I'm having a course on control theory and some problems require linearization. It's not that we focus much on it. It's usually just the first question of a bigger problem so I'm just looking for a simple way to do it. Let's take this for example: $$\ddot x +\dot x^2+e^x=u$$
I want to linearize around $\ddot x_0=\dot x_0=x_0=0\\u_0=1$.
From what I found online( which seemed a bit complicated for me) I think you can linearize by using taylor expansion and ignoring second order and above terms.
Let the left part of the equation be $F(\ddot x,\dot x,x)$ and the right part $G(u)$ Then,it goes as follows : $$F(\ddot x_0,\dot x_0,x_0)+\frac{\partial F}{\partial\ddot x}|_{\ddot x_0,\dot x_0,x_0}δ\ddot x+\frac{\partial F}{\partial\dot x}|_{\ddot x_0,\dot x_0,x_0}δ\dot x+\frac{\partial F}{\partial x}|_{\ddot x_0,\dot x_0,x_0}δx=G(u_0)+\frac{\partial G}{\partial u}|_{u_0}\\1+δ\ddot x+2\dot x0+δx=1+δu \\δ\ddot x+δx=δu$$ Is this correct?