If this is the case then my previous comment is nonsense and completely irrelevant to this post (oops).
It doesn't look like he is modeling a tube amplifier at all.
Of course! There are a lot of okay entry-level control theory books, but the really good books are a bit more advanced. The /r/controltheory wiki here has some good book suggestions (in particular the WikiBooks book on control theory), but I'd really recommend watching Steve Brunton's Control Theory Bootcamp on youtube to get a good overview of intro grad level control. Brian Douglass (also on youtube) has also a bunch of great videos on control theory, if you are interested in diving deeper into specific topics.
I used Chen's "Linear Systems: Theory and Design" as my intro book, but it's not exactly the most riveting. My favourite book now is Ian Postlethwaite and Sigurd Skogestad's "Multivariable Feedback Control: Analysis and Design" (apparently control theorists really like colons in their titles).
Now none of these books will use anything beyond advanced linear algebra and functional analysis, so for the nonlinear control that uses the fancier differential geometry, I'd recommend Bullo and Lewis and "Nonlinear Systems" by Khalil. Note that Khalil has another book called "Nonlinear Control", which is just Nonlinear Systems but cut in half. Don't get that one.
Control theory also intersects with optimization (they share the same arXiv classification), so for optimization I'd recommend Convex Optimization by Boyd and Vanderberghe. It's really a fantastic book. Calculus of variations is also essential for studying optimal control.
For your second question, I guess it depends if you want academic or industry positions. I can happily say that right now the job market for control theory is super hot in both. Aerospace and car companies are hiring controls people to do autonomous car stuff and spacecraft GNC (think the spaceX rocket landing), and a few of the car companies even opened up industrial labs where academics can do research and publish papers. It's pretty good. I'm graduating this year, and I managed to line up a few tenure track job interviews. I think like 40 R1-level places were hiring controls people, mostly for autonomous systems work.
That being said, you should definitely study something you are interested in. I have the fortunate problem of being interested in literally everything, so I kind of picked research topics that were hot for academic jobs. I wouldn't focus so much on choosing between "pure" and "applied", because the line is very blurred sometimes, and I think control theory definitely fills a large span of what people consider "pure" and "applied". So I think you are right in that you can study some very pure math topics, and then use those to do controls work. For example, my mathematical interest from undergrad was graph theory, and now all my controls papers that I write are using neat things like spectral and algebraic graph theory. Other things like spacecraft controls uses stuff like Clifford algebras to do the quaternion computations rigorously.
One control-theory-esque thing that is very hot in math departments right now is optimal mass transport. The math department at my university interviewed two faculty candidates doing OMT work. If you are interested, I'd recommend the books by Cedric Villani. The connection to control theory was done by Brenier and Benamou.
When you learn about your graduate admissions, if you want I can take a look at the faculty and see who does more theoretical control theory stuff and make recommendations. Its completely normal to be indecisive, especially if you are an undergrad about to start grad school. Definitely explore a bit, both on the math and the controls side, and feel free to message me if you have more questions. Good luck!
I linked to a paper that demonstrated that all of the damage functions are made up. You linked to Nicholas Stern... who agreed that they're all made up and don't work. He gave some reasons why it could be worse than some made up damage functions. I gave some reasons why it could be better than some made up damage functions. When my claim is, "None of the models are good, so we should probably have more uncertainty," can you give me a single piece of evidence for why I should have more certainty?! Especially given that, again, your references agree that our models suuuuuuuuck.
If you want a slightly more specific reference for timescale separation, see chapters 10 and 11 of Khalil. I don't know how to say it's "more credible" than others without credential dropping. Climate scientists study the climate. Economists study the economy. They both use dynamical models to the extent they are able, but they don't spend their time thinking just about the mathematics and theory of dynamical systems. I do. That's what my PhD is in, and that's what I'm currently doing published and publishable research in.
I completely understand why people use the models and damage functions that they do - because they can do it, and they need to do something. When you're in the research business, you understand that. People do the things that are possible in order to move toward a more full understanding... but the limitations of the work isn't always translated to the public at large, and frankly, sometimes the researchers doing the work don't always understand some of the tools they're using. That's definitely the case when it comes to climate scientists/economists using made up damage functions to predict the effects of climate change.
Normally, climate scientists primarily deal entirely with large scale, long timescale phenomena... and they probably do well with it. I'm not qualified to critique that work, because I'm not qualified in climate. Economists normally deal with much faster systems... and they probably do well with it. I'm not qualified to critique that work, because I'm not qualified in economics. But when you put the two massively different timescale dynamical systems together and try to make conclusions, I'm eminently qualified to say, "Uh, guys, you can't just do that with a made up damage function. It doesn't work. It's the wrong way round." I've published multi-timescale dynamics work in multiple areas (e.g., flying vehicles, neuron circuits, aerodynamics, pure theory). I don't always understand the underlying systems (e.g., the neuroscientists tell me the neurosciency stuff), but I do the dynamics.
The theory is in Khalil. The explanation is in my earlier link... or your earlier link. If you're expecting me to have a flurry of papers of people talking about this particular issue, I don't. I have the one economist who poked his head up and got close to the fundamental dynamics problem. If you have an actual reason or citation arguing this point, let me know. If you're really just arguing, "But economists seem to agree with me in the abstract and for some reason that makes my position on this issue more credible," I'm going to respectfully, and comfortably, disagree.