Gridding discretization-based multiple stability switching delay search algorithm: The movement of a human being on a controlled swaying bow.

Libor Pekař,Radek Matušů,Roman Prokop

doi:10.1371/journal.pone.0178950

Abstract

Delay represents a significant phenomenon in the dynamics of many human-related systems—including biological ones. It has i.a. a decisive impact on system stability, and the study of this influence is often mathematically demanding. This paper presents a computationally simple numerical gridding algorithm for the determination of stability margin delay values in multiple-delay linear systems. The characteristic quasi-polynomial—the roots of which decide about stability—is subjected to iterative discretization by means of pre-warped bilinear transformation. Then, a linear and a quadratic interpolation are applied to obtain the associated characteristic polynomial with integer powers. The roots of the associated characteristic polynomial are closely related to the estimation of roots of the original characteristic quasi-polynomial which agrees with the system′s eigenvalues. Since the stability border is crossed by the leading one, the switching root locus is enhanced using the Regula Falsi interpolation method. Our methodology is implemented on—and verified by—a numerical bio-cybernetic example of the stabilization of a human-being′s movement on a controlled swaying bow. The advantage of the proposed novel algorithm lies in the possibility of the rapid computation of polynomial zeros by means of standard programs for technical computing; in the low level of mathematical knowledge required; and, in the sufficiently high precision of the roots loci estimation. The relationship to the direct search QuasiPolynomial (mapping) Rootfinder algorithm and computational complexity are discussed as well. This algorithm is also applicable for systems with non-commensurate delays.

Highlights

The stability of Time Delay Systems (TDS) has been a challenging and intensively studied research topic over the past few decades [1,2,3,4]
Motivated by the above deficiency—and by the endeavor to use -accessible standard programs for technical computing (e.g. MATLAB1) by engineers and practitioners, this paper focuses on deriving a computationally simple—yet sufficiently rapid and accurate gridding algorithm for searching for delay margins in both commensurate and non-commensurate TDS with an arbitrary number of independent constant delays
Prior to the presentation of the proposed Delay Dependent Stability (DDS) algorithm and its implementation on stability margin detection when remote controlling a skater on a swaying bow, some additional methods have to be derived and computational ideas formulated

Summary

Introduction

The stability of Time Delay Systems (TDS) has been a challenging and intensively studied research topic over the past few decades [1,2,3,4]. It is, i.a., given by the fact that delay appears. Stability switching search algorithm: A human on a swaying bow projekty-eu/detail?id=141580. The authors are members of the team that received the funding but they are not responsible persons. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript

Methods

Results

Conclusion