The MID property for a second-order neutral time-delay differential equation

Abstract

This paper considers the Multiplicity-Induced-Dominancy (MID) property for second order neutral time-delay differential equations. Necessary and sufficient conditions for the existence of a root of maximal multiplicity are given in terms of this root and the parameters (including the delay) of the given equation. Links with dominancy of this root and with the exponential stability property of the solution of the considered equations are given. Finally, we illustrate the obtained results on the classical oscillator control problem.

On the Spectrum Distribution of Parametric Second-Order Delay Differential Equations: Perspectives in Partial Pole Placement

“…The MID property consists of the conditions under which a given multiple root of a quasipolynomial function is necessarily dominant. Notice that in the generic quasipolynomial function case, the real root of maximal multiplicity is necessarily the dominant (GMID), see for instance [7,8] for first and second order time-delay equation. However, multiple roots with intermediate admissible multiplicities may be dominant or not.…”

“…Most of these results are actually particular cases of a more general result on the GMID property from [29] for generic retarded DDEs of order n with a delayed "term" (polynomial) of order n − 1, which relies on ties between quasipolynomials with a real root of maximal multiplicity and the Kummer confluent hypergeometric function in terms of the location of the characteristic roots. Owing to this link, the GMID property has been completely characterized in [10] and extended to neutral DDEs of orders 1 and 2 in [8,26,30], as well as to the case of complex conjugate roots of maximal multiplicity in [31]. Also, such an idea has been extended to assign an appropriate number of negative roots rather than assigning a multiple root, allowing to a property called Coexistant Real Roots Inducing Dominancy (CRRID), see for instance [6].…”

Time-delay control of quadrotor unmanned aerial vehicles: a multiplicity-induced-dominancy-based approach

“…The MID property refers to conditions under which multiple roots of the characteristic function match the spectral abscissa. Some recent works have shown that a spectral value of maximal multiplicity of a time-delay system is necessarily real and corresponds to the spectral abscissa, a property called generic multiplicity-induced-dominancy , or GMID for short, see for instance Mazanti et al (2021) and Benarab et al (2020). However, in the case of a root of strictly intermediate multiplicity, one has to seek for conditions on the system’s free parameters (typically the control parameters) for the MID to hold.…”

“…The MID property has already been suggested to solve some low-order cases (Hayes, 1950) and some other phenomena described by linear time-delay differential equations (Boussaada and Niculescu, 2016; Boussaada et al, 2018; Boussaada et al, 2020b; Mazanti et al, 2020). Recent results in this direction provide necessary and sufficient conditions for roots of maximal multiplicity in reduced-order time-delay systems of retarded (Boussaada et al (2020a); Mazanti et al, 2020) and neutral types (Ma et al, 2020; Benarab et al, 2020). Nevertheless, the application of such findings on the domain of aerial robots control, as far as it is concerned to the authors, has not been specifically considered.…”