Perhaps this is obvious to many but this introduction perfectly fits to the operational amplifier analysis in circuit theory courses. Current undergrad textbooks cover these topics (details of opamp’s) very lightly; but, this is a profession for many in analog circuits or amplifier design. (Ideal op-amp has an open-loop gain of infinity (which is A here) and we assume zero-propagation delay (no dynamics). The resistive op-amp analysis is exactly as above in principle. We use piecewise linear models to approximate the sigmoid function above.) Since UC Berkeley is the starting point of the electronics revolution, I prefer to stop here. You may check this old book for more details: https://ocw.mit.edu/ans7870/RES/RES.6-010/MITRES_6-010S13_comchaptrs.pdf
The P term is the gradient of an assumed quadratic potential function centered at the goal, and the D term is damping, so those sound somewhat analogous. However the more holistic comparison to the ML situation is the closed-loop value function with the PID controller, for which you’d have to know more about the system.
Yes, indeed. Here the idea is that a cheap sensor and a cheap threshold-based on-off controller beat more expensive solutions that would enforce continuity. (Washers, dryers, and dishwashers use such controls, for example.)
Perhaps this is obvious to many but this introduction perfectly fits to the operational amplifier analysis in circuit theory courses. Current undergrad textbooks cover these topics (details of opamp’s) very lightly; but, this is a profession for many in analog circuits or amplifier design. (Ideal op-amp has an open-loop gain of infinity (which is A here) and we assume zero-propagation delay (no dynamics). The resistive op-amp analysis is exactly as above in principle. We use piecewise linear models to approximate the sigmoid function above.) Since UC Berkeley is the starting point of the electronics revolution, I prefer to stop here. You may check this old book for more details: https://ocw.mit.edu/ans7870/RES/RES.6-010/MITRES_6-010S13_comchaptrs.pdf
The P term is the gradient of an assumed quadratic potential function centered at the goal, and the D term is damping, so those sound somewhat analogous. However the more holistic comparison to the ML situation is the closed-loop value function with the PID controller, for which you’d have to know more about the system.
Non-differentiability isn't so bad (we deal with this both in optimization and control).
But yes, discontinuities can get dicey.
And yet discontinuous controls (say, of the on-off type) are often cheaper to instrument. Irmgard Flügge-Lotz was one of the pioneers of discontinuous automatic control (https://en.wikipedia.org/wiki/Irmgard_Fl%C3%BCgge-Lotz); in fact, she wrote a book on it: https://press.princeton.edu/books/hardcover/9780691653259/discontinuous-automatic-control.
yes, but isn't it interesting that there the discretization is accurate and inside the controller (as opposed to in the plant)?
Yes, indeed. Here the idea is that a cheap sensor and a cheap threshold-based on-off controller beat more expensive solutions that would enforce continuity. (Washers, dryers, and dishwashers use such controls, for example.)