No 8-Instant ZTD (Zhang Time Discretization) Formula with Quintic Precision or Higher as Proved

Abstract – Since 2014, Zhang time discretization (ZTD, also termed Zhang et al. discretization) formulas have been put forward as a new method for time discretization by Zhang et al. Originally, ZTD formulas were used to obtain discrete Zhang neural network (ZNN) model from continuous ZNN one. ZTD formulas were also used later to discretize other continuous-time systems. So far, ZTD formulas with various number of instants have been proposed and applied, and the highest precisions of the $2$-instant ZTD formula to $7$-instant ZTD formulas have been discovered. Instinctively, we want to find out the highest precision of $8$-instant ZTD formulas. During our investigation, we ascertain that any convergent 8-instant ZTD formula cannot possess quintic precision or higher. Combined with previous work, we conclude that any $8$-instant ZTD formula converges with a truncation error in proportion to the $4$-th power of the sampling interval or greater. The conclusion is proved in our paper.

Index Terms – Zhang time discretization formulas, Taylor expansions, Tustin transform, Routh stability criterion, Jury table.