A detectability criterion and data assimilation for non-linear differential equations

In this paper we propose a new sequential data assimilation method for non-linear ordinary differential equations with compact state space. The method is designed so that the Lyapunov exponents of the corresponding estimation error dynamics are negative, i.e. the estimation error decays exponentially fast. The latter is shown to be the case for generic regular flow maps if and only if the observation matrix H satisfies detectability conditions: the rank of H must be at least as great as the number of nonnegative Lyapunov exponents of the underlying attractor. Numerical experiments illustrate the exponential convergence of the method and the sharpness of the theory for the case of Lorenz96 and Burgers equations with incomplete and noisy observations

Infinite horizon control and minimax observer design for linear DAEs

In this paper we construct an infinite horizon minimax state observer for a linear stationary differential-algebraic equation (DAE) with uncertain but bounded input and noisy output. We do not assume regularity or existence of a (unique) solution for any initial state of the DAE. Our approach is based on a generalization of Kalman's duality principle. The latter allows us to transform minimax state estimation problem into a dual control problem for the adjoint DAE: the state estimate in the original problem becomes the control input for the dual problem and the cost function of the latter is, in fact, the worst-case estimation error. Using geometric control theory, we construct an optimal control in the feed-back form and represent it as an output of a stable LTI system. The latter gives the minimax state estimator. In addition, we obtain a solution of infinite-horizon linear quadratic optimal control problem for DAEs.Comment: This is an extended version of the paper which is to appear in the proceedings of the 52nd IEEE Conference on Decision and Control, Florence, Italy, December 10-13, 201

Reduced minimax state estimation

A reduced minimax state estimation approach is proposed for high-dimensional models. It is based on the reduction of the ordinary differential equation with high state space dimension to the low-dimensional Differential-Algebraic Equation (DAE) and on the subsequent application of the minimax state estimation to the resulting DAE. The DAE is composed of a reduced state equation and of a linear algebraic constraint. The later allows to bound linear combinations of the reduced state's components in order to prevent possible instabilities, originating from the model reduction. The method is robust as it can handle model and observational errors in any shape, provided they are bounded. We derive a minimax algorithm adapted to computations in high-dimension. It allows to compute both the state estimate and the reachability set in the reduced space.Nous introduisons une méthode de filtrage dédiée aux modèles de grande dimension et fondée sur une approche minimax réduite. La méthode repose sur une reformulation du problème de grande dimension en une équation différentielle algébrique de petite dimension sur laquelle un filtre minimax est appliqué. L'équation différentielle algébrique se décompose en une équation sur un état réduit et une contrainte algébrique linéaire. Cette dernier permet de borner des combinaisons linéaires des composantes du vecteur d'état réduit, ce qui élimine des instabilités potentiellement induites par la réduction. La méthode est robuste dans le sens où elle permet de traiter n'importe quelle erreur modèle et n'importe quelle erreur d'observation, pourvu que ces dernières soient bornées. Nous proposons une forme algorithmique qui permet d'appliquer le filtre à des modèles de grande dimension. L'algorithme calcule l'estimateur minimax ainsi que l'ensemble des états admissibles