"But even basic control theory – what you'd need to actually stabilize a dynamic system as complex as a helicopter – fills a textbook or two, and the field is still evolving."

I am not that good at math, but I feel like if I was trying to solve that problem, it should be as simple as finding someone smart and saying "use a Kalman filter"...

In one sense I can't say it's "simple" or "basic" because I can't do it, but I'm under the impression that it is a manageable solved problem that can be summarized in four words. Is that wrong?

Yes, in exactly the same sense that "building a bridge is simple because I can say 'use a truss' to a smart person" is wrong. Things having names doesn't automatically make them simple.

There are – like I said – entire textbooks on the subject – and why it is complex – if you are genuinely interested. I mean, heck, there's two textbooks alone about one particular system I randomly found on Wikipedia [1].

However to directly address your misconception, Kalman filters are not control systems. They are related to control theory, and are used to provide inputs to control systems. But control theory is its own thing, and there is no "general" solution to correctly control any given system.

[1] https://en.wikipedia.org/wiki/Furuta_pendulum

Regarding your pendulum example - wouldn't it be significantly simpler if you had a closed-loop system with sensors feeding back information? What I was suggesting about Kalman filters was that instead of using a "proper" analysis, is it possibly more practical to just add some cheap sensors and combine their output to tell the actuator what to do?

In a lot of fields, elegant analysis has given way to cheap brute-force application of computers that is easier to generalize - that's what I'm fumbling at here.

After all, if you or I try to balance a pencil on our fingertip, we are not doing the math on that wikipedia page in our heads. We are in fact using something more akin to the brute force method - sense, synthesize, act, repeat.

I'm not an expert in controls, but it's an area that interests me and I've taken a couple classes. It's true that the theory of linear, time-invariant systems is pretty complete. In that very limited framework, you can basically make a system behave in any way you want. The interesting thing is all the various infuriating ways in which real systems fail to be linear. Those nonlinearities can be accounted for, but now you no longer have a turn-key solution. Furthermore, there really is no such thing as the optimal controller. It all depends on the objective function you are trying to optimize. How much do you value a fast response time versus strong disturbance-rejection? Would operating the system that way cause undue wear and tear over the service lifetime? Would it cause danger or discomfort to human passengers (if any)? How sensitive is system behavior to uncertainty in model parameters? It's obvious that the design objectives involved in building an air-to-air missile are very different from those involved in building an elevator. But explicating them and translating them into an actual controller requires a great deal of care.