In this brief, a new one-step-ahead numerical differentiation rule called six-instant g-cube finite difference (6IgCFD) 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 6IgCFD 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(g⁴) 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.

This research first proposes the general expression of Zhang et al. discretization (ZeaD) formulas to provide an effective general framework for finding various ZeaD formulas by the idea of high-order derivative simultaneous elimination. Then, to solve the problem of future equality-constrained nonlinear optimization (ECNO) with various noises, a specific ZeaD formula originating from the general ZeaD formula is further studied for the discretization of a noise-perturbed continuous-time advanced zeroing neurodynamic model. Subsequently, the resulting noise-perturbed discrete-time advanced zeroing neurodynamic (NP-DTAZN) algorithm is proposed for the real-time solution to the future ECNO problem with various noises suppressed simultaneously. Moreover, theoretical and numerical results are presented to show the convergence and precision of the proposed NP-DTAZN algorithm in the perturbation of various noises. Finally, comparative numerical and physical experiments based on a Kinova JACO² robot manipulator are conducted to further substantiate the efficacy, superiority, and practicability of the proposed NP-DTAZN algorithm for solving the future ECNO problem with various noises.

In this paper, a high-precision general discretization formula using six time instants is first proposed to approximate the first-order derivative. Then, such a formula is studied to discretize two continuous-time neurodynamic models, both of which are derived by applying the neurodynamic approaches based on neural networks (i.e., zeroing neurodynamics and gradient neurodynamics). Originating from the general six-instant discretization (6ID) formula, a specific 6ID formula is further presented. Subsequently, two new discrete-time neurodynamic algorithms, i.e., 6ID-type discrete-time zeroing neurodynamic (DTZN) algorithm and 6ID-type discrete-time gradient neurodynamic (DTGN) algorithm, are proposed and investigated for online future matrix inversion (OFMI). In addition to analyzing the usual nonsingular situation of the coefficient, this paper investigates the sometimes-singular situation of the coefficient for OFMI. Finally, two illustrative numerical examples, including an application to the inverse-kinematic control of a PUMA560 robot manipulator, are provided to show respective characteristics and advantages of the proposed 6ID-type DTZN and DTGN algorithms for OFMI in different situations, where the coefficient matrix to be inverted is always-nonsingular or sometimes-singular during time evolution.

In this article, a novel and challenging problem called future different-level system of nonlinear inequality and linear equation (FDLSNILE) is proposed and investigated. To solve FDLSNILE, the corresponding continuous different-level system of nonlinear inequality and linear equation (CDLSNILE) is first analyzed, and then, a continuous combined zeroing neural network (CCZNN) model for solving CDLSNILE is proposed. To obtain a discrete combined zeroing neural network (DCZNN) model for solving FDLSNILE, a high-precision general 7-instant Zhang et al. discretization (ZeaD) formula for the first-order time derivative approximation is proposed. Furthermore, by applying the general 7-instant ZeaD formula to discretize the CCZNN model, a general 7-instant DCZNN (7IDCZNN) model is thus proposed for solving FDLSNILE. For comparison, by using three conventional ZeaD formulas, three conventional DCZNN models are also developed. Meanwhile, theoretical analyses and results guarantee the efficacy and superiority of the general 7IDCZNN model compared with the other three conventional DCZNN models for solving FDLSNILE. Finally, several comparative numerical experiments, including the motion control of a 5-link redundant manipulator, are provided to substantiate the efficacy and superiority of the general 7-instant ZeaD formula and the corresponding 7IDCZNN model.

In this paper, a four-point one-step-ahead finite-difference formula is presented, which obtains higher computational precision in approximating the first-order derivative. Then, the formula is used for the discretization of the continuous-time Zhang neural network (CTZNN), and it can greatly overcome the limitation of the conventional formulas in CTZNN discretization. Based on this formula, a new-type discrete-time Zhang neural network (DTZNN) model is proposed and investigated for time-variant matrix pseudoinversion. Numerical experiments further validate the feasibility, effectiveness, and superiority of the proposed new-type DTZNN model for solving the time-variant matrix pseudoinversion. Moreover, the proposed new-type DTZNN model is applied to the control of a robot manipulator. Physical experiment performed on a four-link planar robot manipulator is presented to demonstrate physical realizability and effectiveness of the proposed new-type DTZNN model.

In this paper, we derive the general explicit linear four-step (ELFS) rule with fifth-order precision systematically, together with a group of specific ELFS rules provided. Afterwards, we formulate and investigate a new and challenging discrete-time dynamic problem with relatively complex structure and future unknownness, which is simply termed future system of bounded inequalities and nonlinear equations (SBINE). With the aid of the general ELFS rule, the general ELFS-type discretized zeroing neural network (DZNN) model is proposed to solve the future SBINE. Moreover, theoretical and numerical results are presented to show the validity and high precision of the proposed general ELFS-type DZNN model. Finally, comparative numerical experiments based on a wheeled mobile robot containing several additional constraints are further performed to substantiate the applicability, validity, and superiority of the proposed general ELFS-type DZNN model.