Actually, the concept of hydroponic farming dates all the way back to the 1940-ies (as do atmospheric water generators).
You have to know that was on a tiny scale. The reason hydroponics are rarely used today is the expense. While it will admittedly work, more traditional methods also work and cost less per ton of food.
Ever heard of Omega gardens that grow food hydroponically in hollow cylinders which rotate around their own axis containing a light source in the middle?
Good God, how many kilowatt hours is that going to consume in the process of creating enough food for everyone? Modern farms do use energy of course, pumps and John Deere. But nothing like what you're purposing. Power isn't free, sunlight is though.
And what purpose is served by putting the plants in a 44 story tower? If you were to go with this (again silly) idea, a 44 acre, one story greenhouse building would be far easier to construct and maintain. It could be sealed providing many of the advantages you apparently want, and have natural sunlight part of the time.. Much cheaper than a 44 story building with a one acre footprint.
A 44 story building would cost well over a 100 million dollars. A one story building with the same floor space, less.
Better still, why not grow your crops out under the open sky.
Robotic arms were invented in 1956
Because putting an additional 3 percent of the American workforce on unemployment is a GOOD