Modified gradient neural networks for solving the time-varying Sylvester equation with adaptive coefficients and elimination of matrix inversion

Abstract In scientific and engineering fields, the solutions to many problems can be transformed into finding the solutions to Sylvester equation, for which various computational methods (e.g., recurrent neural network, RNN) have been presented and investigated. RNN models are frequently used to solve computational problems due to the prevalent exploitation of the gradient-based RNN. However, the overlong convergent time and the too large residual error restrict the widespread applications of the RNN model in solving time-varying problems. Further, a special type of RNN named zeroing neural network (ZNN) is able to solve the time-varying Sylvester equation, which breaks the limitations mentioned above, but fails to handle complex time-varying problems owing to the sharp increment of the calculated amount in matrix inversion involved. To remedy the limitation, a modified gradient-based RNN (MGRNN) model is proposed to generate more accurate computational solutions with less convergent time and adaptive coefficients for solving the time-varying Sylvester equation, which replaces the matrix inversion problem with the matrix transposition problem. Besides, theoretical analyses and mathematical verifications are presented to validate the efficiency and superiority of the proposed MGRNN model compared with the traditional gradient-based recurrent neural network (GRNN) and ZNN models. Furthermore, simulation experiments are conducted to substantiate the properties of the newly proposed MGRNN model for solving the time-varying Sylvester equation.