New Multi-Step Iterative Methods for Solving Systems of Nonlinear Equations and Their Application on GNSS Pseudorange Equations

• Main Authors: Kalyanasundaram Madhu, Arul Elango, René Jr Landry, Mo’tassem Al-arydah
• Format: Article
• Language: English
• Published: MDPI AG 2020-10-01
• Series: Sensors
• Subjects: navigation equations; higher-order; convergence order; multi-step method; GNSS; GDOP
• Online Access: https://www.mdpi.com/1424-8220/20/21/5976

A two-step fifth and a multi-step <inline-formula><math display="inline"><semantics><mrow><mn>5</mn><mo>+</mo><mn>3</mn><mi>r</mi></mrow></semantics></math></inline-formula> order iterative method are derived, <inline-formula><math display="inline"><semantics><mrow><mi>r</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula> for finding the solution of system of nonlinear equations. The new two-step fifth order method requires two functions, two first order derivatives, and the multi-step methods needs a additional function per step. The performance of this method has been tested with finding solutions to several test problems then applied to solving pseudorange nonlinear equations on Global Navigation Satellite Signal (GNSS). To solve the problem, at least four satellite’s measurements are needed to locate the user position and receiver time offset. In this work, a number of satellites from 4 to 8 are considered such that the number of equations is more than the number of unknown variables to calculate the user position. Moreover, the Geometrical Dilution of Precision (GDOP) values are computed based on the satellite selection algorithm (fuzzy logic method) which could be able to bring the best suitable combination of satellites. We have restricted the number of satellites to 4 to 6 for solving the pseudorange equations to get better GDOP value even after increasing the number of satellites beyond six also yields a 0.4075 GDOP value. Actually, the conventional methods utilized in the position calculation module of the GNSS receiver typically converge with six iterations for finding the user position whereas the proposed method takes only three iterations which really decreases the computation time which provide quicker position calculation. A practical study was done to evaluate the computation efficiency index (CE) and efficiency index (IE) of the new model. From the simulation outcomes, it has been noted that the new method is more efficient and converges 33% faster than the conventional iterative methods with good accuracy of 92%.