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?