Class 01 notebook: u_eq and u_e

[ivanightingale]

In this section

LearningToControlClass/class01/class01_intro.jl

Lines 834 to 875 in 507e0b2

	**Non--Linear Systems are often approximated by Linear Systems (locally).**
	"
	# ╔═╡ e860d92b-cc8f-479b-a0fc-e5f7a11ae1fd
	Foldable(md" $\dot{x} = f(x,u) \; \implies \; A=? \; B=?$", md"""
	Suppose now that we apply our dynamics equation to an input:
	```math
	u(t) = u_{eq} + \delta u(t), \quad t \ge 0
	```
	where $u_{eq}$ is an fixed input and $\delta u(t)$ is a perturbation function such that the input is close
	but not equal to $u_{eq}$ and similarly we perturb the initial condition:
	```math
	x(0) = x_e + \delta x(0)
	```
	We will define the deviation from the reference state as:
	```math
	\delta x(t) = x(t) - x_e, \quad t \ge 0
	```
	To determine the evolution of $\delta x(t)$, we can expand the dynamics around the reference point using a Taylor expansion:
	```math
	\dot{\delta x}(t) = f(x_e + \delta x(t), u_{eq} + \delta u(t))
	```
	```math
	=\frac{\partial f}{\partial x}\bigg|_{(x_e, u_{eq})} \delta x(t) + \frac{\partial f}{\partial u}\bigg|_{(x_e, u_{eq})} \delta u(t) + \mathcal{O}(\|\delta x\|^2) + \mathcal{O}(\|\delta u\|^2)
	```
	Considering just the first-order terms we obtain:
	```math
	A= \frac{\partial f}{\partial x}|_{(x_e,u_e)}
	, \quad B= \frac{\partial f}{\partial u}|_{(x_e,u_e)}
	```
	**Attention!** The linearization describes perturbations around the reference $(x_e,u_e)$; it is valid only while $\|\delta x\|$ and $\|\delta u\|$ remain small.
	""")

There appears to be a notation $u_{eq}$ while in the final formulae it seems to be come $u_e$. $u_{eq}$ could be renamed as $u_e$ given that there is also an $x_e$, or they could be replaced with $u_{eq}$ and $x_{eq}$.