Many practical systems are characterized by non-linear dynamics. Linearization allows us to determine a model for non-linear dynamics that is effective at predicting and controlling system dynamics near its equilibrium points. This form of linearization, which applies in the close neighborhood of equilibrium points, is called small-signal linearization.
Given a non-linear state-space form model for a system’s dynamics, we must first solve for the systems equilibrium points. That is, those combinations of input and state for which the state-trajectory is constant. Lacking other constraints, we may have to arbitrarily choose values for state or input in order to solve for the equilibrium points.
$$ \dot{\bold x}=\bold f(\bold x,u) $$
$$ \dot{\bold x}=0=\bold f(\bold x_0,u_0)\Rarr \bold x_0,u_0 $$
Next, we expand our nonlinear model in terms of perturbation equations. These express the input and state in terms of their variation from an equilibrium point.
$$ \bold x=\bold x_0+\delta\bold x,~u=u_0+\delta u $$
This is, in principle, expanding our nonlinear model to the first two terms of its Taylor series. This is composed of the model evaluated at the equilibrium point, plus two terms to account for the variation of our model’s output with respect to state and input. Given an analytical form of our model, these are its partial derivatives with respect to state and input respectively.
$$ \dot{\bold x}_0+\delta\dot{\bold x}\approx\bold f(\bold x_0,u_0)+\bold F\delta\bold x+\bold G\delta u $$
$$ \dot{\bold x}0+\delta\dot{\bold x}\approx\bold f(\bold x_0,u_0)+\frac{\partial\bold f}{\partial\bold x}|{(\bold x_0,u_0)}\delta\bold x+\frac{\partial\bold f}{\partial u}|_{(\bold x_0,u_0)}\delta u $$
Once a linearized expression has been obtained, we cancel the terms representing the dynamics at the equilibrium point, these going to zero. Then, we may express a linear state-space form model, one in which the state is the perturbation from equilibrium.
$$ \delta\dot{\bold x}\approx\frac{\partial\bold f}{\partial\bold x}|{(\bold x_0,u_0)}\delta\bold x+\frac{\partial\bold f}{\partial u}|{(\bold x_0,u_0)}\delta u $$