Zeroing Neural Networks, an Introduction to, a Survey of, and Predictive Computations for Discretized Time-varying Matrix Problems

This paper is designed to increase the knowledge and computational know-how for time-varying matrix problems and Zeroing Neural Networks in the West. Zeroing Neural Networks (ZNN) were invented for time-varying matrix problems 20 years ago in China and almost all of their advances have been made in and still come from its birthplace. ZNN methods have become a backbone for solving discretized sensor driven time-varying matrix problems in real-time, in theory and in on-chip applications for robots, in control theory and in engineering. They have become the method of choice for many time-varying matrix problems that benefit from or require efficient, accurate and predictive real-time computations. The typical discretized ZNN algorithm needs seven distinct steps for its initial set-up. The construction of discretized ZNN algorithms starts from a model with its associated error equation and the stipulation that the error function decrease exponentially fast. The error function differential equation is then mated with a convergent look-ahead finite difference formula to create a distinctly new multistep-style solver that predicts the future state of the system reliably from current and earlier state and solutions data. Matlab codes for discretized ZNN algorithms typically consist of one linear equations solve and one recursion of already available data per time step. This makes discretized ZNN based algorithms highly competitive with ordinary differential equation initial value path following or homotopy methods that are designed to work adaptively and gives ZNN different characteristics and applicabilities from multistep ODE initial value solvers. Discretized ZNN methods can solve problems given by sensor data with constant sampling gaps or from functional equations.