Commande optimaleLa théorie de la commande optimale permet de déterminer la commande d'un système qui minimise (ou maximise) un critère de performance, éventuellement sous des contraintes pouvant porter sur la commande ou sur l'état du système. Cette théorie est une généralisation du calcul des variations. Elle comporte deux volets : le principe du maximum (ou du minimum, suivant la manière dont on définit l'hamiltonien) dû à Lev Pontriaguine et à ses collaborateurs de l'institut de mathématiques Steklov , et l'équation de Hamilton-Jacobi-Bellman, généralisation de l'équation de Hamilton-Jacobi, et conséquence directe de la programmation dynamique initiée aux États-Unis par Richard Bellman.
Théorie du contrôleEn mathématiques et en sciences de l'ingénieur, la théorie du contrôle a comme objet l'étude du comportement de systèmes dynamiques paramétrés en fonction des trajectoires de leurs paramètres. On se place dans un ensemble, l'espace d'état sur lequel on définit une dynamique, c'est-à-dire une loi mathématiques caractérisant l'évolution de variables (dites variables d'état) au sein de cet ensemble. Le déroulement du temps est modélisé par un entier .
Linear time-invariant systemIn system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined below. These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (x ∗ h)(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication).
Système à minimum de phaseEn traitement du signal et en théorie du contrôle, un système linéaire ne dépendant pas du temps est dit à minimum de phase si ce système et son inverse sont stables et causaux. On parle aussi de filtre à minimum de phase. Pour un système discret, en supposant que la fonction de transfert est rationnelle, ce système est à minimum de phase si et seulement si tous les pôles et zéros de sont à l'intérieur du disque unité. Pour un système continu, la condition pour que ce système soit à minimum de phase est que les pôles et zéros de transmission appartiennent au demi-plan gauche du plan complexe.
Residual sum of squaresIn statistics, the residual sum of squares (RSS), also known as the sum of squared residuals (SSR) or the sum of squared estimate of errors (SSE), is the sum of the squares of residuals (deviations predicted from actual empirical values of data). It is a measure of the discrepancy between the data and an estimation model, such as a linear regression. A small RSS indicates a tight fit of the model to the data. It is used as an optimality criterion in parameter selection and model selection.
Lack-of-fit sum of squaresIn statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares. The pure-error sum of squares is the sum of squared deviations of each value of the dependent variable from the average value over all observations sharing its independent variable value(s).
Partition of sums of squaresThe partition of sums of squares is a concept that permeates much of inferential statistics and descriptive statistics. More properly, it is the partitioning of sums of squared deviations or errors. Mathematically, the sum of squared deviations is an unscaled, or unadjusted measure of dispersion (also called variability). When scaled for the number of degrees of freedom, it estimates the variance, or spread of the observations about their mean value.
Explained sum of squaresIn statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression (SSR – not to be confused with the residual sum of squares (RSS) or sum of squares of errors), is a quantity used in describing how well a model, often a regression model, represents the data being modelled.
Total sum of squaresIn statistical data analysis the total sum of squares (TSS or SST) is a quantity that appears as part of a standard way of presenting results of such analyses. For a set of observations, , it is defined as the sum over all squared differences between the observations and their overall mean .: For wide classes of linear models, the total sum of squares equals the explained sum of squares plus the residual sum of squares. For proof of this in the multivariate OLS case, see partitioning in the general OLS model.
Nonlinear systemIn mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.
