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A Symbolic Computation Approach Towards the Asymptotic Stability Analysis of Differential Systems with Commensurate Delays

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Part of the Advances in Delays and Dynamics book series (ADVSDD,volume 10)

Abstract

A fundamental issue in perturbation analysis is the study of how the asymptotic stability of a linear differential time-delay system varies with respect to small variations of the delay parameter. A classical approach for studying this problem for retarded type linear differential time-delay systems consists in computing the set of critical pairs of their quasipolynomials, namely the values of the delay parameter and the roots of the corresponding quasipolynomials that lie on the imaginary axis, and then in analyzing the variation of these roots with respect to small variations of the delay. Following this approach and using recent methods for solving polynomial systems, we propose a certified and efficient symbolic-numeric algorithm for computing the set of critical pairs of a quasipolynomial. Moreover, using recent algorithmic results developed by the computer algebra community, we present an efficient algorithm for the computation of Newton–Puiseux series at a critical pair. As explained in Li et al. (Analytic Curve Frequency-Sweeping Stability Tests for Systems with Commensurate Delays. Springer, Berlin 2015), these series play an important role in the stability analysis with respect to variations of the delay.

This work was supported by the ANR-13-BS03-0005 (MSDOS).

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Correspondence to Alban Quadrat .

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Bouzidi, Y., Poteaux, A., Quadrat, A. (2019). A Symbolic Computation Approach Towards the Asymptotic Stability Analysis of Differential Systems with Commensurate Delays. In: Valmorbida, G., Seuret, A., Boussaada, I., Sipahi, R. (eds) Delays and Interconnections: Methodology, Algorithms and Applications. Advances in Delays and Dynamics, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-030-11554-8_11

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