The work concerns the "observability problem"--the reconstruction of a dynamic process's full state from a partially measured state-- for nonlinear dynamic systems. The Extended Kalman Filter (EKF) is a widely-used observer for such nonlinear systems. However it suffers from a lack of theoretical justifications and displays poor performance when the estimated state is far from the real state, e.g. due to large perturbations, a poor initial state estimate, etc. . . We propose a solution to these problems, the Adaptive High-Gain (EKF). Observability theory reveals the existence of special representations characterizing nonlinear systems having the observability property. Such representations are called observability normal forms. A EKF variant based on the usage of a single scalar parameter, combined with an observability normal form, leads to an observer, the High-Gain EKF, with improved performance when the estimated state is far from the actual state. Its convergence for any initial estimated state is proven. Unfortunately, and contrary to the EKF, this latter observer is very sensitive to measurement noise. Our observer combines the behaviors of the EKF and of the high-gain EKF. Our aim is to take advantage of both efficiency with respect to noise smoothing and reactivity to large estimation errors. In order to achieve this, the parameter that is the heart of the high-gain technique is made adaptive. Voila, the Adaptive High-Gain EKF. A measure of the quality of the estimation is needed in order to drive the adaptation. We propose such an index and prove the relevance of its usage. We provide a proof of convergence for the resulting observer, and the final algorithm is demonstrated via both simulations and a real-time implementation. Finally, extensions to multiple output and to continuous-discrete systems are given.

Le travail porte sur la problématique de l’observation des systèmes — la reconstruction de l’état complet d’un système dynamique à partir d'une mesure partielle de cet état. Nous considérons spécifiquement les systèmes non linéaires. Le filtre de Kalman étendu (EKF) est l’un des observateurs les plus utilisés à cette fin. Il souffre cependant d’une performance moindre lorsque l'état estimé n’est pas dans un voisinage de l'état réel. La convergence de l’observateur dans ce cas n’est pas prouvée. Nous proposons une solution à ce problème : l’EKF à grand gain adaptatif. La théorie de l’observabilité fait apparaître l’existence de représentations caractérisant les systèmes dit observables. C’est la forme normale d’observabilité. L’EKF à grand gain est une variante de l’EKF que l’on construit à base d’un paramètre scalaire. La convergence de cet observateur pour un système sous sa forme normale d’observabilité est démontrée pour toute erreur d’estimation initiale. Cependant, contrairement à l’EKF, cet algorithme est très sensible au bruit de mesure. Notre objectif est de combiner l’efficacit´e de l’EKF en termes de lissage du bruit, et la r´eactivit´e de l’EKF grand-gain face aux erreurs d’estimation. Afin de parvenir à ce résultat nous rendons adaptatif le paramètre central de la méthode grand gain. Ainsi est constitué l’EKF à grand gain adaptatif. Le processus d’adaptation doit être guidé par une mesure de la qualité de l’estimation. Nous proposons un tel indice et prouvons sa pertinence. Nous établissons une preuve de la convergence de notre observateur, puis nous l’illustrons à l’aide d’une série de simulations ainsi qu’une implémentation en temps réel dur. Enfin nous proposons des extensions au résultat initial : dans le cas de systèmes multi-sorties et dans le cas continu-discret. / The work concerns the “observability problem”—the reconstruction of a dynamic process’s full state from a partially measured state— for nonlinear dynamic systems. The Extended Kalman Filter (EKF) is a widely-used observer for such nonlinear systems. However it suffers from a lack of theoretical justifications and displays poor performance when the estimated state is far from the real state, e.g. due to large perturbations, a poor initial state estimate, etc. . . We propose a solution to these problems, the Adaptive High-Gain (EKF). Observability theory reveals the existence of special representations characterizing nonlinear systems having the observability property. Such representations are called observability normal forms. A EKF variant based on the usage of a single scalar parameter, combined with an observability normal form, leads to an observer, the High-Gain EKF, with improved performance when the estimated state is far from the actual state. Its convergence for any initial estimated state is proven. Unfortunately, and contrary to the EKF, this latter observer is very sensitive to measurement noise. Our observer combines the behaviors of the EKF and of the high-gain EKF. Our aim is to take advantage of both efficiency with respect to noise smoothing and reactivity to large estimation errors. In order to achieve this, the parameter that is the heart of the high-gain technique is made adaptive. Voila, the Adaptive High-Gain EKF. A measure of the quality of the estimation is needed in order to drive the adaptation. We propose such an index and prove the relevance of its usage. We provide a proof of convergence for the resulting observer, and the final algorithm is demonstrated via both simulations and a real-time implementation. Finally, extensions to multiple output and to continuous-discrete systems are given.