Numerical Algorithms

, Volume 81, Issue 3, pp 1043–1065 | Cite as

New five-step DTZD algorithm for future nonlinear minimization with quartic steady-state error pattern

  • Binbin QiuEmail author
  • Yunong Zhang
  • Jinjin Guo
  • Zhi Yang
  • Xiaodong Li
Original Paper


In this paper, a new five-step discrete-time zeroing dynamics (DTZD) algorithm, discretized from a continuous-time zeroing dynamics (CTZD) model, is proposed and investigated for online future nonlinear minimization (OFNM), i.e., online discrete-time dynamic nonlinear minimization. For approximating more accurately the first-order derivative and discretizing more effectively the CTZD model, a six-node g-cube discretization (6Ng CD) formula with higher precision is presented to obtain the new five-step DTZD algorithm. Besides, the corresponding theoretical result shows that the proposed five-step DTZD algorithm is with a quartic steady-state error pattern, i.e., O(g4) pattern, with g denoting the sampling gap. Moreover, a general DTZD algorithm is constructed by applying the general linear multistep method, and a specific DTZD algorithm based on the 4th-order Adams-Bashforth method (termed DTZD-AB algorithm for short) is further developed for OFNM. Several numerical experiments are conducted to substantiate the efficacy, accuracy, and superiority of the proposed five-step DTZD algorithm (as well as the DTZD-AB algorithm) for solving the OFNM problem, as compared with the one-step and three-step DTZD algorithms developed and investigated in previous works.


Six-node g-cube discretization (6Ng CD) formula Discrete-time zeroing dynamics (DTZD) Five-step DTZD algorithm Online future nonlinear minimization (OFNM) Quartic steady-state error pattern 


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The authors thank the editors and anonymous reviewers for their valuable suggestions and constructive comments, which helped to improve the presentation and quality of the paper.

Funding information

This work is supported by the National Natural Science Foundation of China (with number 61473323), by the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, China (with number 2013A07), and also by the Laboratory Open Fund of Sun Yat-sen University (with number 20160209).


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Data and Computer ScienceSun Yat-sen University (SYSU)GuangzhouChina
  2. 2.Key Laboratory of Autonomous Systems and Networked ControlMinistry of EducationGuangzhouChina
  3. 3.SYSU-CMU Shunde International Joint Research InstituteFoshanChina
  4. 4.Key Laboratory of Machine Intelligence and Advanced ComputingMinistry of EducationGuangzhouChina

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