General four-step discrete-time zeroing and derivative dynamics applied to time-varying nonlinear optimization

Time-varying nonlinear optimization (TVNO) problems are considered as important issues in various scientific disciplines and industrial applications. In this paper, the continuous-time derivative dynamics (CTDD) model is developed for obtaining the real-time solutions of TVNO problems. Furthermore, aiming to remedy the weaknesses of CTDD model, a continuous-time zeroing dynamics (CTZD) model is presented and investigated. For potential digital hardware realization, by using bilinear transform, a general four-step Zhang et al discretization (ZeaD) formula is proposed and applied to the discretization of both CTDD and CTZD models. A general four-step discrete-time derivative dynamics (general four-step DTDD) model and a general four-step discrete-time zeroing dynamics (general four-step DTZD) model are proposed on the basis of this general four-step ZeaD formula. Further theoretical analyses indicate that the general four-step DTZD model is zero-stable, consistent and convergent with the truncation error of O(g4), which denotes a vector with every entries being O(g4) with g denoting the sampling period. Theoretical analyses also indicate that the maximal steady-state residual error (MSSRE) has an O(g4) pattern confirmedly. The efficacy and accuracy of the general four-step DTDD and DTZD models are further illustrated by numerical examples.