State Determination in EKF using Singular Value Decomposition for PPP Integrity Monitor System

Precise Point Positioning (PPP) utilizes temporal relations and detailed error models to estimate user positions from code and carrier phase pseudorange measurements that can achieve centimeter for static and decimeter level of accuracy for dynamic receivers. We built a PPP integrity monitor system using measurements from multiple ground-based GNSS receivers and formulated a centralized fault-tolerant PPP filter that can detect multiple satellite clock faults and ephemeris faults from multiple satellites simultaneously without applying Solution Separation on the satellites. This filter is capable of providing integrity to the PPP users. In our prior work, we used a standard Extended-Kalman Filter (EKF) to perform the estimation. However, due to the unstable nature of the matrix inversion in the computation of the Kalman Gain, the user range error estimates in the centralized filter may sometimes deteriorate or even diverge because of the numerical issues in computing Kalman Gain in computer. In this paper, in order to address the issue of numerical instability, we reformulate the Discrete-Time EKF as a least square optimization problem, where the cost function is a quadratic. We minimize the cost function with Singular Value Decomposition (SVD) where the unobservable states and least significant measurements can be identified and excluded with the eigen values from SVD. The newly formed SVD-transformed EKF is applicable to any estimation problem that utilizes EKF, and in this paper it is used for estimating the user range error induced by satellite clock fault and ephemeris fault.