Future Linear Matrix Equation of Generalized Sylvester Type Solved by Zeroing Neural Dynamics and 5-Instant ZeaD Formula

Abstract

In this paper, a new discrete-time zeroing dynamics (or say, Zhang dynamics, ZD) model is proposed, analyzed and investigated for solving generalized-Sylvester-type future linear matrix equation (GS-type FLME). First of all, based on ZD design formula, a continuous-time ZD (CTZD) model is proposed for solving generalized-Sylvester-type continuous-time linear matrix equation (i.e., GS-type CTLME). Secondly, a novel one-step-ahead discretization formula (termed Zhang et al. discretization formula, ZeaD formula) is presented for the first-order derivative approximation with higher computational precision. Then, by exploiting the presented ZeaD formula to discretize the CTZD model, a novel discrete-time ZD (DTZD) model, i.e., ZeaD-type DTZD-E model, is proposed, analyzed and investigated for solving GS-type FLME. Theoretical analyses on the convergence and precision of the proposed DTZD models are presented. Comparative numerical experimental results further substantiate the efficacy and superiority of proposed ZeaD-type DTZD model for solving the GS-type FLME.

Appendix

The following four basic results are presented, which are general concepts and theoretical results about linear multistep methods and difference equations [10].

Result 1: An N-step method $\sum \nolimits _{j = 0}^N {\alpha _j \sigma _{k + j}}=g\sum \nolimits _{j = 0}^N \beta _j \delta _{k + j}$ can be checked for 0-stability by determining the roots of its characteristic polynomial $\varphi (\varsigma ) = \sum \nolimits _{j = 0}^N \alpha _j \varsigma ^j$. If all roots of $\varphi (\varsigma )=0$ satisfy $|\varsigma |\le 1$ with $|\varsigma |=1$ being simple, then the N-step method is 0-stable (i.e., has 0-stability).

Result 2: An N-step method is said to be consistent (i.e., has consistency) of order p if the truncation error for the exact solution is of order $O(g^{p+1})$ where $p>0$.

Result 3: An N-step method is convergent, i.e., $\sigma _{[t/g]}\rightarrow \sigma ^*(t)$, for all $t\in [0,t_{ f }]$, as $g \rightarrow 0$, if and only if the method is 0-stable and consistent. That is, 0-stability plus consistency means convergence, which is also known as Dahlquist equivalence theorem.

Result 4: A 0-stable consistent method converges with the order of its truncation error.