New Discrete-Time ZNN Models for Least-Squares Solution of Dynamic Linear Equation System With Time-Varying Rank-Deficient Coefficient

• Published in: IEEE transaction on neural networks and learning systems 2018-11, Vol.29 (11), p.5767-5776
• Main Authors: Qiu, Binbin, Zhang, Yunong, Yang, Zhi
• Format: Article
• Language: English
• Subjects: Analytical models; Coefficients; Computational modeling; Computer applications; Dynamic linear equation system; Finite difference method; Formulas (mathematics); Heuristic algorithms; Integrated circuit modeling; Least squares method; least-squares solution; Linear equations; Mathematical model; Mathematical models; Neural networks; new-type discrete-time Zhang neural network (DTZNN) models; Numerical differentiation; Numerical models; six-instant <italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">g -cube finite difference (6I<italic xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">g CFD) formula; time-varying rank-deficient coefficient; Time-varying systems
• ISSN: 2162-237X; 2162-2388
• DOI: 10.1109/TNNLS.2018.2805810

In this brief, a new one-step-ahead numerical differentiation rule called six-instant g -cube finite difference (6I g CFD) formula is proposed for the first-order derivative approximation with higher precision than existing finite difference formulas (i.e., Euler and Taylor types). Subsequently, by exploiting the proposed 6I g CFD formula to discretize the continuous-time Zhang neural network model, two new-type discrete-time ZNN (DTZNN) models, namely, new-type DTZNNK and DTZNNU models, are designed and generalized to compute the least-squares solution of dynamic linear equation system with time-varying rank-deficient coefficient in real time, which is quite different from the existing ZNN-related studies on solving continuous-time and discrete-time (dynamic or static) linear equation systems in the context of full-rank coefficients. Specifically, the corresponding dynamic normal equation system, of which the solution exactly corresponds to the least-squares solution of dynamic linear equation system, is elegantly introduced to solve such a rank-deficient least-squares problem efficiently and accurately. Theoretical analyses show that the maximal steady-state residual errors of the two new-type DTZNN models have an O(g^{4}) pattern, where g denotes the sampling gap. Comparative numerical experimental results further substantiate the superior computational performance of the new-type DTZNN models to solve the rank-deficient least-squares problem of dynamic linear equation systems.