Neural network-based discrete-time Z-type model of high accuracy in noisy environments for solving dynamic system of linear equations

To solve dynamic system of linear equations with square or rectangular system matrices in real time, a discrete-time Z-type model based on neural network is proposed and investigated. It is developed from and studied with the aid of a unified continuous-time Z-type model. Note that the framework of such a unified continuous-time Z-type model is generic and has a wide range of applications, such as robotic redundancy resolution with quadratic programming formulations. To do so, a one-step-ahead numerical differentiation formula and its optimal sampling-gap rule in noisy environments are presented. We compare the Z-type model extensively with E-type and N-type models. Theoretical results on stability and convergence are provided which show that the maximal steady-state residual errors of the Z-type, E-type and N-type models have orders \(O(\tau ^3)\), \(O(\tau ^2)\) and \(O(\tau )\), respectively, where \(\tau \) is the sampling gap. We also prove that the residual error of any static method that does not exploit the time-derivative information of a time-dependent system of linear equations has order \(O(\tau )\) when applied to solve discrete real-time dynamic system of linear equations. Finally, several illustrative numerical experiments in noisy environments as well as two application examples to the inverse-kinematics control of redundant manipulators are provided and illustrated. Our analysis substantiates the efficacy of the Z-type model for solving the dynamic system of linear equations in real time.