Abstract-In this paper, a Moving Horizon Estimator with pre-estimation (MHE-PE) is proposed for discrete-time nonlinear systems under bounded noise. While the classical Moving Horizon Estimator (MHE) compensates for model errors by estimating the process noise sequence over the horizon via optimization, the MHE-PE does it using an auxiliary estimator. The MHE-PE is shown to require significantly less computation time compared to the MHE, while providing the same order of magnitude of estimation errors. The stability of the estimation errors of the MHE-PE is also proven and an upper bound on its estimation errors is derived. Performances of the MHE-PE is illustrated via a simulation example of pressure estimation in a gas-phase reversible reaction.

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Trajectory estimation during atmospheric re entry of ballistic objects such as space debris is a very complex problem due to high variations of their ballistic coefficients. In general, the characteristics of the tracked object are not accurately known and an assumption on the dynamics of the ballistic coefficient has to be made in the estimation model. The designed estimator must hence prove to be robust enough to such model uncertainties, and to bad initialization if no good prior information on the initial position, velocity, and the characteristics of the object is available. Robustness of a Moving Horizon Estimator (MHE) is studied in this paper and compared to several other filters: Extended Kalman Filter (EKF), Unscented Kalman Filter (UKF) and Regularized Particle Filter (RPF). The performances of the filters are analysed in terms of convergence percentage, accuracy, robustness to bad initialization, and computation time, via Monte Carlo simulations of trajectories of several space debris. Contrary to the classical tracking problem of supersonic ballistic objects for which RPF has been proven to be efficient in the literature, it is shown that its performance are overcome by MHE for the space debris tracking problem considered in this paper.

Space debris tracking during atmospheric re-entry is a very complex problem due to high variations with time of the ballistic coefficient. The nature of these variations is generally unknown and an assumption has to be made in the estimation model which can result in high model errors. An estimator which is robust against model errors is therefore required. In previous work done by the authors, Moving Horizon Estimation (MHE) has been shown to outperform other classical nonlinear estimators in terms of accuracy and robustness against poor initialization for a simplified 1D case of space debris tracking during the re-entry. However, the large computation time of the MHE prevents its implementation for the 3D cases. Recently, the Moving Horizon Estimation with Pre-Estimation (MHE-PE) which requires much less computation time than the classical MHE while keeping its accuracy and robustness has been proposed. This paper therefore implements the MHE-PE to solve the 3D space debris tracking problem during the re-entry. Its performances are compared to some classical nonlinear estimators in terms of non-divergence percentage, accuracy and computation time through Monte Carlo simulations.

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