## Contribution de Vivien MALLET & Sergiy ZHUK:

*Reduced-order minimax state estimation*

*Reduced-order minimax state estimation*

*We introduce a new filtering approach for high-dimensional numerical systems. It is based on a reduction of the high-dimensional system to some low-dimensional Differential-Algebraic Equation (DAE), and on the application of linear minimax filtering to the resulting DAE. In the minimax approach, the model error and the observational error can be deterministic or stochastic, and of any shape provided they have bounded energy. In practice, it is assumed that the errors belong to some ellipsoid. Based on this information, the algorithm describes a reachability set that contains all states consistent with the model, the observations and the assumptions on the errors. In the non-reduced version of the filter, the estimator of the state is taken as Chebyshev center of the reachability set. Note that the non-reduced version of the filter coincides with the Kalman filter provided there is no systematic error and the description of the ellipsoid is reinterpreted in terms of variances. The non-reduced filter is intractable because it involves solving a high-dimensional matrix Riccati equation. The formulation of the filter for DAE allows to estimate only a part of the state or a projection of the state onto some subspace (e.g., computed from a proper orthogonal decomposition). The reduced-model error is decomposed into the projection of the model error onto the subspace and a commutation error between the projection operator and the model. The reachability set is provided in the subspace. Computing the reachability set and the estimator for the reduced state is tractable whenever the dimension N of the reduced space is low enough--the algorithm involves N calls to the tangent linear model. The performance of the filter will be illustrated in the talk, with a large quadratic model or with full air quality simulations if available at the time.*