New 5-Step Discrete-Time Zeroing Neuronet for Time-Dependent Matrix Square Root Finding

In this paper, for the purpose of finding square root of a time-dependent matrix, a new discrete-time zeroing neuronet (DTZN) is proposed. Firstly, the problem of square root finding of time-dependent matrix is formulated. Then, an explicit continuous-time zeroing neuronet (CTZN) is derived from the problem formulation equation via vectorization technique. Furthermore, based on Taylor expansion, we present a 5-Step Zhang time-discretization (ZTD) formula. The ZTD is used to approximate the 1st-order derivative of the object, of which the truncational error is proportional to the cube of the sampling period. Finally, the 5-Step DTZN for solving the square root of a time-dependent matrix is acquired by using the presented 5- Step ZTD formula to discretize the CTZN. Theoretical analyses shown stable and convergent performance of the proposed 5- Step DTZN for solving the square root of a time-dependent matrix. Computer experiments results present the stability and convergence of the obtained DTZN for solving square root of time-dependent matrix with the maximum steady-state residual errors proportional to the fourth power of sampling period. By comparison with the DTZNs using Euler formula and the previous 5-Step discretization formula, the proposed 5-Step DTZN has an advantage in residual error. In addition, the influences of step size and sampling period are illustrated by computer experiments results.