From 7cb1f4efa4a144cdd5d39fe55d5a94367c5e3330 Mon Sep 17 00:00:00 2001 From: Andrew Date: Fri, 8 Aug 2025 17:23:58 -0400 Subject: [PATCH 1/2] update --- README.md | 12 +- class01/background_materials/math_basics.jl | 1 + .../optimization_basics.jl | 6 + .../optimization_motivation.jl | 7 +- class01/class01_intro.jl | 150 +++++++++++++++--- class01/indecision_tree.png | Bin 0 -> 219279 bytes class01/nlp_robot_arm.png | Bin 0 -> 2083875 bytes class04/class04.md | 3 +- 8 files changed, 150 insertions(+), 29 deletions(-) create mode 100644 class01/indecision_tree.png create mode 100644 class01/nlp_robot_arm.png diff --git a/README.md b/README.md index bacb317..c7fc4f8 100644 --- a/README.md +++ b/README.md @@ -30,16 +30,16 @@ This student-led course explores modern techniques for controlling — and learn | 1 | 08/22/2025 | Lecture — Andrew Rosemberg | Course map; why PDE-constrained **optimization**; tooling overview; stability & state-space dynamics; Lyapunov; discretization issues | | | 2 | 08/29/2025 | Lecture | Numerical **optimization** for control (grad/SQP/QP); ALM vs. interior-point vs. penalty methods | | | 3 | 09/05/2025 | Lecture | Pontryagin’s Maximum Principle; shooting & multiple shooting; LQR, Riccati, QP viewpoint (finite / infinite horizon) | | -| 4 | 09/12/2025 | Lecture | Dynamic Programming & Model-Predictive Control | | +| 4 | 09/12/2025 | **External seminar 1** - Joaquim Dias Garcia| Dynamic Programming & Model-Predictive Control | | | 5 | 09/19/2025 | Lecture | **Nonlinear** trajectory **optimization**; collocation; implicit integration | | -| 6 | 09/26/2025 | **External seminar 1** - Henrique Ferrolho | Trajectory **optimization** on robots in Julia Robotics | | +| 6 | 09/26/2025 | **External seminar 2** - Henrique Ferrolho | Trajectory **optimization** on robots in Julia Robotics | | | 7 | 10/03/2025 | Lecture | Essentials of PDEs for control engineers; weak forms; FEM/FDM review | | -| 8 | 10/10/2025 | **External seminar 2** TBD (speaker to be confirmed) | Topology **optimization** | | -| 9 | 10/17/2025 | **External seminar 3 — François Pacaud** | GPU-accelerated optimal control | | +| 8 | 10/10/2025 | **External seminar 3** TBD (speaker to be confirmed) | Topology **optimization** | | +| 9 | 10/17/2025 | **External seminar 4** — François Pacaud | GPU-accelerated optimal control | | |10 | 10/24/2025 | Lecture - Michael Klamkin | Physics-Informed Neural Networks (PINNs): formulation & pitfalls | | -|11 | 10/31/2025 | **External seminar 4** - Chris Rackauckas | Neural Differential Equations: PINNs + classical solvers | | +|11 | 10/31/2025 | **External seminar 5** - Chris Rackauckas | Neural Differential Equations: PINNs + classical solvers | | |12 | 11/07/2025 | Lecture - Pedro Paulo | Neural operators (FNO, Galerkin Transformer); large-scale surrogates | | -|13 | 11/14/2025 | **External seminar 5** - Charlelie Laurent | Scalable PINNs / neural operators; CFD & weather applications | | +|13 | 11/14/2025 | **External seminar 6** - Charlelie Laurent | Scalable PINNs / neural operators; CFD & weather applications | | |14 | 11/21/2025 | Lecture | Robust control & min-max DDP (incl. PDE cases); chance constraints; Data-driven control & RL-in-the-loop | | ## Reference Material diff --git a/class01/background_materials/math_basics.jl b/class01/background_materials/math_basics.jl index 7f4d100..36bf58f 100644 --- a/class01/background_materials/math_basics.jl +++ b/class01/background_materials/math_basics.jl @@ -37,6 +37,7 @@ md" | | | | |-----------:|:--|:------------------| | Lecturer | : | Rosemberg, Andrew | +| Date | : | 28 of July, 2025 | # Background Math (_Welcome to Pluto!_) diff --git a/class01/background_materials/optimization_basics.jl b/class01/background_materials/optimization_basics.jl index 6e6589e..9757985 100644 --- a/class01/background_materials/optimization_basics.jl +++ b/class01/background_materials/optimization_basics.jl @@ -34,6 +34,12 @@ end # ╔═╡ 0df8b65a-0527-4545-bf11-00e9912bced0 md""" + +| | | | +|-----------:|:--|:------------------| +| Lecturer | : | Rosemberg, Andrew | +| Date | : | 28 of July, 2025 | + # Background – Modeling Optimization Problems in JuMP 🏗️ This short Pluto notebook walks you through three small optimisation models of increasing diff --git a/class01/background_materials/optimization_motivation.jl b/class01/background_materials/optimization_motivation.jl index 2bda5ec..cb1b0d5 100644 --- a/class01/background_materials/optimization_motivation.jl +++ b/class01/background_materials/optimization_motivation.jl @@ -251,9 +251,10 @@ using Unitful # ╔═╡ 533cbe59-a206-418e-a397-b21c68e314da md" -Author: Andrew Rosemberg - -Date: 28 of July, 2025 +| | | | +|-----------:|:--|:------------------| +| Lecturer | : | Rosemberg, Andrew | +| Date | : | 28 of July, 2025 | " # ╔═╡ 01c44cc2-68d5-11f0-2860-05c9ffbde13a diff --git a/class01/class01_intro.jl b/class01/class01_intro.jl index f63d4da..4054304 100644 --- a/class01/class01_intro.jl +++ b/class01/class01_intro.jl @@ -18,6 +18,7 @@ end # ╔═╡ 13b12c00-6d6e-11f0-3780-a16e73360478 begin + class_dir = @__DIR__ import Pkg Pkg.activate(".") Pkg.instantiate() @@ -28,6 +29,7 @@ begin using HypertextLiteral using PlutoTeachingTools using ShortCodes, MarkdownLiteral + import Images: load end # ╔═╡ b6ba1231-2942-4f06-8252-22f02553bb57 @@ -39,6 +41,14 @@ using ForwardDiff # ╔═╡ ec473e69-d5ec-4d6a-b868-b89dadb85705 ChooseDisplayMode() +# ╔═╡ 8d7a34ef-5a2d-41a8-ac55-39ab00d7e432 +md" +| | | | +|-----------:|:--|:------------------| +| Lecturer | : | Rosemberg, Andrew | +| Date | : | 28 of July, 2025 | +" + # ╔═╡ 1f774f46-d57d-4668-8204-dc83d50d8c94 md"# Intro - Optimal Control and Learning @@ -46,35 +56,35 @@ In this course, we are interested in problems with the following structure: ```math \begin{equation} -\!\!\!\!\!\!\!\!\min_{\substack{(\mathbf y_1,\mathbf x_1)\\\mathrm{s.t.}}} +\!\!\!\!\!\!\!\!\min_{\substack{(\mathbf u_1,\mathbf x_1)\\\mathrm{s.t.}}} \!\underset{% - \phantom{\substack{(\mathbf y_1,\mathbf x_1)\\\mathrm{s.t.}}}% - \!\!\!\!\!\!\!\!\!\!(\mathbf y_1,\mathbf x_1)\in\mathcal X_1(\mathbf x_0)% + \phantom{\substack{(\mathbf u_1,\mathbf x_1)\\\mathrm{s.t.}}}% + \!\!\!\!\!\!\!\!\!\!(\mathbf u_1,\mathbf x_1)\in\mathcal X_1(\mathbf x_0)% }{% - \!\!\!\!f(\mathbf x_1,\mathbf y_1)% + \!\!\!\!c(\mathbf x_1,\mathbf y_1)% } +\mathbb{E}_1\Bigl[ \quad \cdots \;+\;\mathbb{E}_t\Bigl[ - \min_{\substack{(\mathbf y_t,\mathbf x_t)\\\mathrm{s.t.}}} + \min_{\substack{(\mathbf u_t,\mathbf x_t)\\\mathrm{s.t.}}} \!\underset{% - \phantom{\substack{(\mathbf y_t,\mathbf x_t)\\\mathrm{s.t.}}}% - \!\!\!\!(\mathbf y_t,\mathbf x_t)\in\mathcal X_t(\mathbf x_{t-1},w_t)% + \phantom{\substack{(\mathbf u_t,\mathbf x_t)\\\mathrm{s.t.}}}% + \!\!\!\!(\mathbf u_t,\mathbf x_t)\in\mathcal X_t(\mathbf x_{t-1},w_t)% }{% - \!\!\!\!\!\!\!\!\!\!f(\mathbf x_t,\mathbf y_t)% + \!\!\!\!\!\!\!\!\!\!c(\mathbf x_t,\mathbf u_t)% } +\mathbb{E}_{t+1}[\cdots] \Bigr]. \end{equation} ``` -which minimizes a first stage cost function $f(\mathbf{x}_1, -\mathbf{y}_1)$ and the expected value of future costs over possible +which minimizes a first stage cost function $c(\mathbf{x}_1, +\mathbf{u}_1)$ and the expected value of future costs over possible values of the exogenous stochastic variable $\{w_{t}\}_{t=2}^{T} \in \Omega$. Here, $\mathbf{x}_0$ is the initial system state and the -control decisions $\mathbf{y}_t$ are obtained at every period $t$ +control decisions $\mathbf{u}_t$ are obtained at every period $t$ under a feasible region defined by the incoming state $\mathbf{x}_{t-1}$ and the realized uncertainty $w_t$. $\mathbf{E}_t$ represents the expected value over future uncertainties $\{w_{\tau}\}_{\tau=t}^{T}$. This optimization program assumes that the system is entirely defined by @@ -87,7 +97,7 @@ constraints can be generally posed as: \begin{align} &\mathcal{X}_t(\mathbf{x}_{t-1}, w_t)= \begin{cases} - \mathcal{T}(\mathbf{x}_{t-1}, w_t, \mathbf{y}_t) = \mathbf{x}_t \\ + f(\mathbf{x}_{t-1}, w_t, \mathbf{u}_t) = \mathbf{x}_t \\ h(\mathbf{x}_t, \mathbf{y}_t) \geq 0 \end{cases} \end{align} @@ -98,23 +108,115 @@ constraints can be generally posed as: md""" where the outgoing state of the system $\mathbf{x}_t$ is a transformation based on the incoming state, the realized uncertainty, -and the control variables. $h(\mathbf{x}_t, \mathbf{y}_t) \geq 0$ -captures the state constraints. Markov Decision Process (MDPs) refer -to $\mathcal{T}$ as the "transition kernel" of the system. State and +and the control variables. In the Markov Decision Process (MDP) framework, we refer to $f$ as the "transition kernel" of the system. State and control variables are restricted further by additional constraints captured by $h(\mathbf{x}_t, \mathbf{y}_t) \geq 0$. We -consider policies that map the past information into decisions. In +consider policies that map the past information into decisions: $\pi_t : (\mathbf{x}_{t-1}, w_t) \rightarrow \mathbf{x}_t$. In period $t$, an optimal policy is given by the solution of the dynamic equations: ```math \begin{align} - V_{t}(\mathbf{x}_{t-1}, w_t) = &\min_{\mathbf{x}_t, \mathbf{y}_t} \quad \! \! f(\mathbf{x}_t, \mathbf{y}_t) + \mathbf{E}[V_{t+1}(\mathbf{x}_t, w_{t+1})] \\ - & \text{ s.t. } \quad\mathbf{x}_t = \mathcal{T}(\mathbf{x}_{t-1}, w_t, \mathbf{y}_t) \nonumber \\ - & \quad \quad \quad \! \! h(\mathbf{x}_t, \mathbf{y}_t) \geq 0. \nonumber + V_{t}(\mathbf{x}_{t-1}, w_t) = &\min_{\mathbf{x}_t, \mathbf{u}_t} \quad \! \! c(\mathbf{x}_t, \mathbf{u}_t) + \mathbf{E}_{t+1}[V_{t+1}(\mathbf{x}_t, w_{t+1})] \\ + & \text{ s.t. } \quad\mathbf{x}_t = f(\mathbf{x}_{t-1}, w_t, \mathbf{u}_t) \nonumber \\ + & \quad \quad \quad \! \! h(\mathbf{x}_t, \mathbf{u}_t) \geq 0. \nonumber \end{align} ``` +```math +\implies \pi_t^{*}(\mathbf{x}_{t-1}, w_t) \in \{\mathbf{x}_t \;|\; \exists u_t \;:\; c(\mathbf{x}_t, \mathbf{u}_t) + \mathbf{E}_{t+1}[V_{t+1}(\mathbf{x}_t, w_{t+1})] = V_{t}(\mathbf{x}_{t-1}, w_t) \} +``` + +""" + +# ╔═╡ 1d7092cd-0044-4d38-962a-ce3214c48c24 +md""" +Function $V_{t}(\mathbf{x}_{t-1}, w_t)$ is refered to as the value function. To find the optimal policy for the $1^{\text{st}}$ stage, we need to find the optimal policy for the entire horizon $\{t=2,\cdots,T\}$ or at least estimate the "optimal" value function. +""" + +# ╔═╡ 60ba261a-f2eb-4b45-ad6d-b6042926ccab +load(joinpath(class_dir, "indecision_tree.png")) + +# ╔═╡ 15709f7b-943e-4190-8f40-0cfdb8772183 +md""" +Notice that the number of "nodes" to be evaluated (either decisions or their cost) grows exponetially with the number of stages. This the the *Curse of dimensionality* +in stochastic programming. + +""" + +# ╔═╡ 5d7a4408-21ff-41ec-b004-4b0a9f04bb4f +question_box(md"Can you name a few ways to try and/or solve this problem?") + +# ╔═╡ c08f511e-b91d-4d17-a286-96469c31568a +md"## Example: Robotic Arm Manipulation" + +# ╔═╡ b3129bcb-c24a-4faa-a5cf-f69ce518ea87 +load(joinpath(class_dir, "nlp_robot_arm.png")) + +# ╔═╡ c1f43c8d-0616-4572-bb48-dbb71e40adda +md""" +The tip of the second link is computed using the direct geometric model: + +```math +p(\theta_{1},\theta_{2}) \;=\; +\begin{cases} +x = L_{1}\,\sin\theta_{1} \;+\; L_{2}\,\sin\!\bigl(\theta_{1}+\theta_{2}\bigr),\\[6pt] +y = L_{1}\,\cos\theta_{1} \;+\; L_{2}\,\cos\!\bigl(\theta_{1}+\theta_{2}\bigr). +\end{cases} +\tag{1} +``` +""" + +# ╔═╡ 57d896ca-221a-4cfc-b37a-be9898fac923 +begin +md""" +**State** +```math + \mathbf{x}_t=\begin{bmatrix}\theta_{1,t}&\theta_{2,t}&\dot\theta_{1,t}&\dot\theta_{2,t}\end{bmatrix}^{\!\top} +``` + +**Control** +```math + \mathbf{u}_t=\boldsymbol\tau_t=\begin{bmatrix}\tau_{1,t}&\tau_{2,t}\end{bmatrix}^{\!\top} +``` + +**Dynamics** (Euler sample time Δt) +```math + \mathbf{x}_{t+1}=f_d(\mathbf{x}_t,\mathbf{u}_t) + \;\;\equiv\; + \begin{bmatrix} + \boldsymbol\theta_t+\Delta t\,\dot{\boldsymbol\theta}_t\\[2pt] + \dot{\boldsymbol\theta}_t+\Delta t\,\mathcal{M}^{-1}(\boldsymbol\theta_t)(B(\boldsymbol\theta_t)\boldsymbol\tau_t + F(w_t) - C(\boldsymbol\theta_t,\boldsymbol{\dot\theta})\bigr) + \end{bmatrix} +``` + +**Stage cost** + +```math +c(\mathbf{x}_t,\mathbf{u}_t)= +\underbrace{\|p(\boldsymbol\theta_t)-p_{\text{target}}\|_2^{2}}_{\text{tracking}} ++\;\lambda_\tau\|\boldsymbol\tau_t\|_2^{2}\;, +\qquad \lambda_\tau>0 . +``` + +Terminal cost +$V_T(\mathbf{x}_T)=\|p(\boldsymbol\theta_T)-p_{\text{target}}\|_2^{2}$. + +**Constraints** + +```math +h(\mathbf{x}_t,\mathbf{u}_t)\ge 0\;:\; +\begin{cases} +\theta_{\min}\le\boldsymbol\theta_t\le\theta_{\max} &\text{(joint limits)}\\ +|\dot{\boldsymbol\theta}_t|\le\dot\theta_{\max} &\text{(velocity limits)}\\ +|\boldsymbol\tau_t|\le\tau_{\max} &\text{(actuator limits)} +\end{cases} +``` + """ +end + +# ╔═╡ e2d3d160-d3b6-41f2-a8bc-2878ba71e78c + # ╔═╡ 52005382-177b-4a11-a914-49a5ffc412a3 section_outline(md"A Crash Course:",md" (Continuous-Time) Dynamics @@ -1029,8 +1131,18 @@ end # ╔═╡ Cell order: # ╟─13b12c00-6d6e-11f0-3780-a16e73360478 # ╟─ec473e69-d5ec-4d6a-b868-b89dadb85705 +# ╟─8d7a34ef-5a2d-41a8-ac55-39ab00d7e432 # ╟─1f774f46-d57d-4668-8204-dc83d50d8c94 # ╟─a0f71960-c97c-40d1-8f78-4b1860d2e0a2 +# ╟─1d7092cd-0044-4d38-962a-ce3214c48c24 +# ╟─60ba261a-f2eb-4b45-ad6d-b6042926ccab +# ╟─15709f7b-943e-4190-8f40-0cfdb8772183 +# ╟─5d7a4408-21ff-41ec-b004-4b0a9f04bb4f +# ╟─c08f511e-b91d-4d17-a286-96469c31568a +# ╟─b3129bcb-c24a-4faa-a5cf-f69ce518ea87 +# ╟─c1f43c8d-0616-4572-bb48-dbb71e40adda +# ╟─57d896ca-221a-4cfc-b37a-be9898fac923 +# ╠═e2d3d160-d3b6-41f2-a8bc-2878ba71e78c # ╟─52005382-177b-4a11-a914-49a5ffc412a3 # 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b/class01/class01_intro.jl index 4054304..7e3f76f 100644 --- a/class01/class01_intro.jl +++ b/class01/class01_intro.jl @@ -463,16 +463,41 @@ Foldable(md"All mechanical systems can be written this way. Why?", md""" Manipulator Dynamics Equations are a way of rewriting the Euler--Lagrange equations. -> In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. +#### 🚀 Detour: The Principle of Least Action 🚀 +In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. + +> The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange. + +In classical mechanics: + ```math L = \underbrace{\frac{1}{2} v^{\top}M(q)v}_{\text{Kinematic Energy}} - \underbrace{U(q)}_{\text{Potential Energy}} ``` -What can you say about $M(q)$? When do we have a problem inverting it? +A curve ($q^\star(t)$) is physically realised iff it is a stationary +point of ($\mathcal{S}$) : + +```math +\delta\mathcal{S}=0 +\;\;\Longrightarrow\;\; +\frac{d}{dt}\!\bigl(\tfrac{\partial L}{\partial\dot q}\bigr) +- \frac{\partial L}{\partial q}=0 +\quad\Longrightarrow\quad +M(q)\,\ddot q + C(q,\dot q)\,\dot q + \nabla U(q)=0 . +``` """) +# ╔═╡ f3d155c6-5384-481a-8373-582e753ea8d6 +question_box(md"What can you say about $M(q)$? When do we have a problem inverting it?") + +# ╔═╡ b9aeab8a-f8ea-4310-8568-5d6bda0bb4d3 +question_box(md"Can you derive the stationary condition?") + +# ╔═╡ e1dc6ecf-4e62-415a-a620-0731953c5ab4 + + # ╔═╡ 5f35a169-887f-477f-b010-167627f7ce4c md"## Linear Systems @@ -1160,6 +1185,9 @@ end # ╟─f10927fe-d392-4374-bad1-ab5ac85b8116 # ╟─b8b206ef-cdc5-4cc9-9b55-70d711ba2a9e # ╟─a09de9e4-7ecc-4d23-9135-384077f0c03f +# ╟─f3d155c6-5384-481a-8373-582e753ea8d6 +# ╟─b9aeab8a-f8ea-4310-8568-5d6bda0bb4d3 +# ╠═e1dc6ecf-4e62-415a-a620-0731953c5ab4 # ╟─5f35a169-887f-477f-b010-167627f7ce4c # ╟─e860d92b-cc8f-479b-a0fc-e5f7a11ae1fd # ╟─bb4bfa72-bf69-41f5-b017-7cbf31653bae