In practice, what is the difference between the “nominal plant model” and the “plant model”?

Question

I am reading a book on nonlinear control and disturbance rejection, and in many examples, they distinguish between the "plant model" $G_p(s)$ and the "nominal plant model" $G_n(s)$. They also add that the discrepancy between these two models adds uncertainty that is lumped with disturbance. They specifically called this "unmodeled dynamics", or "modeling errors".

Nominal model was defined as the system's dynamics in the absence of modeling errors. On another hand, it was described as "the ideal model".

In another example, they provided expressions for $G_p(s)$ and $G_n(s)$.

I am very confused:

1. Is the nominal model the desired model? (i.e. do we want to reach it?)
2. Is the nominal model an imprecise idealization of the plant to simplify the mathematics?
3. Do we know both $G_p(s)$ and $G_n(s)$? In the book both were provided. If we know both, then what is the point of idealization and/or modeling errors?
4. How does it work in practice? Do implementations by researchers and engineers target $G_p(s)$ or $G_n(s)$?

I am really curious about both the theoretical and implementation sides. Any insight is immensely appreciated.

Answer 1

You design your control for the nominal model, i.e., for the model where you assume that you know all the parameters, or where you use some averages of what the parameters should be.

In practice, you use only this model, and no other model is available. However, the real plant (for sure) differs from your model: it can have different values of the parameters (parametric uncertainties) or some unmodelled dynamics, e.g., the senor's inertia. These uncertainties define how and why the real plant is not the same as the nominal one that you have used for control design, and why your real signals are not as good as the simulations results.

At this moment, you ask: well, can I check in advance how does my controller deal with all these uncertainties? Yes, to do so you need a model of the plant that is different from the nominal one, e.g., it considers possible saturations. Then you test you nominal controller (the one that you have designed for the nominal plant) for various plant models with different parameters and so on. It gives you some ideas about the sensitivity of your controller with respect to uncertainties. And finally, you apply your controller to the real plant and hope it will work.

To summarize, you use the nominal model for control design, you use plant models to test your design against uncertainties, and you apply your controller to a real plant.