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Measurement Equations (measurement + equation)

Distribution by Scientific Domains

Engineering	75%

Selected Abstracts

The Kalman filter for the pedologist's tool kit

EUROPEAN JOURNAL OF SOIL SCIENCE, Issue 6 2006
R. Webster
Summary The Kalman filter is a tool designed primarily to estimate the values of the ,state' of a dynamic system in time. There are two main equations. These are the state equation, which describes the behaviour of the state over time, and the measurement equation, which describes at what times and in what manner the state is observed. For the discrete Kalman filter, discussed in this paper, the state equation is a stochastic difference equation that incorporates a random component for noise in the system and that may include external forcing. The measurement equation is defined such that it can handle indirect measurements, gaps in the sequence of measurements and measurement errors. The Kalman filter operates recursively to predict forwards one step at a time the state of the system from the previously predicted state and the next measurement. Its predictions are optimal in the sense that they have minimum variance among all unbiased predictors, and in this respect the filter behaves like kriging. The equations can also be applied in reverse order to estimate the state variable at all time points from a complete series of measurements, including past, present and future measurements. This process is known as smoothing. This paper describes the ,predictor,corrector' algorithm for the Kalman filter and smoother with all the equations in full, and it illustrates the method with examples on the dynamics of groundwater level in the soil. The height of the water table at any one time depends partly on the height at previous times and partly on the precipitation excess. Measurements of the height of water table and their errors are incorporated into the measurement equation to improve prediction. Results show how diminishing the measurement error increases the accuracy of the predictions, and estimates achieved with the Kalman smoother are even more accurate. Le filtre de Kalman comme outil pour le pédologue Résumé Le filtre de Kalman est un outil conçu essentiellement pour estimer les valeurs de l'état d'un système dynamique dans le temps. Il comprend deux équations principales. Celles-ci sont l'équation d'état, qui décrit l'évolution de l'état pendant le temps, et l'équation de mesure qui decrit à quel instants et de quelle façon on observe l'état. Pour le filtre discret de Kalman, décrit dans cet article, l'équation d'état est une équation stochastique différentielle qui comprend une composante aléatoire pour le bruit dans le système et qui peut inclure une force extérieure. On définit l'équation de mesure de façon à ce qu'elle puisse traiter des mesures indirectes, des vides dans des séquences de mesures et des erreurs de mesure. Le filtre de Kalman fonctionne récursivement pour prédire en avance une démarche à temps l'état du système de la démarche prédite antérieure plus l'observation prochaine. Ses prédictions sont optimales dans le sens qu'elles minimisent la variance parmi toutes les prédictions non-biasées, et à cet égard le filtre se comporte comme le krigeage. On peut appliquer, aussi, les équations dans l'ordre inverse pour estimer la variable d'état à toutes pointes à toutes les instants d'une série complète d'observations, y compris les observations du passé, du présent et du futur. Ce processus est connu comme ,smoothing'. Cet article décrit l'algorithme ,predictor,corrector' du filtre de Kalman et le ,smoother' avec toutes les équations entières. Il illustre cette méthode avec des exemples de la dynamique du niveau de la nappe phréatique dans le sol. Le niveau de la nappe à un instant particulier dépend en partie du niveau aux instants précédents et en partie de l'excès de la précipitation. L'équation d'état fournit la relation générale entre les deux variables et les prédictions. On incorpore les mesures du niveau de la nappe et leurs erreurs pour améliorer les prédictions. Les résultats mettent en évidence que lorsqu'on diminue l'erreur de mesure la précision des prédictions augmente, et aussi que les estimations avec le ,smoother' de Kalman sont encore plus précises. [source]

Active mode observation of switching systems based on set-valued estimation of the continuous state

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 14 2009
M. Baglietto
Abstract Mode observability is addressed for a class of discrete-time linear systems that may switch in an unknown and unpredictable way among different modes taken from a finite set. The possible a priori knowledge on the continuous state of the system and the presence of unknown but bounded noises affecting both the system and the measurement equations are explicitly taken into account. The mode observation is performed ,actively': control sequences (discerning control sequences) are searched, which allow to identify the switching sequence on the basis of the observations. Conditions that characterize discerning controls in a finite-horizon setting are obtained. Moreover, a procedure is proposed in order to derive ,persistently discerning' control sequences (over an infinite horizon). A numerical example is reported to clarify the approach. Copyright © 2008 John Wiley & Sons, Ltd. [source]

The evaluation of high occupancy vehicle lanes on sun yat-sen freeway in Taiwan

JOURNAL OF ADVANCED TRANSPORTATION, Issue 2 2005
Rong-Chang Jou
Abstract This study proposes a methodological framework to incorporate latent factors, including direct and indirect perceptions, as the explanatory variables in a discrete choice models using revealed preference and stated preference data sets. The methodology requires the estimation of a model system comprising of a discrete choice model and the structural and measurement equations of a latent variable model. The application involves the evaluation of responses to the new high occupancy vehicle (HOV) lanes on the Sun Yat-Sen Freeway in Taiwan. The results obtained from this study provide valuable insights into the planning and assessment of HOV lanes. [source]

LMI APPROACH TO ROBUST FILTERING FOR DISCRETE TIME-DELAY SYSTEMS WITH NONLINEAR DISTURBANCES

ASIAN JOURNAL OF CONTROL, Issue 2 2005
Huijun Gao
ABSTRACT This paper investigates the problem of robust filtering for a class of uncertain nonlinear discrete-time systems with multiple state delays. It is assumed that the parameter uncertainties appearing in all the system matrices reside in a polytope, and that the nonlinearities entering into both the state and measurement equations satisfy global Lipschitz conditions. Attention is focused on the design of robust full-order and reduced-order filters guaranteeing a prescribed noise attenuation level in an H, or l2 - l, sense with respect to all energy-bounded noise disturbances for all admissible uncertainties and time delays. Both delay-dependent and independent approaches are developed by using linear matrix inequality (LMI) techniques, which are applicable to systems either with or without a priori information on the size of delays. [source]