“Matrix CookBook”<\/a><\/p>\n <\/p>\n
%matplotlib inline\nimport numpy as np\nfrom scipy.linalg import eigh\nfrom scipy.stats import multivariate_normal\nimport matplotlib.pyplot as plt\nfrom matplotlib.patches import Ellipse\nfrom sklearn.datasets import make_spd_matrix\n\nplt.ion()\ndef compute_ellipse(cov):\nw, u = eigh(cov)\nstd = np.sqrt(w)\nangle = np.arctan2(u[1, 0], u[0, 0])\nreturn 3 * std[0], 3 * std[1], angle * 180. \/ np.pi\n\ndef plot_gaussian(mu, cov):\n"""Plot the mean and covariance of a 2D Gaussian distribution.\n\nParameters\n----------\nmu : array_like, shape (2,)\nMean vector of the distribution.\ncov: array_like, shape (2, 2)\nCovariance matrix of the distribution.\n"""\n\n# Plot the mean vector.\nsc = plt.scatter(mu[0], mu[1], label='Mean')\nwidth, height, angle = compute_ellipse(cov[:2, :2])\n# Plot the covariance matrix as an ellipse.\nellipse = Ellipse(xy=mu, width=width, height=height, angle=angle,\nalpha=0.2, label='Covariance')\nplt.gca().add_artist(ellipse)\nreturn sc, ellipse\n\nC = make_spd_matrix(2, random_state=1)\nmu = np.array([1, 2])\n\nX = multivariate_normal(mu, C).rvs(1000)\n\nplt.scatter(*X.T, s=1)\nplot_gaussian(mu, C);\n<\/code><\/pre>\n
Distribution jointe<\/h3>\n D\u00e9composons notre \u00e9tat
<\/span> <\/span>Avec
Marginalisation<\/h3>\n <\/span> <\/span>Conditionnement<\/h3>\n Si on connait la valeur de
<\/span> <\/span>Combinaison lin\u00e9aire<\/h3>\n Si on a deux variable al\u00e9atoires ind\u00e9pendantes<\/strong> :<\/p>\n <\/span> <\/span> <\/span> <\/span>Alors la combinaison lin\u00e9aire des deux est distribu\u00e9e comme:<\/p>\n
<\/span> <\/span>Propagation de la dynamique du syst\u00e8me<\/h2>\n Les filtres de Kalman mod\u00e9lisent le syst\u00e8me en temps discret et l’\u00e9volution entre deux pas de temps est donn\u00e9e par la loi lin\u00e9aire :<\/p>\n
<\/span> <\/span>avec :
<\/span> <\/span>Exemple<\/h2>\nPoint mobile dans le plan<\/h3>\n Appliquons cela \u00e0 un objet ponctuel en mouvement sans frottement dans un plan.<\/p>\n
L’\u00e9tat de l’objet \u00e0 un instant
<\/p>\n
Le contr\u00f4le sur l’objet se fait par l’interm\u00e9diaire de l’\u00e9quivalent d’une pouss\u00e9e :<\/p>\n
<\/span> <\/span>On peut donc appliquer les \u00e9quations du mouvement de Newton pour mod\u00e9liser l’\u00e9volution du mobile entre deux instants proches. Si on consid\u00e8re que l’acc\u00e9l\u00e9ration est constante entre
<\/span> <\/span>Soit la vitesse initiale \u00e0 laquelle on ajoute la variation de vitesse due \u00e0 la pouss\u00e9e.<\/p>\n
Et le d\u00e9placement peut se calculer par:<\/p>\n
<\/span> <\/span>O\u00f9 l’on retrouve la position \u00e0 l’instant pr\u00e9c\u00e9dent, plus la variation de poistion due \u00e0 la vitesse initiale, plus la variation de position li\u00e9e \u00e0 la pouss\u00e9e.<\/p>\n
Mise en \u00e9quation de la propagation du filtre de Kalman<\/h3>\n Pour se caler dans le formalisme des filtre de Kalman, je suppose que l’\u00e9tat du syst\u00e8me, au pas de temps
<\/span> <\/span>On peut ensuite s\u00e9parer dans les \u00e9quations de mouvement. Les parties qui d\u00e9pendent de l’\u00e9tat pr\u00e9c\u00e9dent (position
<\/span> <\/span>Le contr\u00f4le corresond ici \u00e0 la pouss\u00e9e:<\/p>\n
<\/span> <\/span>Et l’\u00e9volution due au contr\u00f4le est donn\u00e9e par:<\/p>\n
<\/span> <\/span>Simulons un tel systeme:<\/p>\n
delta_t = 0.1\nt = np.arange(0, 10., delta_t)\n\n# L'acc\u00e9l\u00e9ration est modif\u00e9e \u00e0 3 reprises pendant 10 pas de temps dans des directions diff\u00e9rentes\nacceleration = np.zeros((len(t), 2))\nacceleration[10: 20] = (0, 0.4)\nacceleration[30: 40] = (0, -0.6)\nacceleration[50: 60] = (0.1, 0.3)\n\n# Matrice d'\u00e9volution de l'\u00e9tat\nF = np.identity(4)\nF[0, 2] = delta_t\nF[1, 3] = delta_t\n\n# Matrice donnant l'effet du contr\u00f4le sur syst\u00e8me\nG = np.zeros((4, 2))\nG[0, 0] = G[1, 1] = 0.5 * delta_t**2\nG[2, 0] = G[3, 1] = delta_t\n\n# Estimation du bruit li\u00e9 au processus d'\u00e9volution\nQ = np.diag([0.001, 0.001, 0.002, 0.002]) ** 2\n\n# Etat initial\nstate_actual = np.zeros((len(t), 4))\nstate_actual[0] = (0, 0, 0.1, 0)\n\nstate_mean = np.zeros((len(t), 4))\nstate_mean[0] = (0, 0, 0.1, 0)\n\n# Evolution dans le temps\nnp.random.seed(0)\nfor i in range(1, len(t)):\nstate_actual[i] = multivariate_normal(F @ state_actual[i - 1] + G @ acceleration[i], Q).rvs()\nstate_mean[i] = F @ state_mean[i - 1] + G @ acceleration[i]\n<\/code><\/pre>\nfrom itertools import chain\nfrom IPython.display import HTML\nfrom matplotlib.animation import FuncAnimation\n\nclass AnimResult:\ndef __init__(self, state_actual, acceleration, state_mean, covariance=None, observations=None):\nself.state_actual = state_actual\nself.acceleration = acceleration\nself.state_mean = state_mean\nself.covariance = covariance\nself.observations = observations\n\ndef plot_init(self):\nplt.plot(*state_mean.T[:2], "--", color="tab:blue", label="Trajectoire estim\u00e9e")\nplt.plot(*state_actual.T[:2], "-", color="tab:orange", label="Trajectoire simul\u00e9e")\nif self.observations is not None:\nself.plot_observations()\n\ndef plot_acceleration(self):\nif self.acceleration is None:\nself.accel_arrow = None\nelse:\nself.accel_arrow = plt.arrow(\n*chain(self.state_mean[0, :2],\n0.5 * self.acceleration[0]),\nwidth=0.01,\nvisible=np.any(self.acceleration[i] != 0)\n)\n\ndef plot_estimate(self):\nself.actual_scatter = plt.scatter(*self.state_actual[[0], :2].T, color="tab:orange", label="Position simul\u00e9e")\n\nif self.covariance is None:\nself.mean_scatter = plt.scatter(*self.state_mean[[0], :2].T, color="tab:blue", label="Position estim\u00e9e")\nself.cov_ellipse = None\nelse:\nself.mean_scatter, self.cov_ellipse = plot_gaussian(self.state_mean[0, :2], self.covariance[0, :2, :2])\n\ndef update(self, i):\nself.actual_scatter.set_offsets(*self.state_actual[[i], :2])\n\nself.mean_scatter.set_offsets(*self.state_mean[[i], :2])\n\nif self.cov_ellipse is not None:\nwidth, height, angle = compute_ellipse(self.covariance[i, :2, :2])\nself.cov_ellipse.set_center(self.state_mean[i, :2])\nself.cov_ellipse.set_width(width)\nself.cov_ellipse.set_height(height)\nself.cov_ellipse.set_angle(angle)\n\nif self.acceleration is not None:\n\nself.accel_arrow.set_data(\nx=self.state_mean[i, 0],\ny=self.state_mean[i, 1],\ndx=0.5 * self.acceleration[i, 0],\ndy=0.5 * self.acceleration[i, 1]\n)\nself.accel_arrow.set_visible(np.any(self.acceleration[i] != 0))\n\nif self.observations is not None:\nself.update_observations(i)\n\ndef plot_observations(self, i):\npass\n\ndef update_observations(self, i):\npass\n\ndef animate(self):\nself.plot_init()\nself.plot_estimate()\nplt.legend()\nself.plot_acceleration()\n\nani = FuncAnimation(\nplt.gcf(), self.update, frames=np.arange(len(t)), interval=100\n)\nreturn HTML(ani.to_html5_video())\n<\/code><\/pre>\nplt.ioff()\nplt.figure()\nplt.gca().set_aspect("equal")\nAnimResult(state_actual=state_actual, acceleration=acceleration, state_mean=state_mean).animate()\n<\/code><\/pre>\n <\/p>\n
\n
<\/p>\n
Propagation de l’incertitude<\/h2>\n Puisque nous n’avons que des lois normales et des transformations lin\u00e9aires, l’erreur peut \u00e9galement \u00eatre propag\u00e9e avec, en notant
<\/span> <\/span>Puisque:<\/p>\n
<\/span> <\/span># Etat initial\nstate_actual = np.zeros((len(t), 4))\nstate_actual[0] = (0, 0, 0.1, 0)\n\nstate_mean = np.zeros((len(t), 4))\nstate_mean[0] = (0, 0, 0.1, 0)\n\nstate_cov = np.zeros((len(t), 4, 4))\nstate_cov[0] = Q * 5 ** 2\n\n# Evolution dans le temps\nnp.random.seed(0)\nfor i in range(1, len(t)):\nstate_actual[i] = multivariate_normal(F @ state_actual[i - 1] + G @ acceleration[i], Q).rvs()\nstate_mean[i] = F @ state_mean[i - 1] + G @ acceleration[i]\nstate_cov[i] = F @ state_cov[i - 1] @ F.T + Q\n\nplt.ioff()\nplt.figure()\nplt.gca().set_aspect("equal")\nAnimResult(state_actual=state_actual, acceleration=acceleration, state_mean=state_mean, covariance=state_cov).animate()\n<\/code><\/pre>\n <\/p>\n
<\/p>\n
On voit donc ici le mouvement estim\u00e9 du syst\u00e8me connaissant uniquement l’\u00e9tat initial et les contr\u00f4les appliqu\u00e9s sur le syst\u00e8me. L’incertitude sur l’\u00e9tat (ici, on ne visualise que l’incertitude sur la position) augmente continuellement au cours du temps.<\/p>\n
Pour la diminuer, on peut avoir recours \u00e0 des observations.<\/p>\n
Observations<\/h2>\nTh\u00e9orie<\/h3>\n Par ailleurs, une observation se mod\u00e9lise par:<\/p>\n
<\/span> <\/span>avec:<\/p>\n
\n <\/span> <\/span>Pour calculer l’information gagn\u00e9e apr\u00e8s une observation, on part de la distribution jointe de l’\u00e9tat courant et du r\u00e9sultat de l’observation:<\/p>\n
<\/span> <\/span>On peut ensuite appliquer les formules associ\u00e9es aux gaussiennes multivari\u00e9es:<\/p>\n
<\/span> <\/span> <\/span> <\/span> <\/span> <\/span>Soit ici, en supposant qu’on a observ\u00e9
<\/span> <\/span> <\/span> <\/span>Simulons ici une observation correspondant \u00e0 la mesure de la position et de la vitesse selon l’axe
<\/span> <\/span>cela donne:<\/p>\n
from scipy.stats import multivariate_normal\n\n# Etat initial\nstate_actual = np.zeros((len(t), 4))\nstate_actual[0] = (0, 0, 0.1, 0)\n\nstate_mean = np.zeros((len(t), 4))\nstate_mean[0] = (0, 0, 0.1, 0)\n\nstate_cov = np.zeros((len(t), 4, 4))\nstate_cov[0] = Q * 5 ** 2\n\n# Transformation leading to the observation\nH = np.array([\n[1, 0, 0, 0],\n[0, 0, 1, 0]\n])\n\n# Observation noise\nR = (0.01 ** 2, 0.1 ** 2) * np.identity(2)\n\nobservation_times = [40, 60]\nobservation_results = {}\n\n# Evolution dans le temps\nnp.random.seed(0)\nfor i in range(1, len(t)):\nstate_actual[i] = multivariate_normal(F @ state_actual[i - 1] + G @ acceleration[i], Q).rvs()\nstate_mean[i] = F @ state_mean[i - 1] + G @ acceleration[i]\nstate_cov[i] = F @ state_cov[i - 1] @ F.T + Q\n\nif i in observation_times:\n# Observation process\nobservation_results[i] = obs_i = multivariate_normal(H @ state_actual[i], R).rvs()\n\n# update the estimates\nz_i = H @ state_mean[i]\ncov_state_obs = state_cov[i] @ H.T\ncov_obs_obs = H @ state_cov[i] @ H.T + R\n\nstate_mean[i] += cov_state_obs @ np.linalg.solve(cov_obs_obs, obs_i - z_i)\nstate_cov[i] -= cov_state_obs @ np.linalg.solve(cov_obs_obs, cov_state_obs.T)\n\nclass AnimLinearObs(AnimResult):\ndef plot_observations(self):\nymin, ymax = plt.ylim()\ndx = 3 * np.sqrt(R.flatten())[0]\nself.observation_lines = {}\nself.observation_fill = {}\n\nfor i, obs_i in self.observations.items():\nxi = obs_i[0]\nself.observation_lines[i] = plt.vlines(xi, ymin, ymax, color="tab:green", linestyle="--", alpha=0.1)\nself.observation_fill[i] = plt.fill_between([xi - dx, xi + dx], [ymin, ymin], [ymax, ymax], color="lightgreen", alpha=0.1)\nplt.ylim(ymin, ymax)\n\ndef update_observations(self, i):\nfor iobs, line in self.observation_lines.items():\nline.set_alpha(0.8 if i == iobs else 0.1)\nself.observation_fill[iobs].set_alpha(0.8 if i == iobs else 0.1)\n\nplt.ioff()\nplt.figure()\nplt.gca().set_aspect("equal")\nAnimLinearObs(state_actual, acceleration, state_mean, state_cov, observation_results).animate()\n<\/code><\/pre>\n <\/p>\n
<\/p>\n
On peut voir la contraction de l’incertitude selon
Extended Kalman Filters<\/h2>\n Si les lois d’\u00e9volution et d’observation ne sont plus lin\u00e9aires, il est tout de m\u00eame possible d’appliquer le filtre de Kalman en le lin\u00e9arisant localement:<\/p>\n
<\/span> <\/span> <\/span> <\/span>Il suffit alors de remplacer les formules de mise \u00e0 jour des covariances en utilisant la jacobienne des fonctions
<\/span> <\/span> <\/span> <\/span>Ici, on simule une observation, o\u00f9 l’on peut mesurer la distance et l’angle d’observation depuis l’origine du rep\u00e8re:<\/p>\n
<\/span> <\/span>La loi de mouvement reste inchang\u00e9e, donc la jacobienne est identique \u00e0 la matrice
import jax\nimport jax.numpy as jnp\n\ndef movement(state, control):\nreturn F @ state + G @ control\n\ndef observation(state):\nx, y = state[:2]\nr = jnp.sqrt(x ** 2 + y ** 2)\nangle = jnp.arctan2(y, x)\nreturn jnp.array([r, angle * 180 \/ np.pi])\n\nR = (np.identity(2) * (0.025, 0.5)) ** 2\n\n# Etat initial\nstate_actual = np.zeros((len(t), 4))\nstate_actual[0] = (0, 0, 0.1, 0)\n\nstate_mean = np.zeros((len(t), 4))\nstate_mean[0] = (0, 0, 0.1, 0)\n\nstate_cov = np.zeros((len(t), 4, 4))\nstate_cov[0] = Q * 5 ** 2\n\nobservation_results = {}\n\n# Evolution dans le temps\nnp.random.seed(0)\nfor i in range(1, len(t)):\nstate_actual[i] = multivariate_normal(movement(state_actual[i - 1], acceleration[i]), Q).rvs()\nstate_mean[i] = movement(state_mean[i - 1], acceleration[i])\nF_i = jax.jacobian(movement)(state_mean[i - 1], acceleration[i])\n\nstate_cov[i] = F_i @ state_cov[i - 1] @ F_i.T + Q\n\nif i in observation_times:\n# Observation process\nobservation_results[i] = obs_i = multivariate_normal(observation(state_actual[i]), R).rvs()\n\n# update the estimates\nz_i = observation(state_mean[i])\nH_i = jax.jacobian(observation)(state_mean[i])\ncov_state_obs = state_cov[i] @ H_i.T\ncov_obs_obs = H_i @ state_cov[i] @ H_i.T + R\n\nstate_mean[i] += cov_state_obs @ np.linalg.solve(cov_obs_obs, obs_i - z_i)\nstate_cov[i] -= cov_state_obs @ np.linalg.solve(cov_obs_obs, cov_state_obs.T)\n\ndef draw_polar_section(r, angle, dr, dangle, circle_grid=10):\nx0 = r * np.cos(angle \/ 180. * np.pi)\ny0 = r * np.sin(angle \/ 180. * np.pi)\n\nrs = np.array([ r - dr, r + dr ] + [r + dr ] * circle_grid + [r - dr] + [r - dr] * circle_grid)\nangles = np.array(\n[angle - dangle, angle - dangle] +\n[(angle - dangle) + (i + 1) \/ circle_grid * 2 * dangle for i in range(circle_grid)] +\n[angle + dangle ] +\n[(angle + dangle) - (i + 1) \/ circle_grid * 2 * dangle for i in range(circle_grid)]\n)\n\npolygon_xs = rs * np.cos(angles \/ 180. * np.pi)\npolygon_ys = rs * np.sin(angles \/ 180. * np.pi)\n\nreturn plt.scatter([x0], [y0], c="tab:green", alpha=0.1), plt.fill(polygon_xs, polygon_ys, alpha=0.1, color="lightgreen")\n\nclass AnimNonLinearObs(AnimResult):\ndef plot_observations(self):\ndr, dangle = 3 * np.sqrt(R.diagonal())\n\nself.observation_patches = {\ni: draw_polar_section(*obs_i, dr, dangle) for i, obs_i in self.observations.items()\n}\n\ndef update_observations(self, i):\nfor iobs, (obs_center, obs_poly) in self.observation_patches.items():\nalpha = 0.8 if i == iobs else 0.1\nobs_center.set_alpha(alpha)\nfor p in obs_poly: p.set_alpha(alpha)\n\nplt.ioff()\nplt.figure()\nplt.gca().set_aspect("equal")\nAnimNonLinearObs(state_actual, acceleration, state_mean, state_cov, observation_results).animate()\n<\/code><\/pre>\n <\/p>\n
<\/p>\n
Filtres de Kalman sans parfum (Unscented Kalman Filter<\/em>)<\/h2>\nLa transformation sans parfum<\/h3>\n L’id\u00e9e de cette transformation et d’estimer l’effet de transformations non-lin\u00e9aires en utilisant des \u00e9chantillons bien choisis appel\u00e9s “points sigma”.<\/p>\n
On fait passer ces points \u00e0 travers les transformations d’\u00e9volution temporelle et d’observation et on estime ensuite, \u00e0 partir de ces points, les param\u00e8tres de la distribution r\u00e9sultante.<\/p>\n
L’id\u00e9e est de construire un ensemble de paires de points qui pointent dans les directions principales de la distribution Gaussienne. Si on d\u00e9compose la covariance en :<\/p>\n
<\/span> <\/span>Alors, on construit les points sigma comme suit\u00a0:<\/p>\n
<\/span> <\/span>\u00e0 chaque point est associ\u00e9 un poids:<\/p>\n
<\/span> <\/span>o\u00f9
%matplotlib inline\nimport numpy as np\nfrom scipy.linalg import eigh\nfrom math import gcd\n\ndef sigma_points(mu, sigma, lambda_):\nn_dims = len(mu)\n\nw, u = eigh(sigma)\nw = np.maximum(w, 0)\nL = (u * np.sqrt(w)).T\n\nX = np.vstack(\n[ mu ] +\n[ mu + np.sqrt(n_dims + lambda_) * L[i] for i in range(n_dims) ] +\n[ mu - np.sqrt(n_dims + lambda_) * L[i] for i in range(n_dims) ]\n)\n\nw = np.hstack([lambda_ \/ (n_dims + lambda_)] + [0.5 \/ (n_dims + lambda_)] * (2 * n_dims))\n\nreturn X, w\n\nC = np.array([[1.0, 0.8], [0.8, 1]])\nmu = np.array([1, 2])\n\nX_sigma, w_sigma = sigma_points(mu, C, lambda_=1)\n\nplt.ion()\nplt.figure()\nplot_gaussian(mu, C);\nplt.scatter(*X_sigma.T, s=10);\n<\/code><\/pre>\n
Application to the Kalman Filter<\/h3>\n On passe ces points \u00e0 travers la transformation d’\u00e9volution temporelle et \u00e0 travers la fonction d’observation et on estime ensuite les diff\u00e9rentes quantit\u00e9 d’int\u00e9r\u00eat en int\u00e9grant les poids:<\/p>\n
<\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span> <\/span># Apply on sigma points matrix instead of single vector\ndef movement(state, control):\nreturn state @ F.T + G @ control\n\ndef observation(state):\nx, y = state.T[:2]\nr = np.sqrt(x ** 2 + y ** 2)\nangle = np.arctan2(y, x)\nreturn np.vstack([r, angle * 180 \/ np.pi]).T\n\n# Etat initial\nstate_actual = np.zeros((len(t), 4))\nstate_actual[0] = (0, 0, 0.1, 0)\n\nstate_mean = np.zeros((len(t), 4))\nstate_mean[0] = (0, 0, 0.1, 0)\n\nstate_cov = np.zeros((len(t), 4, 4))\nstate_cov[0] = Q * 5 ** 2\n\nobservation_results = {}\nlambda_ = 1e-1\n\n# Evolution dans le temps\nnp.random.seed(0)\nfor i in range(1, len(t)):\n\nstate_actual[i] = multivariate_normal(movement(state_actual[[i - 1]], acceleration[i])[0], Q).rvs()\n\nstate_sigma, weight_sigma = sigma_points(state_mean[i - 1], state_cov[i - 1], lambda_)\nstate_sigma_tr = movement(state_sigma, acceleration[i])\n\nstate_mean[i] = np.einsum("ij, i -> j", state_sigma_tr, weight_sigma)\nstate_diff = state_sigma_tr - state_mean[i]\nstate_cov[i] = np.einsum("ij, ik, i -> jk", state_diff, state_diff, weight_sigma)\n\nif i in observation_times:\n# Observation process\nobservation_results[i] = obs_i = multivariate_normal(observation(state_actual[[i]])[0], R).rvs()\n\n# update the estimates\n\nz_sigma = observation(state_sigma_tr)\nz_i = np.einsum("ij, i -> j", z_sigma, weight_sigma)\n\nz_diff = z_sigma - z_i\n\ncov_state_obs = np.einsum("ij, ik, i -> jk", state_diff, z_diff, weight_sigma)\ncov_obs_obs = np.einsum("ij, ik, i -> jk", z_diff, z_diff, weight_sigma)\n\nstate_mean[i] += cov_state_obs @ np.linalg.solve(cov_obs_obs, obs_i - z_i)\nstate_cov[i] -= cov_state_obs @ np.linalg.solve(cov_obs_obs, cov_state_obs.T)\n\nplt.ioff()\nplt.figure()\nplt.gca().set_aspect("equal")\nAnimNonLinearObs(state_actual, acceleration, state_mean, state_cov, observation_results).animate()\n<\/code><\/pre>\n <\/p>\n
Les filtres de Kalman Les filtres de Kalman permettent d’estimer l’\u00e9tat d’un syst\u00e8me dynamique pour lequel on connait : les r\u00e8gles \/ lois d’\u00e9volution les actions qui lui sont appliqu\u00e9es une s\u00e9rie d’observations \/ mesures potentiellement incompl\u00e8tes ou bruit\u00e9es. Dans sa version la plus simple, l’ensemble des transformations du syst\u00e8me ainsi que les observations r\u00e9alis\u00e9es […]<\/p>\n","protected":false},"author":3,"featured_media":17235,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[65],"tags":[],"class_list":["post-17325","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-machine-learning"],"acf":[],"yoast_head":"\n
Les filtres de Kalman - Probayes<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Les filtres de Kalman - Probayes\" \/>\n<meta property=\"og:description\" content=\"Les filtres de Kalman Les filtres de Kalman permettent d’estimer l’\u00e9tat d’un syst\u00e8me dynamique pour lequel on connait : les r\u00e8gles \/ lois d’\u00e9volution les actions qui lui sont appliqu\u00e9es une s\u00e9rie d’observations \/ mesures potentiellement incompl\u00e8tes ou bruit\u00e9es. Dans sa version la plus simple, l’ensemble des transformations du syst\u00e8me ainsi que les observations r\u00e9alis\u00e9es […]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\" \/>\n<meta property=\"og:site_name\" content=\"Probayes\" \/>\n<meta property=\"article:published_time\" content=\"2024-05-03T13:12:55+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-09-04T14:12:00+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-1024x479.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1024\" \/>\n\t<meta property=\"og:image:height\" content=\"479\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Laetitia Fauconnet-Viegas\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Laetitia Fauconnet-Viegas\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"17 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\"},\"author\":{\"name\":\"Laetitia Fauconnet-Viegas\",\"@id\":\"https:\/\/www.probayes.com\/#\/schema\/person\/8ed56e24b0173efb2cb3d1df68dcf635\"},\"headline\":\"Les filtres de Kalman\",\"datePublished\":\"2024-05-03T13:12:55+00:00\",\"dateModified\":\"2024-09-04T14:12:00+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\"},\"wordCount\":2071,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/www.probayes.com\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg\",\"articleSection\":[\"Machine Learning\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\",\"url\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\",\"name\":\"Les filtres de Kalman - Probayes\",\"isPartOf\":{\"@id\":\"https:\/\/www.probayes.com\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg\",\"datePublished\":\"2024-05-03T13:12:55+00:00\",\"dateModified\":\"2024-09-04T14:12:00+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage\",\"url\":\"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg\",\"contentUrl\":\"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg\",\"width\":7000,\"height\":3276},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/www.probayes.com\/en\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Les filtres de Kalman\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/www.probayes.com\/#website\",\"url\":\"https:\/\/www.probayes.com\/\",\"name\":\"Probayes\",\"description\":\"Ensemble, construisons vos solutions d'Intelligence Artificielle\",\"publisher\":{\"@id\":\"https:\/\/www.probayes.com\/#organization\"},\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/www.probayes.com\/#organization\",\"name\":\"Probayes\",\"url\":\"https:\/\/www.probayes.com\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.probayes.com\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/www.probayes.com\/wp-content\/uploads\/2020\/03\/Probayes_baseline_FdTransp.png\",\"contentUrl\":\"https:\/\/www.probayes.com\/wp-content\/uploads\/2020\/03\/Probayes_baseline_FdTransp.png\",\"width\":1494,\"height\":543,\"caption\":\"Probayes\"},\"image\":{\"@id\":\"https:\/\/www.probayes.com\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/www.probayes.com\/#\/schema\/person\/8ed56e24b0173efb2cb3d1df68dcf635\",\"name\":\"Laetitia Fauconnet-Viegas\",\"url\":\"https:\/\/www.probayes.com\/en\/author\/laetitia\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Les filtres de Kalman - Probayes","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/","og_locale":"en_US","og_type":"article","og_title":"Les filtres de Kalman - Probayes","og_description":"Les filtres de Kalman Les filtres de Kalman permettent d’estimer l’\u00e9tat d’un syst\u00e8me dynamique pour lequel on connait : les r\u00e8gles \/ lois d’\u00e9volution les actions qui lui sont appliqu\u00e9es une s\u00e9rie d’observations \/ mesures potentiellement incompl\u00e8tes ou bruit\u00e9es. Dans sa version la plus simple, l’ensemble des transformations du syst\u00e8me ainsi que les observations r\u00e9alis\u00e9es […]","og_url":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/","og_site_name":"Probayes","article_published_time":"2024-05-03T13:12:55+00:00","article_modified_time":"2024-09-04T14:12:00+00:00","og_image":[{"width":1024,"height":479,"url":"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-1024x479.jpg","type":"image\/jpeg"}],"author":"Laetitia Fauconnet-Viegas","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Laetitia Fauconnet-Viegas","Est. reading time":"17 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#article","isPartOf":{"@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/"},"author":{"name":"Laetitia Fauconnet-Viegas","@id":"https:\/\/www.probayes.com\/#\/schema\/person\/8ed56e24b0173efb2cb3d1df68dcf635"},"headline":"Les filtres de Kalman","datePublished":"2024-05-03T13:12:55+00:00","dateModified":"2024-09-04T14:12:00+00:00","mainEntityOfPage":{"@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/"},"wordCount":2071,"commentCount":0,"publisher":{"@id":"https:\/\/www.probayes.com\/#organization"},"image":{"@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage"},"thumbnailUrl":"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg","articleSection":["Machine Learning"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/","url":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/","name":"Les filtres de Kalman - Probayes","isPartOf":{"@id":"https:\/\/www.probayes.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage"},"image":{"@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage"},"thumbnailUrl":"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg","datePublished":"2024-05-03T13:12:55+00:00","dateModified":"2024-09-04T14:12:00+00:00","breadcrumb":{"@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#primaryimage","url":"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg","contentUrl":"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg","width":7000,"height":3276},{"@type":"BreadcrumbList","@id":"https:\/\/www.probayes.com\/2024\/05\/03\/les-filtres-de-kalman\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.probayes.com\/en\/"},{"@type":"ListItem","position":2,"name":"Les filtres de Kalman"}]},{"@type":"WebSite","@id":"https:\/\/www.probayes.com\/#website","url":"https:\/\/www.probayes.com\/","name":"Probayes","description":"Ensemble, construisons vos solutions d'Intelligence Artificielle","publisher":{"@id":"https:\/\/www.probayes.com\/#organization"},"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/www.probayes.com\/#organization","name":"Probayes","url":"https:\/\/www.probayes.com\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.probayes.com\/#\/schema\/logo\/image\/","url":"https:\/\/www.probayes.com\/wp-content\/uploads\/2020\/03\/Probayes_baseline_FdTransp.png","contentUrl":"https:\/\/www.probayes.com\/wp-content\/uploads\/2020\/03\/Probayes_baseline_FdTransp.png","width":1494,"height":543,"caption":"Probayes"},"image":{"@id":"https:\/\/www.probayes.com\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/www.probayes.com\/#\/schema\/person\/8ed56e24b0173efb2cb3d1df68dcf635","name":"Laetitia Fauconnet-Viegas","url":"https:\/\/www.probayes.com\/en\/author\/laetitia\/"}]}},"jetpack_featured_media_url":"https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg","jetpack_sharing_enabled":true,"rttpg_featured_image_url":{"full":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg",7000,3276,false],"landscape":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg",7000,3276,false],"portraits":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087.jpg",7000,3276,false],"thumbnail":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-150x150.jpg",150,150,true],"medium":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-300x140.jpg",300,140,true],"large":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-1024x479.jpg",640,299,true],"1536x1536":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-1536x719.jpg",1536,719,true],"2048x2048":["https:\/\/www.probayes.com\/wp-content\/uploads\/2024\/04\/shutterstock_2122874087-2048x958.jpg",2048,958,true]},"rttpg_author":{"display_name":"Laetitia Fauconnet-Viegas","author_link":"https:\/\/www.probayes.com\/en\/author\/laetitia\/"},"rttpg_comment":0,"rttpg_category":"<a href=\"https:\/\/www.probayes.com\/category\/blog\/machine-learning\/\" rel=\"category tag\">Machine Learning<\/a>","rttpg_excerpt":"Les filtres de Kalman Les filtres de Kalman permettent d’estimer l’\u00e9tat d’un syst\u00e8me dynamique pour lequel on connait : les r\u00e8gles \/ lois d’\u00e9volution les actions qui lui sont appliqu\u00e9es une s\u00e9rie d’observations \/ mesures potentiellement incompl\u00e8tes ou bruit\u00e9es. Dans sa version la plus simple, l’ensemble des transformations du syst\u00e8me ainsi que les observations r\u00e9alis\u00e9es…","_links":{"self":[{"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/posts\/17325","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/comments?post=17325"}],"version-history":[{"count":17,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/posts\/17325\/revisions"}],"predecessor-version":[{"id":17991,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/posts\/17325\/revisions\/17991"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/media\/17235"}],"wp:attachment":[{"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/media?parent=17325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/categories?post=17325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.probayes.com\/en\/wp-json\/wp\/v2\/tags?post=17325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"t\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-b4e3cbf5d4c5c6d9b702dd139f14c147_l3.png\" "\\"Rendered")
est ![\"\boldsymbol x_t\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-678543a61104f694cc5fe1d3f600cb06_l3.png\" "\\"Rendered")
.<\/p>\n![\"\hat {\boldsymbol x}_t\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-06111a3c66d4e0b92f5cd7882a80b427_l3.png\" "\\"Rendered")
est connue avec une certaine incertitude repr\u00e9sent\u00e9e par une matrice de covariance ![\"P_t\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-2550a93f909ef5a7ee29f3b9729716dd_l3.png\" "\\"Rendered")
.<\/p>\n![\"\[\boldsymbol x_t \sim \mathcal N(\hat{\boldsymbol x}_t, P_t)\]\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-ac30b219cedbb786abf9ff32be8e52f5_l3.png\" "\\"Rendered")
<\/p>\n![\"png\"](\"output_2_0.png\")
<\/p>\n![\"\boldsymbol x\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-8b8cc37476872ad5a1e8e8d5b972b6fa_l3.png\" "\\"Rendered")
distribu\u00e9e selon une loi Gaussienne, en deux sous-parties, la distribution jointe des deux sous partie est :<\/p>\n![\"\[\boldsymbol x = \pmatrix{](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-13a402180a6f59432852ce2452335cfe_l3.png\" "\\"Rendered")
<\/p>\n![\"\Sigma_{21} = \Sigma_{12}^T\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-927e1297055177d3d117c8a47ff24154_l3.png\" "\\"Rendered")
.<\/p>\n![\"\[\boldsymbol x_1 \sim](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-43291d86392f6ccba326b43b89d887d7_l3.png\" "\\"Rendered")
<\/p>\n![\"\boldsymbol x_2\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-138a7f897e2b68e326fb9b846c1609c0_l3.png\" "\\"Rendered")
alors:<\/p>\n![\"\[\boldsymbol x_{1|2} \sim](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-35f911691edd4091e2ddc90802355cba_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\boldsymbol x_2 \sim](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-809ecfebe1465e60b21d6458595f1fc4_l3.png\" "\\"Rendered")
<\/p>\n![\"\[A\boldsymbol x_1 + B \boldsymbol x_2 + \boldsymbol c \sim](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-8154626392c8455e14b2f83eba9974ee_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\boldsymbol x_{k + 1|k} = F_k \boldsymbol x_{k|k} + G_k \boldsymbol u_k + \boldsymbol w_k\]\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-5f7122899fceec8eb1149f2994508a85_l3.png\" "\\"Rendered")
<\/p>\n![\"\boldsymbol x_{k|k}\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-6009c3e3552f37c88bb3f902039cfb4f_l3.png\" "\\"Rendered")
: L’estimtation de l’\u00e9tat au pas ![\"k\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" "\\"Rendered")
apr\u00e8s avoir pris en compte les \u00e9ventuelles observations![\"\boldsymbol x_{k + 1|k}\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-651fe4c728b7a1d6c2953b6f3cc50a62_l3.png\" "\\"Rendered")
: L’estimation de l’\u00e9tat au pas ![\"k+1\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-1e22674f5099474b9902f53541dfc8dc_l3.png\" "\\"Rendered")
par propagation de la dynamique du syst\u00e8me depuis l’estimation au pas pr\u00e9c\u00e9dent.![\"F_k\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-e26a0f3c0fdc578c8d4eb2ace7efecd0_l3.png\" "\\"Rendered")
: matrice de transition qui repr\u00e9sente l’\u00e9volution “naturelle” du syst\u00e8me (pas n\u00e9cessairement constante)![\"\boldsymbol u_k\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-bbca96028e3650abe532456fa48a6807_l3.png\" "\\"Rendered")
: une commande (ou contr\u00f4le) appliqu\u00e9e au syst\u00e8me![\"G_k\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-ca53d5313f9fafe4c15c73a5aa111dde_l3.png\" "\\"Rendered")
: l’effet de la commande sur le syst\u00e8me (pas n\u00e9cessairement constante)![\"\boldsymbol w_k\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-a8c9a49e5dcda2c2a01544669c5d2bae_l3.png\" "\\"Rendered")
: le bruit du processus, c’est-\u00e0-dire ce qu’on ne mod\u00e9lise pas bien.<\/p>\n![\"\[\boldsymbol w_k \sim \mathcal N(\boldsymbol 0, Q_k)\]\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-7200f284b5d98dacff93aaf076b15b86_l3.png\" "\\"Rendered")
<\/p>\n![\"\dot{\boldsymbol x}_t\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-2d6e0aab539c4943227296d8f50e1c11_l3.png\" "\\"Rendered")
:<\/p>\n![\"\[\boldsymbol a_t = \frac{1}{m} \boldsymbol f_t\]\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-a1d758902e472d376f7d8103567606fd_l3.png\" "\\"Rendered")
<\/p>\n![\"t + \Delta t\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-1e8c44c8aa70692470ffaedac557d29c_l3.png\" "\\"Rendered")
, la vitesse va \u00eatre donn\u00e9es par:<\/p>\n![\"\[\dot{\boldsymbol x}_{t + \Delta_t} = \boldsymbol a_t \Delta t + \dot{\boldsymbol x}_t\]\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-f4f47286244e71927b022e4c10269c70_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\boldsymbol x_{t + \Delta_t} = \frac{1}{2}\boldsymbol a_t (\Delta t)^2 + \dot{\boldsymbol x}_t \Delta t + \boldsymbol x_t\]\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-f86d620801407d2c848ec7763e9bed3f_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\boldsymbol x_{k|k} = \left(\begin{array}{c}](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-ce2dd90cecd224d448d93d7f9d27d257_l3.png\" "\\"Rendered")
<\/p>\n![\"\boldsymbol{\dot x}\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-e760334ce442a70fdd1c13dc4a5a16cb_l3.png\" "\\"Rendered")
) et les parties qui d\u00e9pendent du contr\u00f4le ![\"\boldsymbol u_k = \boldsymbol a_k\"](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-1b9c224319aaa8588b6a4e918cde39cc_l3.png\" "\\"Rendered")
(ici l’acc\u00e9l\u00e9ration) et on peut ainsi construire les diff\u00e9rents termes de l’\u00e9tape de propagation:<\/p>\n![\"\[F_k = \left(\begin{array}{cc}](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-f8ba42984ef5c5f0678e783884f45139_l3.png\" "\\"Rendered")
<\/p>\n![\"\[\boldsymbol u_k = \boldsymbol a_k = \pmatrix{a_{xk} \ a_{yk}}= \frac{1}{m}](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-4f7a5db6409d027ca58d4ddee492b2a6_l3.png\" "\\"Rendered")
<\/p>\n![\"\[G_k = \left(\begin{array}{c}](\"https:\/\/www.probayes.com\/wp-content\/ql-cache\/quicklatex.com-6b296ee8fc3e67a3320bc095a0ffa105_l3.png\" "\\"Rendered")
<\/p>\n