Augmented Nonlinear Least Squares Estimation With Applications to Localization

Least squares (LS) estimation is simple yet effective for parameter estimation. Most real-world problems are nonlinear. In practice, nonlinear LS problems are linearized first and then solved by a linear LS estimator. However, since these methods solve a nonlinear LS problem by a linear way, there is room for improvement only if estimators being nonlinear in the original measurement. In this article, an LS approach is proposed for nonlinear LS problems, along with theoretical analysis and justification. First, for a general nonlinear measurement model with any form of noise, a unified linearization (UL) approach is proposed under certain assumptions and a linearized least squares (LLS) estimator is defined based on the UL model. Second, an augmented nonlinear least squares (ANLS) estimator is proposed based on LLS estimation. In the ANLS estimator, the original measurement is augmented by its nonlinear conversion intended to further utilize the nonlinear measurement. Moreover, we prove that the ANLS estimator outperforms LLS estimation-based estimators. Finally, the UL and the ANLS estimator are applied to source localization with time difference of arrival and/or frequency difference of arrival measurement, and a two-step ANLS localization algorithm is developed. The performance of the proposed algorithm is evaluated via simulation of some typical localization scenarios.