Discrete generalized-Sylvester matrix equation solved by RNN with a novel direct discretization numerical method - Shuai Li (Numerical Algorithms, 2022)
In the fields of artificial intelligence and control engineering, generalized-Sylvester matrix equation is considered as an important mathematic problem, and its solving process is usually viewed as a challenge that deserves particular attention. In this paper, a creative discrete-form recurrent neural network (RNN) model is developed, analyzed and investigated for solving discrete-form time-variant generalized-Sylvester matrix equation (DF-TV-GSME), which is derived by a direct discretization numerical method. Specifically, first of all, DF-TV-GSME, which includes the well-known Lyapunov matrix equation and Sylvester matrix equation, is presented as the target problem of this research. Secondly, for solving such problem, different from the traditional discrete-form RNN design philosophy, second-order Taylor expansion is applied to derive the discrete-form RNN model. This creative process avoids involving the continuous time-variant environment and continuous-form model. Then, theoretical properties analyses are presented, which present the convergence and precision of the discrete-form RNN model. Abundant numerical experiments are further carried out with different perspectives of DF-TV-GSME, which further confirm the excellent properties of discrete-form RNN model.