From 40619f717fa8d8548c8996a11fc699c52f170956 Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Thu, 25 Sep 2025 01:21:30 +0000 Subject: [PATCH 1/3] Initial plan From c84457914088b89939e3673215425b280a6b87c5 Mon Sep 17 00:00:00 2001 From: "copilot-swe-agent[bot]" <198982749+Copilot@users.noreply.github.com> Date: Thu, 25 Sep 2025 01:30:56 +0000 Subject: [PATCH 2/3] Fix notation inconsistency: standardize u_eq to u_e in Taylor expansion section Co-authored-by: andrewrosemberg <24196241+andrewrosemberg@users.noreply.github.com> --- class01/class01_intro.jl | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/class01/class01_intro.jl b/class01/class01_intro.jl index 2e7e538..2580bb7 100644 --- a/class01/class01_intro.jl +++ b/class01/class01_intro.jl @@ -840,10 +840,10 @@ 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 +u(t) = u_e + \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: +where $u_e$ is an fixed input and $\delta u(t)$ is a perturbation function such that the input is close +but not equal to $u_e$ and similarly we perturb the initial condition: ```math x(0) = x_e + \delta x(0) @@ -857,10 +857,10 @@ We will define the deviation from the reference state as: 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)) +\dot{\delta x}(t) = f(x_e + \delta x(t), u_e + \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) +=\frac{\partial f}{\partial x}\bigg|_{(x_e, u_e)} \delta x(t) + \frac{\partial f}{\partial u}\bigg|_{(x_e, u_e)} \delta u(t) + \mathcal{O}(\|\delta x\|^2) + \mathcal{O}(\|\delta u\|^2) ``` Considering just the first-order terms we obtain: From 645b8628ee38a6076faaeb3ab508cf2a51ea481a Mon Sep 17 00:00:00 2001 From: Andrew Date: Thu, 25 Sep 2025 13:10:01 -0400 Subject: [PATCH 3/3] update docs --- class01/class01_intro.html | 4 +- class01/class01_intro.jl | 305 ++++++++++++++++++++++++++++++------- 2 files changed, 250 insertions(+), 59 deletions(-) diff --git a/class01/class01_intro.html b/class01/class01_intro.html index a16db25..00d0f7a 100644 --- a/class01/class01_intro.html +++ b/class01/class01_intro.html @@ -3,11 +3,11 @@