Difference between Linearization around an equilibrium point and Linearization around an operating point

Question

In nonlinear conrtrol theory, When making linearization for nonlinear system I have seen two cases: the first is to make linearization around an equilibrium point and the second is around any operating point. I need to understand the difference between these two cases.

Answer 1

There are two cases to consider:

• If you linearize around an equilibrium point, you can draw conclusions about the local stability of the system. I.e, if the linearized system is stable at that equilibrium, then there exists a neighborhood around that point where the nonlinear system is also stable.
• If you linearize around a non-equilibrium point, you can't draw such a conclusion generally.

Linearizing around an operating point is not different than any of those two cases. Usually (if possible), the operating point is chosen such that it is also an equilibrium, because of the mentioned advantage.

So the important distinction is equilibrium or non-equilibrium, not equilibrium and operating point: either the operating point is an equilibrium, then you linearize around an equilibrium anyway. Or the operating point is a non-equilibrium, in which case the linear system usually doesn't tell you if the nonlinear system is locally stable.