Nonlinear estimation based on conversion-sample optimization

For nonlinear estimation, the linear minimum mean square error (LMMSE) estimator using the original measurement is the best of all estimators linear in the measurement. Recently, it has been proven that an LMMSE estimator using a stacked measurement vector composed of the original measurement and its nonlinear conversion (transformation or function) outperforms the original LMMSE estimator. However, to find an optimal conversion for various nonlinear problems is very difficult, because it involves functional optimization. Recognizing that the conversion affects estimation performance only through conversion-dependent quantities needed by the augmented LMMSE-based estimator, this paper proposes an optimized conversion-sample filter (OCF), which optimizes all these quantities to optimize estimation performance. These quantities are much easier to obtain than the optimal conversion itself. Using deterministic sampling, obtaining these quantities amounts to determining the converted values at both the deterministic sample points and the measurement value. In the OCF, these values are obtained optimally and analytically by solving a constrained Rayleigh quotient problem. The proposed OCF has a simple and analytical form and can be generally applied to nonlinear estimation. Simulation results demonstrate the effectiveness of the proposed OCF compared with some popular nonlinear estimators, including several LMMSE-based filters, a recently-proposed uncorrelated conversion based filter, and a particle filter.