Linearization of non linear systems

[theBEAST]

Homework Statement

In my dynamics modelling class, the professor went over an example where we linearize non linear state equations to approximate the behavior. In this case, we are not given an operating point. However, the professor said you can solve for the operating point by setting x' = 0. See the picture below for the notes.

I don't really understand why we can do this, if we set x' = 0, doesn't this just solve for the max and min values of x. For example if x = displacement, then we get the max and min displacements at a certain point. I tried to do some research but I am still confused.

Homework Equations

x' = f(x,u)

The Attempt at a Solution

 

[ehild]

A dynamic system can have equilibrium points, where the velocities ($\dot x_i$-s ) are zero, and investigate the motion of the system around these points: If moving out the system slightly from equilibrium, it returns back or goes away, and there are some other possibilities.

ehild