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Continuous and discrete zeroing dynamics models using JMP function array and design formula for solving time-varying Sylvester-transpose matrix inequality

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Abstract

Zeroing dynamics (ZD) has shown great performance to solve various time-varying problems. In this paper, the problem of time-varying Sylvester-transpose matrix inequality is first investigated. Since it is difficult to solve a matrix inequality with a matrix variable and its transpose by traditional methods, this paper proposes a continuous ZD (CZD) model by employing ZD design formula and JMP function array to solve this challenging problem. Furthermore, for better implementation on digital computers, three discrete ZD (DZD) models are proposed by using three different discretization formulas with different precision, i.e., the Euler-forward formula, the 6-instant Zhang et al discretization (ZeaD) formula and the 7-instant ZeaD formula. What is more, theoretical truncation error analyses and numerical experiments substantiate the convergence, efficacy and superiority of the DZD models for solving time-varying Sylvester-transpose matrix inequality.

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Funding

This work is supported by the National Natural Science Foundation of China (with number 61976230) and also by the Shenzhen Science and Technology Plan Project (with number JCYJ20170818154936083).

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Authors and Affiliations

  1. School of Data and Computer Science, Sun Yat-sen University, Guangzhou, 510006, People’s Republic of China

    Yunong Zhang & Yihong Ling

  2. Research Institute of Sun Yat-sen University in Shenzhen, Shenzhen, 518057, People’s Republic of China

    Yunong Zhang, Xiao Liu, Yihong Ling, Min Yang & Huanchang Huang

  3. School of Electronics and Information Technology, Sun Yat-sen University, Guangzhou, 510006, People’s Republic of China

    Xiao Liu, Min Yang & Huanchang Huang

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  1. Yunong Zhang
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Correspondence to Yunong Zhang.

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Appendix:

Appendix:

As for an N-step algorithm, the following four lemmas [57, 58] are presented as general concepts and theoretical results for ZD research.

Lemma 2

An N-step method depicted by \(\sum \nolimits _{j = 0}^{N} {\alpha _{j} x_{k + j}}=g\sum \nolimits _{j = 0}^{N} {\upbeta }_{j} \delta _{k + j}\) can be checked for its 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 φ(ς) = 0 satisfy |ς|≤ 1 with |ς| = 1 being unique, then the N-step method is 0-stable (i.e., has 0-stability).

Lemma 3

If the truncation error for the exact solution is of order O(gp) with p > 0 defined, an N-step method is said to be consistent of order p.

Lemma 4

For all t ∈ [0,tf], as \(g \rightarrow 0\), if and only if the method is 0-stable and consistent, an N-step method is convergent, i.e., \(x_{[t/g]}\rightarrow x^{*}(t)\). That is, 0-stability plus consistency means convergence, which is also known as Dahlquist equivalence theorem.

Lemma 5

If an N-step method is 0-stable and consistent, the N-step method is convergent, with the order of its truncation error.

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Zhang, Y., Liu, X., Ling, Y. et al. Continuous and discrete zeroing dynamics models using JMP function array and design formula for solving time-varying Sylvester-transpose matrix inequality. Numer Algor 86, 1591–1614 (2021). https://doi.org/10.1007/s11075-020-00946-1

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