The computation of matrix pseudoinverses is a recurrent requirement across various scientific computing and engineering domains. The prevailing models for matrix pseudoinverse typically operate under the assumption of a noise-free solution process or presume that any noise present has been effectively addressed prior to computation. However, the concurrent real-time computation of time-varying matrix pseudoinverses holds significant practical utility, while the preemptive preprocessing for noise elimination or reduction may impose supplementary computational overheads on real-time implementations. Different from previous models for solving the pseudoinverse of time-varying matrices, in this paper, a model for solving the pseudoinverse of time-varying matrices using a double-integral structure, called Double-Integral-Enhanced Zeroing Neural Network (DIEZNN) model, is proposed and investigated, which is capable of solving time-varying matrix pseudoinverse while efficiently eliminating the negative effects of linear noise perturbations. The experimental results show that in the presence of linear noise, the DIEZNN model demonstrates better noise suppression performance compared to both the original zeroing neural network model and the Zeroing Neural Network (ZNN) model enhanced with a Li-type activation function. In addition, these models are applied to the control of chaotic system of controllable permanent magnet synchronous motor, which further verifies the superiority of DIEZNN in engineering application.