A Behavioral Approach to GNSS Positioning and DOP Determination

Abstract- In this paper, a behavioral framework is proposed to solve dynamic GNSS positioning problems, which in the mean time may also provide a way to determine the DOP (dilution of precision) of a moving object. The concept of beha-vioral framework was first proposed by Jan C. Willems in a series of papers (J.C. Willems, "From time series to linear system- part I, II, and III, " Automatica Vol. 22, 1986.) as a generic tool for mathematical modeling of dynamical systems. In the proposed approach, a GNSS positioning problem is firstly described by a kernel representation and then the problem can be solved by a structured total least-squares (STLS) algorithm. STLS algorithm is a modified version of the traditional total least-squares (TLS) method. It can be shown that the STLS algorithm is able to provide better performance than the TLS al-gorithm for the problems that possess a particular structure. In the case of the present paper, it is shown that the GNSS posi-tioning problem has a Hankel structure (i.e., the geometric matrix of the pseudorange equation is Hankel), therefore the prob-lem can be solved by an STLS algorithm subject to a Hankel structure. On the other hand, a formula for calculating DOP of a dynamic positioning problem is also provided. The proposed method is able to calculate the DOP value for multiple time epochs, in compared with the conventional DOP formula which can only be derived at a single time epoch, so as to reflect the inter-dependency between consecutive time epochs