In control theory, why do we linearize around the equilibrium for a nonlinear system? For example, in these notes:

In the first example with the pendulum, they define the equilibrium as where the pendulum is at the vertical position (x=0), with a angular velocity of 0 (x'=0) and the input torque is 0 (u=0). Why do we want to study the behaviour of a system at rest? Why not linearize somewhere else. Couldn't linearization be performed at a state where the system is not at rest in case you wanted to calculate for information in such a configuration? 

I believe the answer to your question is simply Hartman-Grobman Theorem. Wikipedia states the theorem wonderfully so I just quote:

Basically the theorem states that the behaviour of a dynamical system near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point provided that no eigenvalue of the linearization has its real part equal to 0. Therefore when dealing with such fixed points one can use the simpler linearization of the system to analyze its behaviour.

We don't have any clue if linearization around some random point will provide insights about the local behavior of the system.
