diff --git a/README.md b/README.md
index dbf2dda22f0ff2d59063ae1867ef1c6be1be3160..363606b1267826970c97cf41010c0f7397525c31 100644
--- a/README.md
+++ b/README.md
@@ -18,10 +18,12 @@ Méthodes indirectes et directes en contrôle optimal
* [Sujet 2 : tp/simple-shooting.ipynb](tp/simple-shooting.ipynb) - Tir simple indirect
* [Sujet 3 : tp/direct-indirect.ipynb](tp/direct-indirect.ipynb) - Méthode directe et tir avec structure sur le contrôle
+* [Sujet 4 : tp/mpc-navigation.ipynb](tp/mpc-navigation.ipynb) - Méthode MPC pour la navigation marine
-Equation différentielle ordinaire
+Calcul différentiel et équation différentielle ordinaire
-* [Sujet 4 : tp/runge-kutta.ipynb](tp/runge-kutta.ipynb) - Méthodes de Runge-Kutta
+* [Sujet 5 : tp/derivative.ipynb](tp/derivative.ipynb) - Calcul de dérivées
+* [Sujet 6 : tp/runge-kutta.ipynb](tp/runge-kutta.ipynb) - Méthodes de Runge-Kutta
**Notes de cours supplémentaires - ressources**
diff --git a/tp/derivative.ipynb b/tp/derivative.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..c03c237a6c51ebd38d388e582fb8c73022548d2f
--- /dev/null
+++ b/tp/derivative.ipynb
@@ -0,0 +1,495 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "[<img src=\"https://gitlab.irit.fr/toc/etu-n7/controle-optimal/-/raw/master/ressources/Logo-toulouse-inp-N7.png\" alt=\"N7\" height=\"100\"/>](https://gitlab.irit.fr/toc/etu-n7/controle-optimal)\n",
+ "\n",
+ "# Calcul de dérivées\n",
+ "\n",
+ "- Date : 2025\n",
+ "- Durée approximative : 1h15\n",
+ "\n",
+ "## Introduction\n",
+ "\n",
+ "Il existe plusieurs façon de calculer une dérivée sur un calculateur : \n",
+ "\n",
+ "- par différences finies;\n",
+ "- par différentiation complexe;\n",
+ "- en utilisant la différentiation automatique;\n",
+ "- en utilisant le calcul formel et un générateur de code.\n",
+ "\n",
+ "Nous allons étudier ici quelques cas.\n",
+ "\n",
+ "On notera $\\Vert{\\cdot}\\Vert$ la norme euclidienne usuelle et $\\mathcal{N}(0,1)$ une variable aléatoire Gaussienne centrée réduite."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# load packages\n",
+ "using DualNumbers\n",
+ "using DifferentialEquations\n",
+ "using ForwardDiff\n",
+ "using LinearAlgebra\n",
+ "using Plots\n",
+ "using Plots.Measures\n",
+ "using Printf"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Dérivées par différences finies avant\n",
+ "\n",
+ "Soit $f$ une fonction lisse de $\\mathbb{R}^{n}$ dans $\\mathbb{R}^{m}$, $x$ un point de $\\mathbb{R}^{n}$ et $v$ un vecteur de $\\mathbb{R}^{n}$. Si on note $g \\colon h \\mapsto f(x+hv)$, alors on a d'après la formule de Taylor-Young :\n",
+ "$$\n",
+ " g(h) = \\sum_{i=0}^{n} \\frac{h^i}{i!} g^{(i)}(0) + R_n(h), \\quad R_n(h) = o(h^n),\n",
+ "$$\n",
+ "ou d'après Taylor-Lagrange, \n",
+ "$$\n",
+ " \\| R_n(h) \\| \\leq \\frac{M_n h^{n+1}}{(n+1)!},\n",
+ "$$\n",
+ "de même, \n",
+ "$$\n",
+ " g(-h) = \\sum_{i=0}^{n} \\frac{(-h)^i}{i!} g^{(i)}(0) + R_n(h).\n",
+ "$$\n",
+ "\n",
+ "La méthode des différences finies avants consiste à approcher la différentielle de $f$ en $x$ dans la direction $v$ par la formule suivante : \n",
+ "$$\n",
+ " \\frac{f(x+hv) - f(x)}{h} = \n",
+ " \\frac{g(h)-g(0)}{h} = g'(0) + \\frac{h}{2} g^{2}(0) + \\frac{h^2}{6} g^{(3)}(0) + o(h^2).\n",
+ "$$\n",
+ "L'approximation ainsi obtenue de $ g'(0) = f'(x) \\cdot v \\in \\mathbb{R}^m$ est d'ordre 1 si $g^{(2)}(0) \\neq 0$ ou au moins d'ordre 2 sinon. \n",
+ "\n",
+ "**Remarque.** Sur machine, Les calculs se font en virgule flottante. On note epsilon machine, le plus petit nombre $\\mathrm{eps}_\\mathrm{mach}$ tel que $1+\\mathrm{eps}_\\mathrm{mach}\\ne 1$. Cette quantité dépend des machines et de l'encodage des données. Pour l'optenir en `Julia` il suffit de taper `eps()`. On peut indiquer la précision numérique : `eps(Float64)` ou `eps(Float64(1))` pour les flottants codés sur 64 bits et `eps(Float32(1))` sur 32 bits.\n",
+ "\n",
+ "Notons $\\mathrm{num}(g,\\, h)$ la valeur de $g(h)$ calculée numériquement et supposons que l'on puisse majorer l'erreur relative numérique par : \n",
+ "$$\n",
+ " \\left\\| \\mathrm{num}(g,h) - g(h) \\right\\| := \\| e_h\\| \\leq \\mathrm{eps}_\\mathrm{mach} L_f,\n",
+ "$$\n",
+ "ou $L_f$ est une constante qui dépend de la valeur de $f$ sur le domaine d'intérêt. Ainsi on a : \n",
+ "\\begin{align*}\n",
+ " \\left\\| \\frac{\\mathrm{num}(g,h) - \\mathrm{num}(g,0)}{h} - g'(0) \\right\\|\n",
+ " &= \\left\\| \\frac{g(h) + e_h - g(0) - e_0}{h} - g'(0) \\right\\|, \\\\[1em]\n",
+ " &= \\left\\| \\frac{R_1(h)}{h} + \\frac{e_h - e_0}{h} \\right\\|, \\\\[1em]\n",
+ " &\\leq \\left\\| \\frac{R_1(h)}{ h} \\right\\| + \\left\\| \\frac{e_h - e_0}{h} \\right\\|, \\\\[1em]\n",
+ " & \\leq \n",
+ " \\underbrace{ \\frac{M_1 h}{2}}_{{\\text{Erreur d'approximation}}} + \n",
+ " \\underbrace{2 \\frac{\\mathrm{eps}_\\mathrm{mach}L_f}{h}}_{{\\text{Erreur numérique}}}.\n",
+ "\\end{align*} \n",
+ "Le majorant trouvé atteint son minimum en \n",
+ "$$\n",
+ " h_{*} = 2 \\sqrt{\\frac{\\mathrm{eps}_\\mathrm{mach} L_f }{M_1}}.\n",
+ "$$\n",
+ "\n",
+ "En considérant que $L_f \\simeq M_1$, alors le choix se révélant le plus optimal est \n",
+ "$$\n",
+ " h_{*} \\approx \\sqrt{\\mathrm{eps}_\\mathrm{mach}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Dérivées par différences finies centrées\n",
+ "\n",
+ "On peut utiliser un schéma de différences finies centrée pour calculer la dérviée de $g$. \n",
+ "$$\n",
+ " \\frac{f(x+hv) - f(x-hv)}{2h} = \\frac{g(h) - g(-h)}{2h} = \n",
+ " g'(0) + g^{(3)}(0) \\frac{h^2}{6} + \\mathcal{O}(h^4),\n",
+ "$$\n",
+ "l'approximation ainsi obtenue de $f'(x) \\cdot v \\in \\mathbb{R}^{m}$ est d'ordre 2 si $g^{(3)}(0) \\neq 0$ ou au moins d'ordre 4 sinon. À noter que ce schéma nécessite plus d'évaluations de la fonction $f$. On peut montrer comme précédemment que le meilleur $h$ est de l'ordre \n",
+ "$$\n",
+ " h_* \\approx \\sqrt[3]{\\mathrm{eps}_\\mathrm{mach}}.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Dérivées par différentiation complexe\n",
+ "\n",
+ "Les formules des schémas avant et centrée sont sensibles aux calculs de la différence $\\Delta f = f(x+h) - f(x)$ ou $\\Delta f = f(x+h) - f(x-h)$. Pour remédier à ce problème, les [différences finies à pas complexe](https://dl.acm.org/doi/10.1145/838250.838251) ont été introduites.\n",
+ "Si on suppose que la fonction $g$ est holomorphe, c'est-à-dire dérivable au sens complexe,\n",
+ "on peut considérer un pas complexe $ih$. Un développement limité de $g$ en $0$ s'écrit\n",
+ "\n",
+ "$$\n",
+ " f(x+ih v) = g(ih) = g(0) + ih g'(0) - \\frac{h^2}{2} g^{(2)}(0) - i\\frac{h^3}{6} g^{(3)}(0) + o(h^3),\n",
+ "$$\n",
+ "\n",
+ "On considère alors l'approximation : \n",
+ "\n",
+ "$$\n",
+ " f'(x) \\cdot v = g'(0) \\approx \\frac{\\mathrm{Im}(f(x+ihv))}{h}.\n",
+ "$$\n",
+ "\n",
+ "On peut prouver que l'approximation ci-dessus est au moins d'ordre 2 et aussi démontrer que tout pas inférieur à $h_*$ est optimal, avec \n",
+ "$$\n",
+ " h_{*} \\approx \\sqrt{\\mathrm{eps}_{\\mathrm{mach}}}.\n",
+ "$$\n",
+ "\n",
+ "**Remarque.** Utiliser en `Julia` la commande `imag` pour calculer la partie imaginaire d'un nombre complexe et la variable `im` pour représenter l'unité imaginaire."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Dérivées par différentiation automatique via les nombres duaux\n",
+ "\n",
+ "Les nombres duaux s'écrivent sous la forme $a + b\\, \\varepsilon$ avec $(a,b)\\in \\mathbb{R}^2$ et $\\varepsilon^2 = 0$. Nous allons voir comment nous pouvons les utiliser pour calculer des dérivées.\n",
+ "\n",
+ "Soit deux fonctions $f, g \\colon \\mathbb{R} \\to \\mathbb{R}$ dérivables, de dérivées respectives $f'$ et $g'$. On pose\n",
+ "\n",
+ "$$\n",
+ "f(a + b\\, \\varepsilon) := f(a) + f'(a)\\, b\\, \\varepsilon\n",
+ "$$\n",
+ "\n",
+ "et\n",
+ "\n",
+ "$$\n",
+ "g(a + b\\, \\varepsilon) := g(a) + g'(a)\\, b\\, \\varepsilon.\n",
+ "$$\n",
+ "\n",
+ "On a alors automatiquement les propriétés suivantes. Posons $d = x + \\varepsilon$, alors :\n",
+ "\n",
+ "- $(f + g)(d) = (f+g)(x) + (f+g)'(x) \\, \\varepsilon$\n",
+ "- $(fg)(d) = (fg)(x) + (fg)'(x) \\, \\varepsilon$\n",
+ "- $(g \\circ f)(d) = (g \\circ f)(x) + (g \\circ f)'(x) \\, \\varepsilon$\n",
+ "\n",
+ "Voici comment créer un nombre dual en `Julia` et récupérer les parties réelles et duales (avec ce que j'ai défini ci-dessous) :\n",
+ "\n",
+ "```julia\n",
+ "using DualNumbers\n",
+ "\n",
+ "# scalar case\n",
+ "d = 1 + 2ε # ou 1 + 2 * ε ou 1 + ε * 2\n",
+ "real(d) # 1\n",
+ "dual(d) # 2\n",
+ "\n",
+ "# vector case\n",
+ "d = [1, 3] + [2, 4]ε # ou [1, 3] + [2, 4] * ε ou [1, 3] + ε * [2, 4] ou [1+2ε, 3+4ε]\n",
+ "real(d) # [1, 3]\n",
+ "dual(d) # [2, 4]\n",
+ "```\n",
+ "\n",
+ "**Remarque.** On peut aussi utiliser le package `ForwardDiff` pour calculer des dérivées automatiquement. Il est plus performant que `DualNumbers`."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Fonctions auxiliaires"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# available methods\n",
+ "methods = (:forward, :central, :complex, :dual, :forward_ad)\n",
+ "\n",
+ "# type of x or its coordinates\n",
+ "function mytypeof(x::Union{T, Vector{<:T}}) where T<:AbstractFloat\n",
+ " return T\n",
+ "end\n",
+ "\n",
+ "# default step value\n",
+ "function _step(x, v, method)\n",
+ " T = mytypeof(x)\n",
+ " eps_value = T isa AbstractFloat ? eps(T) : eps(1.)\n",
+ " if method == :forward\n",
+ " step = √(eps_value)\n",
+ " elseif method == :central\n",
+ " step = (eps_value)^(1/3)\n",
+ " elseif method == :complex\n",
+ " step = √(eps_value)\n",
+ " else\n",
+ " step = 0.0\n",
+ " end\n",
+ " step *= √(max(1., norm(x))) / √(max(1.0, norm(v)))\n",
+ " return step\n",
+ "end\n",
+ "\n",
+ "# default method value\n",
+ "function _method()\n",
+ " return :forward \n",
+ "end;\n",
+ "\n",
+ "# creation of dual number ε\n",
+ "import Base.*\n",
+ "*(e::Function, x::Union{Number, Vector{<:Number}}) = e(x)\n",
+ "*(x::Union{Number, Vector{<:Number}}, e::Function) = e(x)\n",
+ "ε(x=1) = begin \n",
+ " if x isa Number\n",
+ " return Dual.(0.0, x)\n",
+ " else\n",
+ " return Dual.(zeros(length(x)), x)\n",
+ " end\n",
+ "end\n",
+ "em = ε\n",
+ "dual(x::Union{Dual, Vector{<:Dual}}) = dualpart.(x)\n",
+ "real(x::Union{Dual, Vector{<:Dual}}) = realpart.(x);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## La méthode principale pour le calcul de dérivées\n",
+ "\n",
+ "La fonction `derivative` ci-dessous calcule la dérivée directionnelle\n",
+ "\n",
+ "$$\n",
+ " f'(x) \\cdot v.\n",
+ "$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "## TO COMPLETE\n",
+ "\n",
+ "# compute directional derivative\n",
+ "function derivative(f, x, v; method=_method(), h=_step(x, v, method))\n",
+ " if method ∉ methods \n",
+ " error(\"Choose a valid method in \", methods)\n",
+ " end\n",
+ " if method == :forward\n",
+ " return f(x) # TO UPDATE\n",
+ " elseif method == :central\n",
+ " return f(x) # TO UPDATE\n",
+ " elseif method == :complex\n",
+ " return f(x) # TO UPDATE\n",
+ " elseif method == :dual \n",
+ " return f(x) # TO UPDATE\n",
+ " elseif method == :forward_ad\n",
+ " if x isa Number\n",
+ " return ForwardDiff.derivative(f, x)*v\n",
+ " else\n",
+ " return ForwardDiff.jacobian(f, x)*v\n",
+ " end\n",
+ " end\n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# function to print derivative values and errors\n",
+ "function print_derivatives(f, x, v, sol)\n",
+ "\n",
+ " println(\"Hand derivative: \", sol, \"\\n\")\n",
+ "\n",
+ " for method ∈ methods\n",
+ " dfv = derivative(f, x, v, method=method)\n",
+ " println(\"Method: \", method)\n",
+ " println(\" derivative: \", dfv)\n",
+ " @printf(\" error: %e\\n\", norm(dfv - sol))\n",
+ " if method ∈ (:forward, :central, :complex)\n",
+ " step = _step(x, v, method)\n",
+ " println(\" step: \", step)\n",
+ " @printf(\" error/step: %e\\n\", norm(dfv - sol) / step)\n",
+ " end\n",
+ " println()\n",
+ " end\n",
+ "\n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Exercice 1\n",
+ "\n",
+ "1. Compléter la fonction `derivative` ci-dessus avec les méthodes de différences finies avant, centrée, par différentiation complexe et par différentiation automatique via les nombres duaux.\n",
+ "2. Exécuter le code ci-dessous et vérifier les résultats obtenus."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Scalar case\n",
+ "\n",
+ "# check if the derivatives are correct\n",
+ "f(x) = cos(x)\n",
+ "x = π/4\n",
+ "v = 1.0\n",
+ "\n",
+ "# solution\n",
+ "sol = -sin(x)*v\n",
+ "\n",
+ "# print derivatives and errors for each method\n",
+ "print_derivatives(f, x, v, sol)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Vectorial case\n",
+ "\n",
+ "# check if the derivatives are correct\n",
+ "f(x) = [0.5*(x[1]^2 + x[2]^2); x[1]*x[2]]\n",
+ "x = [1.0, 2.0]\n",
+ "v = [1.0, -1.0]\n",
+ "\n",
+ "# solution\n",
+ "sol = [x[1]*v[1]+x[2]*v[2], x[1]*v[2]+x[2]*v[1]]\n",
+ "\n",
+ "# print derivatives and errors for each method\n",
+ "print_derivatives(f, x, v, sol)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Pas optimal\n",
+ "\n",
+ "On se propose de tester pour la fonction $\\cos$ aux points $x_0=\\pi/3$, $x_1 = 10^6\\times\\pi/3$ et la fonction $\\cos+10^{-8} \\mathcal{N}(0, \\, 1)$ au point $x_0=\\pi/3$ l'erreur entre les différences finies et la dérivée au point considéré en fonction de $h$. On prendra $h=10^{-i}$ pour $i= \\{1,\\ldots,16\\}$ et on tracera ces erreurs dans une échelle logarithmique (en `Julia`, avec le package `Plots` on utilise l'option `scale=:log10`).\n",
+ "\n",
+ "## Exercice 2\n",
+ "\n",
+ "- Visualiser les différentes erreurs en fonction de $h$ pour les différentes méthodes de calcul de dérivées. Commentaires.\n",
+ "- Modifier la précision de $x_0$ et $x_1$ en `Float32`. Commentaires."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# affichage des erreurs en fonction de h\n",
+ "function plot_errors(steps, errors, h_star, title)\n",
+ "\n",
+ " steps_save = steps\n",
+ " ymax = 10^10\n",
+ "\n",
+ " # supprimer les erreurs nulles\n",
+ " non_nul_element = findall(!iszero, errors) \n",
+ " errors = errors[non_nul_element]\n",
+ " steps = steps[non_nul_element]\n",
+ "\n",
+ " # Courbe des erreurs pour les differents steps en bleu\n",
+ " plt = plot((10.).^(-steps), errors, xscale=:log10, yscale=:log10, linecolor=:blue, lw=2, legend=false)\n",
+ "\n",
+ " # régler xlims pour toujours avoir tous les steps de départ\n",
+ " plot!(plt, xlims=(10^(-maximum(steps_save)), 10^(-minimum(steps_save))))\n",
+ "\n",
+ " # ylims toujours entre 10^-16 et ymax\n",
+ " plot!(plt, ylims=(10^(-16), ymax))\n",
+ "\n",
+ " # Ligne verticale pour situer l'erreur optimale h* en rouge\n",
+ " plot!(plt,[h_star, h_star], [10^(-16), ymax], linecolor=:red, lw=1, linestyle=:dash)\n",
+ "\n",
+ " # titre de la figure et xlabel\n",
+ " plot!(plt, xlabel = \"h\", title = title, legend=false, titlefontsize=10)\n",
+ "\n",
+ " # ajouter des marges en bas de la figure pour mieux voir le xlabel \n",
+ " plot!(plt, bottom_margin = 5mm)\n",
+ "\n",
+ " #\n",
+ " return plt\n",
+ "\n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Les differentes fonctions et la dérivée theorique\n",
+ "fun1(x) = cos(x)\n",
+ "fun2(x) = cos(x) + 1.e-8*randn()\n",
+ "dfun(x) = -sin(x);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# method \n",
+ "method = :forward # TO PLAY WITH\n",
+ "h_star = √(eps(1.0)) # TO UPDATE ACCORDING TO THE METHOD\n",
+ "\n",
+ "# Points pour lesquels on souhaite effectuer les tests\n",
+ "x0 = π/3\n",
+ "x1 = 1.e6*π/3\n",
+ "\n",
+ "# steps pour faire les tests\n",
+ "steps = range(1, 16, 16)\n",
+ "\n",
+ "# Initialisation des vecteurs d'erreur\n",
+ "err_x0 = zeros(length(steps))\n",
+ "err_x0p = zeros(length(steps))\n",
+ "err_x1 = zeros(length(steps))\n",
+ "\n",
+ "# Calcul des erreurs\n",
+ "for i in 1:length(steps)\n",
+ " h = 10^(-steps[i])\n",
+ " err_x0[i] = abs(derivative(fun1, x0, 1.0, h=h, method=method) - (dfun(x0)))\n",
+ " err_x1[i] = abs(derivative(fun1, x1, 1.0, h=h, method=method) - (dfun(x1)))\n",
+ " err_x0p[i] = abs(derivative(fun2, x0, 1.0, h=h, method=method) - (dfun(x0)))\n",
+ "end\n",
+ "\n",
+ "# Affichage des erreurs\n",
+ "p1 = plot_errors(steps, err_x0, h_star, \"cos(x0)\")\n",
+ "p2 = plot_errors(steps, err_x0p, h_star, \"cos(x0) + perturbation\")\n",
+ "p3 = plot_errors(steps, err_x1, h_star, \"cos(x1)\")\n",
+ "\n",
+ "plot(p1, p2, p3, layout=(1,3), size=(850, 350))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.9.0",
+ "language": "julia",
+ "name": "julia-1.9"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.9.0"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/tp/mpc-navigation.ipynb b/tp/mpc-navigation.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..99424387f8524f94a5c183f75ed0cea320fc4f70
--- /dev/null
+++ b/tp/mpc-navigation.ipynb
@@ -0,0 +1,493 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "660275fc",
+ "metadata": {},
+ "source": [
+ "<div style=\"display:flex\"> \n",
+ " <img \n",
+ " src=\"https://gitlab.irit.fr/toc/etu-n7/controle-optimal/-/raw/master/ressources/Logo-toulouse-inp-N7.png\" \n",
+ " height=\"100\"\n",
+ " > \n",
+ " <img \n",
+ " src=\"https://gitlab.irit.fr/toc/etu-n7/controle-optimal/-/raw/master/ressources/ship.jpg\"\n",
+ " style=\"display: block;\n",
+ " margin-left: auto;\n",
+ " margin-right: auto;\n",
+ " width: 50%;\"\n",
+ " >\n",
+ "</div> \n",
+ "\n",
+ "# Problème de navigation, une approche [MPC](https://en.wikipedia.org/wiki/Model_predictive_control)\n",
+ "\n",
+ "- Date : 2025\n",
+ "- Durée approximative : inconnue\n",
+ "\n",
+ "On considère un navire dans un courant constant $w=(w_x,w_y)$, $\\|w\\| \\lt 1$. \n",
+ "[L'angle de cap](https://fr.wikipedia.org/wiki/Cap_(navigation)) est contrôlé, amenant aux équations différentielles suivantes : \n",
+ "\n",
+ "$$ \\begin{array}{rcl}\n",
+ " \\dot{x}(t) &=& w_x+\\cos\\theta(t),\\quad t \\in [0,t_f]\\\\\n",
+ " \\dot{y}(t) &=& w_y+\\sin\\theta(t),\\\\\n",
+ " \\dot{\\theta}(t) &=& u(t). \n",
+ " \\end{array} $$\n",
+ "\n",
+ "La vitesse angulaire est limitée et normalisée : $\\|u(t)\\| \\leq 1$. Il y a des conditions aux limites au temps initial $t = 0$ et au temps final $t = t_f$, sur la position $(x,y)$ et sur l'angle $\\theta$. L'objectif est de minimiser le temps final. Ce sujet est inspiré de ce [TP](https://github.com/pns-mam/commande/tree/main/tp3) dont le problème vient d'une collaboration entre l'Université Côte d'Azur et l'entreprise française [CGG](https://www.cgg.com) qui s'intéresse aux manoeuvres optimales de très gros navires pour la prospection marine.\n",
+ "\n",
+ "✏️ **Exercice 1.** On supposera $p^0 = -1$, on se place dans le cas normal.\n",
+ "\n",
+ "1. Ecrire le problème de contrôle optimal sous la forme de Mayer.\n",
+ "2. Donner le pseudo-hamiltonien $H(q, p, u)$, où $q = (x, y, \\theta)$ et $p = (p_x, p_y, p_\\theta)$.\n",
+ "3. Calculer l'équation adjointe, c'est-à-dire vérifiée par le vecteur adjoint $p$, donnée par le principe du maximum de Pontryagin.\n",
+ "4. Calculer le contrôle maximisant en fonction de $p_\\theta$ (on pourra l'écrire comme une [fonction multivaluée](https://fr.wikipedia.org/wiki/Fonction_multivaluée)).\n",
+ "5. Calculer le contrôle singulier, c'est-à-dire celui permettant de vérifier $p_\\theta(t) = 0$ sur un intervalle de temps non réduit à un singleton."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "a68c170f",
+ "metadata": {},
+ "source": [
+ "## Données du problème"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8dd6f718",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using OptimalControl, NLPModelsIpopt, Plots, OrdinaryDiffEq, LinearAlgebra, Plots.PlotMeasures\n",
+ "\n",
+ "t0 = 0.\n",
+ "x0 = 0. \n",
+ "y0 = 0.\n",
+ "θ0 = π/7\n",
+ "xf = 4.\n",
+ "yf = 7.\n",
+ "θf = -π/2\n",
+ "\n",
+ "function current(x, y) # current as a function of position\n",
+ " ε = 1e-1\n",
+ " w = [ 0.6, 0.4 ]\n",
+ " δw = ε * [ y, -x ] / sqrt(1+x^2+y^2)\n",
+ " w = w + δw\n",
+ " if (w[1]^2 + w[2]^2 >= 1)\n",
+ " error(\"|w| >= 1\")\n",
+ " end\n",
+ " return w\n",
+ "end\n",
+ "\n",
+ "#\n",
+ "function plot_state!(plt, x, y, θ; color=1)\n",
+ " plot!(plt, [x], [y], marker=:circle, legend=false, color=color, markerstrokecolor=color, markersize=5, z_order=:front)\n",
+ " quiver!(plt, [x], [y], quiver=([cos(θ)], [sin(θ)]), color=color, linewidth=2, z_order=:front)\n",
+ " return plt\n",
+ "end\n",
+ "\n",
+ "function plot_current!(plt; current=current, N=10, scaling=1)\n",
+ " for x ∈ range(xlims(plt)..., N)\n",
+ " for y ∈ range(ylims(plt)..., N)\n",
+ " w = scaling*current(x, y)\n",
+ " quiver!(plt, [x], [y], quiver=([w[1]], [w[2]]), color=:black, linewidth=0.5, z_order=:back)\n",
+ " end\n",
+ " end\n",
+ " return plt\n",
+ "end\n",
+ "\n",
+ "# on affiche dans le plan de phase augmenté les conditions aux limites et le courant\n",
+ "plt = plot(\n",
+ " xlims=(-2, 6), \n",
+ " ylims=(-1, 8), \n",
+ " size=(600, 600), \n",
+ " aspect_ratio=1, \n",
+ " xlabel=\"x\", \n",
+ " ylabel=\"y\", \n",
+ " title=\"Conditions aux limites\",\n",
+ " leftmargin=5mm, \n",
+ " bottommargin=5mm,\n",
+ ")\n",
+ "\n",
+ "plot_state!(plt, x0, y0, θ0; color=2)\n",
+ "plot_state!(plt, xf, yf, θf; color=2)\n",
+ "plot_current!(plt)\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "8b4c8930",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function plot_trajectory!(plt, t, x, y, θ; N=5) # N : nombre de points où l'on va afficher θ\n",
+ "\n",
+ " # trajectoire\n",
+ " plot!(plt, x.(t), y.(t), legend=false, color=1, linewidth=2, z_order=:front)\n",
+ "\n",
+ " if N > 0\n",
+ "\n",
+ " # longueur du trajet\n",
+ " s = 0\n",
+ " for i ∈ 2:length(t)\n",
+ " s += norm([x(t[i]), y(t[i])] - [x(t[i-1]), y(t[i-1])])\n",
+ " end\n",
+ "\n",
+ " # intervalle de longueur\n",
+ " Δs = s/(N+1)\n",
+ " tis = []\n",
+ " s = 0\n",
+ " for i ∈ 2:length(t)\n",
+ " s += norm([x(t[i]), y(t[i])] - [x(t[i-1]), y(t[i-1])])\n",
+ " if s > Δs && length(tis) < N\n",
+ " push!(tis, t[i])\n",
+ " s = 0\n",
+ " end\n",
+ " end\n",
+ "\n",
+ " # affichage des points intermédiaires\n",
+ " for ti ∈ tis\n",
+ " plot_state!(plt, x(ti), y(ti), θ(ti); color=1)\n",
+ " end\n",
+ "\n",
+ " end\n",
+ "\n",
+ " return plt\n",
+ " \n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "0403331e",
+ "metadata": {},
+ "source": [
+ "## Solveur (OptimalControl)\n",
+ "\n",
+ "✏️ **Exercice 2.** Coder le problème de contrôle optimal ci-dessous.\n",
+ "\n",
+ "On pourra s'insiprer du problème de la [rame de métro à temps minimal](https://control-toolbox.org/OptimalControl.jl/stable/tutorial-double-integrator-time.html) décrit dans la documentation du package OptimalControl pour définir notre problème de manoeuvre de navire ci-après.\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "a1933e2c",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function solve(t0, x0, y0, θ0, xf, yf, θf, w; \n",
+ " grid_size=300, tol=1e-8, max_iter=500, print_level=4, display=true)\n",
+ "\n",
+ " # ---------------------------------------\n",
+ " # Définition du problème : TO UPDATE\n",
+ " ocp = @def begin\n",
+ "\n",
+ " end\n",
+ " # ---------------------------------------\n",
+ "\n",
+ " # Initialisation\n",
+ " tf_init = 1.5*norm([xf, yf]-[x0, y0])\n",
+ " x_init(t) = [ x0, y0, θ0 ] * (tf_init-t)/(tf_init-t0) + [xf, yf, θf] * (t-t0)/(tf_init-t0)\n",
+ " u_init = (θf - θ0) / (tf_init-t0)\n",
+ " init = (state=x_init, control=u_init, variable=tf_init)\n",
+ "\n",
+ " # Résolution\n",
+ " sol = OptimalControl.solve(ocp; \n",
+ " init=init,\n",
+ " grid_size=grid_size, \n",
+ " tol=tol, \n",
+ " max_iter=max_iter, \n",
+ " print_level=print_level,\n",
+ " display=display,\n",
+ " disc_method=:euler,\n",
+ " )\n",
+ "\n",
+ " # Récupération des données utiles\n",
+ " t = time_grid(sol)\n",
+ " q = state(sol)\n",
+ " x = t -> q(t)[1]\n",
+ " y = t -> q(t)[2]\n",
+ " θ = t -> q(t)[3]\n",
+ " u = control(sol)\n",
+ " tf = variable(sol)\n",
+ " \n",
+ " return t, x, y, θ, u, tf, iterations(sol), sol.solver_infos.constraints_violation\n",
+ " \n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cc8547d2",
+ "metadata": {},
+ "source": [
+ "## Première résolution\n",
+ "\n",
+ "On considère ici un vent constant et on résout une première fois le problème."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "ffea082a",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# -----------------------------------------------------------------------------------------------------------\n",
+ "t, x, y, θ, u, tf, iter, cons = solve(t0, x0, y0, θ0, xf, yf, θf, current(x0, y0); display=false);\n",
+ "\n",
+ "println(\"Iterations : \", iter)\n",
+ "println(\"Constraints violation : \", cons)\n",
+ "println(\"tf : \", tf)\n",
+ "\n",
+ "# -----------------------------------------------------------------------------------------------------------\n",
+ "# affichage\n",
+ "\n",
+ "# trajectoire\n",
+ "plt_q = plot(xlims=(-2, 6), ylims=(-1, 8), aspect_ratio=1, xlabel=\"x\", ylabel=\"y\")\n",
+ "plot_state!(plt_q, x0, y0, θ0; color=2)\n",
+ "plot_state!(plt_q, xf, yf, θf; color=2)\n",
+ "plot_current!(plt_q; current=(x, y) -> current(x0, y0))\n",
+ "plot_trajectory!(plt_q, t, x, y, θ)\n",
+ "\n",
+ "# contrôle\n",
+ "plt_u = plot(t, u; color=1, legend=false, linewidth=2, xlabel=\"t\", ylabel=\"u\")\n",
+ "\n",
+ "#\n",
+ "plot(plt_q, plt_u; \n",
+ " layout=(1, 2), \n",
+ " size=(1200, 600),\n",
+ " leftmargin=5mm, \n",
+ " bottommargin=5mm,\n",
+ " plot_title=\"Simulation courant constant\"\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "827b18d5",
+ "metadata": {},
+ "source": [
+ "## Simulation du système réel\n",
+ "\n",
+ "Dans la simulation précédente, nous faisons l'hypothèse que le courant est constant. Cependant, d'un point de vue pratique le courant dépend de la position $(x, y)$. Etant donné un modèle de courant, donné par la fonction `current`, nous pouvons simuler la trajectoire réelle du navire, pourvu que l'on ait la condition initiale et le contrôle au cours du temps."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "25422e2e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function realistic_trajectory(tf, t0, x0, y0, θ0, u, current; abstol=1e-12, reltol=1e-12, saveat=[])\n",
+ " \n",
+ " function rhs!(dq, q, dummy, t)\n",
+ " x, y, θ = q\n",
+ " w = current(x, y)\n",
+ " dq[1] = w[1]+cos(θ)\n",
+ " dq[2] = w[2]+sin(θ)\n",
+ " dq[3] = u(t)\n",
+ " end\n",
+ " \n",
+ " q0 = [ x0, y0, θ0 ]\n",
+ " tspan = (t0, tf)\n",
+ " ode = ODEProblem(rhs!, q0, tspan)\n",
+ " sol = OrdinaryDiffEq.solve(ode, Tsit5(), abstol=abstol, reltol=reltol, saveat=saveat)\n",
+ "\n",
+ " t = sol.t\n",
+ " x = t -> sol(t)[1]\n",
+ " y = t -> sol(t)[2]\n",
+ " θ = t -> sol(t)[3]\n",
+ "\n",
+ " return t, x, y, θ\n",
+ " \n",
+ "end;"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "da61110e",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# trajectoire réaliste\n",
+ "t, x, y, θ = realistic_trajectory(tf, t0, x0, y0, θ0, u, current)\n",
+ "\n",
+ "# -----------------------------------------------------------------------------------------------------------\n",
+ "# affichage\n",
+ "\n",
+ "# trajectoire\n",
+ "plt_q = plot(xlims=(-2, 6), ylims=(-1, 8), aspect_ratio=1, xlabel=\"x\", ylabel=\"y\")\n",
+ "plot_state!(plt_q, x0, y0, θ0; color=2)\n",
+ "plot_state!(plt_q, xf, yf, θf; color=2)\n",
+ "plot_current!(plt_q; current=current)\n",
+ "plot_trajectory!(plt_q, t, x, y, θ)\n",
+ "plot_state!(plt_q, x(tf), y(tf), θ(tf); color=3)\n",
+ "\n",
+ "# contrôle\n",
+ "plt_u = plot(t, u; color=1, legend=false, linewidth=2, xlabel=\"t\", ylabel=\"u\")\n",
+ "\n",
+ "#\n",
+ "plot(plt_q, plt_u; \n",
+ " layout=(1, 2), \n",
+ " size=(1200, 600),\n",
+ " leftmargin=5mm, \n",
+ " bottommargin=5mm,\n",
+ " plot_title=\"Simulation avec modèle de courant\"\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "503bd360",
+ "metadata": {},
+ "source": [
+ "## Approche MPC\n",
+ "\n",
+ "En pratique, nous n'avons pas à l'avance les données réelles du courant sur l'ensemble du trajet, c'est pourquoi nous allons recalculer régulièrement le contrôle optimal. L'idée est de mettre à jour le contrôle optimal à intervalle de temps régulier en prenant en compte le courant à la position où le navire se trouve. On est donc amener à résoudre un certain nombre de problème à courant constant, avec celui-ci mis réguilièremet à jour. Ceci est une introduction aux méthodes dites MPC, pour \"Model Predictive Control\" en anglais."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "563c73fd",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "Nmax = 20 # nombre d'itérations max de la méthode MPC\n",
+ "ε = 1e-1 # rayon sur la condition finale pour stopper les calculs\n",
+ "Δt = 1.0 # pas de temps fixe de la méthode MPC\n",
+ "P = 300 # nombre de points de discrétisation pour le solveur\n",
+ "\n",
+ "t1 = t0\n",
+ "x1 = x0\n",
+ "y1 = y0\n",
+ "θ1 = θ0\n",
+ "\n",
+ "data = []\n",
+ "\n",
+ "N = 1\n",
+ "stop = false\n",
+ "\n",
+ "while !stop\n",
+ " \n",
+ " # on récupère le courant à la position actuelle\n",
+ " w = current(x1, y1)\n",
+ "\n",
+ " # on résout le problème\n",
+ " t, x, y, θ, u, tf, iter, cons = solve(t1, x1, y1, θ1, xf, yf, θf, w; grid_size=P, display=false);\n",
+ "\n",
+ " # calcul du temps suivant\n",
+ " if (t1+Δt < tf)\n",
+ " t2 = t1+Δt\n",
+ " else\n",
+ " t2 = tf\n",
+ " println(\"t2=tf: \", t2)\n",
+ " stop = true\n",
+ " end\n",
+ "\n",
+ " # on stocke les données: le temps initial courant, le temps suivant, le contrôle\n",
+ " push!(data, (t2, t1, x(t1), y(t1), θ(t1), u, tf))\n",
+ "\n",
+ " # on met à jour les paramètres de la méthode MPC: on simule la réalité\n",
+ " t, x, y, θ = realistic_trajectory(t2, t1, x1, y1, θ1, u, current)\n",
+ " t1 = t2\n",
+ " x1 = x(t1)\n",
+ " y1 = y(t1)\n",
+ " θ1 = θ(t1)\n",
+ "\n",
+ " # on calcule la distance à la position cible\n",
+ " distance = norm([ x1, y1, θ1 ] - [ xf, yf, θf ])\n",
+ " println(\"N: \", N, \"\\t distance: \", distance, \"\\t iterations: \", iter, \"\\t constraints: \", cons, \"\\t tf: \", tf)\n",
+ " if !((distance > ε) && (N < Nmax))\n",
+ " stop = true\n",
+ " end\n",
+ "\n",
+ " #\n",
+ " N += 1\n",
+ "\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "edd803b2",
+ "metadata": {},
+ "source": [
+ "## Affichage"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "d0292a49",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# trajectoire\n",
+ "plt_q = plot(xlims=(-2, 6), ylims=(-1, 8), aspect_ratio=1, xlabel=\"x\", ylabel=\"y\")\n",
+ "\n",
+ "# condition finale\n",
+ "plot_state!(plt_q, xf, yf, θf; color=2)\n",
+ "\n",
+ "# courant\n",
+ "plot_current!(plt_q; current=current)\n",
+ "\n",
+ "# contrôle\n",
+ "plt_u = plot(xlabel=\"t\", ylabel=\"u\")\n",
+ "\n",
+ "N = 1\n",
+ "\n",
+ "for d ∈ data\n",
+ "\n",
+ " #\n",
+ " t2, t1, x1, y1, θ1, u, tf = d\n",
+ "\n",
+ " # calcule de la trajectoire réelle\n",
+ " t, x, y, θ = realistic_trajectory(t2, t1, x1, y1, θ1, u, current)\n",
+ "\n",
+ " # trajectoire\n",
+ " plot_state!(plt_q, x1, y1, θ1; color=2)\n",
+ " plot_trajectory!(plt_q, t, x, y, θ; N=0)\n",
+ "\n",
+ " # contrôle\n",
+ " plot!(plt_u, t, u; color=1, legend=false, linewidth=2)\n",
+ "\n",
+ " N += 1\n",
+ "\n",
+ "end\n",
+ "\n",
+ "plot_state!(plt_q, x(tf), y(tf), θ(tf); color=3)\n",
+ "\n",
+ "#\n",
+ "plot(plt_q, plt_u; \n",
+ " layout=(1, 2), \n",
+ " size=(1200, 600),\n",
+ " leftmargin=5mm, \n",
+ " bottommargin=5mm,\n",
+ " plot_title=\"Simulation avec modèle de courant\"\n",
+ ")"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.11.3",
+ "language": "julia",
+ "name": "julia-1.11"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.11.3"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/tp/runge-kutta.ipynb b/tp/rk-explicites.ipynb
similarity index 100%
rename from tp/runge-kutta.ipynb
rename to tp/rk-explicites.ipynb