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Zeroing neural network (ZNN) is an effective method to calculate time-varying problems. However, the ZNN and its extensions separately addressed the robustness and the convergence. To simultaneously promote the robustness and finite-time convergence, a nonlinearly activated integration-enhanced ZNN (NIEZNN) model based on a coalescent activation function (C-AF) has been designed for solving the time-varying Sylvester equation in various noise situations. The C-AF with an optimized structure is convenient for simulations and calculations, which promotes NIEZNN accelerates convergence speed without remarkable efficiency loss. The robustness and the finite-time convergence of the NIEZNN model have been proved in theoretical analyses. Furthermore, the upper bounds of convergence time of the NIEZNN model and the noise-attached NIEZNN model have been deduced in theory. At last, numerical comparative results and the application to mobile manipulator have validated the efficiency and superiority of the NIEZNN model based on the designed coalescent activation function.

The solution of a complex-valued matrix pseudoinverse is one of the key steps in various science and engineering fields. Owing to its important roles, researchers had put forward many related algorithms. With the development of research, a time-varying matrix pseudoinverse received more attention than a time-invarying one, as we know that a zeroing neural network (ZNN) is an efficient method to calculate a pseudoinverse of a complex-valued time-varying matrix. Due to the initial ZNN (IZNN) and its extensions lacking a mechanism to deal with both convergence and robustness, that is, most existing research on ZNN models only studied the convergence and robustness, respectively. In order to simultaneously improve the double features (i.e., convergence and robustness) of ZNN in solving a complex-valued time-varying pseudoinverse, this paper puts forward a double features ZNN (DFZNN) model by adopting a specially designed time-varying parameter and a novel nonlinear activation function. Moreover, two nonlinear activation types of complex number are investigated. The global convergence, predefined time convergence, and robustness are proven in theory, and the upper bound of the predefined convergence time is formulated exactly. The results of the numerical simulation verify the theoretical proof, in contrast to the existing complex-valued ZNN models, the DFZNN model has shorter predefined convergence time in a zero noise state, and enhances robustness in different noise states. Both the theoretical and the empirical results show that the DFZNN model has better ability in solving a time-varying complex-valued matrix pseudoinverse. Finally, the proposed DFZNN model is used to track the trajectory of a manipulator, which further verifies the reliability of the model.