The so-called zeroing neural network (ZNN) is an effective recurrent neural network for solving dynamic problems including the dynamic nonlinear equations. There exist numerous unperturbed ZNN models that can converge to the theoretical solution of solvable nonlinear equations in infinity long or finite time. However, when these ZNN models are perturbed by external disturbances, the convergence performance would be dramatically deteriorated. To overcome this issue, this paper for the first time proposes a finite-time convergent ZNN with the noise-rejection capability to endure disturbances and solve dynamic nonlinear equations in finite time. In theory, the finite-time convergence and noise-rejection properties of the finite-time convergent and noise-rejection ZNN (FTNRZNN) are rigorously proved. For potential digital hardware realization, the discrete form of the FTNRZNN model is established based on a recently developed five-step finite difference rule to guarantee a high computational accuracy. The numerical results demonstrate that the discrete-time FTNRZNN can reject constant external noises. When perturbed by dynamic bounded or unbounded linear noises, the discrete-time FTNRZNN achieves the smallest steady-state errors in comparison with those generated by other discrete-time ZNN models that have no or limited ability to handle these noises. Discrete models of the FTNRZNN and the other ZNNs are comparatively applied to redundancy resolution of a robotic arm with superior positioning accuracy of the FTNRZNN verified.

Aiming at time-variant problems solving, a special type of recurrent neural networks, termed zeroing neural network (ZNN), has been proposed, developed, and validated since 2001. Although equality-constrained time-variant quadratic programming (TVQP) has been well solved using the ZNN approach, TVQP problems with inequality constraints involved have not been satisfactorily handled by the existing ZNN models. To overcome this issue, this paper designs a ZNN model with exponential convergence for solving equality- and inequality-constrained TVQP problems. Considering a fast convergence is preferred in some time-critical applications in practice, a predefined-time stabilizer is for the first time utilized to endow the ZNN model with predefined-time convergence, leading to a predefined-time convergent ZNN (PTCZNN) model that exhibits an antecedently- and explicitly-defined convergence time. Theoretical analysis is performed with the convergence of the two ZNN models including the predefined-time convergence of the PTCZNN model rigorously proved. Validations are comparatively conducted to verify the effectiveness and superiority of the PTCZNN model in terms of convergence performance. To demonstrate the potential applications, the PTCZNN model is applied to image fusion and kinematic control of two robotic arms with joint limits considered. The efficacy and applicability of the PTCZNN model are validated by the illustrative examples. This is the first time to develop a ZNN model working as a quadratic programming solver that is applicable to kinematic control of robotic arms with joint constraints handled since the emergence of ZNNs.

Time-variant linear matrix equations (TVLMEs) are ubiquitous in engineering. To solve TVLMEs, various zeroing neural network (ZNN) models have been developed. These ZNNs globally converge to the solution of TVLMEs either in infinity long time or in finite time. However, even the convergence time of a finite-time convergent ZNN is implicit and closely dependent on the initial condition of a problem. This may reduce its applicability to time-critical applications in practice. To overcome this problem, this paper for the first time accelerates a ZNN to fixed-time convergence using a novel activation function. The convergence time of the proposed ZNN can be antecedently defined as an explicit parameter. Theoretically, its fixed-time convergence and robustness properties are rigorously proved. Comparative numerical results substantiate the superior convergence and robustness performance of the fixed-time convergent ZNN for TVLMEs solving. Additionally, the fixed-time convergent ZNN is applied to motion planning of a redundant robotic arm.

Nonlinear optimization problems with dynamical parameters are widely arising in many practical scientific and engineering applications, and various computational models are presented for solving them under the hypothesis of short-time invariance. To eliminate the large lagging error in the solution of the inherently dynamic nonlinear optimization problem, the only way is to estimate the future unknown information by using the present and previous data during the solving process, which is termed the future dynamic nonlinear optimization (FDNO) problem. In this paper, to suppress noises and improve the accuracy in solving FDNO problems, a novel noise-tolerant neural (NTN) algorithm based on zeroing neural dynamics is proposed and investigated. In addition, for reducing algorithm complexity, the quasi-Newton Broyden-Fletcher-Goldfarb-Shanno (BFGS) method is employed to eliminate the intensively computational burden for matrix inversion, termed NTN-BFGS algorithm. Moreover, theoretical analyses are conducted, which show that the proposed algorithms are able to globally converge to a tiny error bound with or without the pollution of noises. Finally, numerical experiments are conducted to validate the superiority of the proposed NTN and NTN-BFGS algorithms for the online solution of FDNO problems.