In practice, what is the difference between the "nominal plant model" and the "plant model"?
Is the nominal model the desired model? (i.e. do we want to reach it?)


In much the same way as I'd like a rich stranger to die and bequeath $100 million to me, yes, it's the desired model.


  Is the nominal model an imprecise idealization of the plant to simplify the mathematics?


The nominal model is somewhere between your best guess of how the plant actually behaves, and a simplified model of how the plant behaves that you guess will be good enough for your purposes.

It's not possible to know exactly how a physical system will work, and in general, any physical system will have essentially infinitely complex dynamics (just count the electrons and multiply by 2...)

So you come up with a model that you hope will solve the problem at hand.

Even if by some magic you could exactly know the real plant behavior at some point, aging and environmental changes will change the behavior -- and if you're making a controller design that needs to work on multiple units, manufacturing variation adds even more variability.


  Do we know both () and ()? In the book both were provided. If we know both, then what is the point of idealization and/or modeling errors?


Nope.  See "best guess", above.


  How does it work in practice? Do implementations by researchers and engineers target () or ()?


You build \$G_n(s)\$ with some notion of how much it's going to miss the mark.  Then you do a robust design.

If I were writing the book, I think I'd carefully avoid the term "actual plant model" or "plant model" in favor of the term "actual plant dynamics" -- because we know the model, but we can never know the actual plant dynamics.