ℓ 1 ${\ell }_1$ norm-based recursive estimation for non-linear systems with non-Gaussian noises

This study addresses the state estimation problem of discrete-time non-linear stochastic systems with non-Gaussian noises, particularly impulsive noises. Instead of minimizing the mean square error of the state estimate, which tends to excessively focus on outliers caused by non-Gaussian noises, the ${\ell }_1$ norm-based non-linear recursive filter (L1KF) is put forward in this paper. Here, minimizing the ${\ell }_1$ norm of model errors is actually to pursue the minimum sum of absolute values of all errors, which is equitable to all model errors rather than paying much attention on outliers. To further improve estimation accuracy, a recursive nonlinear smoother (L1KS) is proposed, based on minimizing the ${\ell }_1$ norm of model errors. The proposed ${\ell }_1$ norm-based filter and smoother are implemented using unscented transformation for statistical linear regression applied to nonlinear models. Additionally, the computational complexity of the proposed method is analysed. Simulation results of tracking a radar target with impulsive noises demonstrate the effectiveness and robustness of the proposed estimator.