On stability crossing curves for general systems with two delays

Abstract

For the general linear scalar time-delay systems of arbitrary order with two delays, this article provides a detailed study on the stability crossing curves consisting of all the delays such that the characteristic quasipolynomial has at least one imaginary zero. The crossing set, consisting of all the frequencies corresponding to all the points in the stability crossing curves, are expressed in terms of simple inequality constraints and can be easily identified from the gain response curves of the coefficient transfer functions of the delay terms. This crossing set forms a finite number of intervals of finite length. The corresponding stability crossing curves form a series of smooth curves except at the points corresponding to multiple zeros and a number of other degenerate cases. These curves may be closed curves, open ended curves, and spiral-like curves oriented horizontally, vertically, or diagonally. The category of curves are determined by which constraints are violated at the two ends of the corresponding intervals of the crossing set. The directions in which the zeros cross the imaginary axis are explicitly expressed. An algorithm may be devised to calculate the maximum delay deviation without changing the number of right half plane zeros of the characteristic quasipolynomial (and preservation of stability as a special case).

Mathematical investigation of diabetically impaired ultradian oscillations in the glucose–insulin regulation

“…Conditions when a 2 = a 4 = 0 have been discussed, for example in [28] using degree theory. We do not make use of a rational transformation to bring the transcendental equation into a polynomial problem (as done, for example in [9,34]). In the generic case, one can solve (21) and (22) for sin z and cos z and upon using that cos 2 z + sin 2 z = 1, we obtain a cubic polynomial for ρ = ω 2 ,…”

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

“…This is confirmed analytically by deriving the minimum feedback gains in the presence of delay variations when nominal delay values are integer multiples of π/ω. By the same arguments as in the previous paragraph, we obtain: at (τ 01 ,τ 02 ) = (2nπ/ω,2mπ/ω), (56) at (τ 01 ,τ 02 ) = [2nπ/ω,(2m + 1)π/ω], and…”

Stability of car following with human memory effects and automatic headway compensation

“…The set of parameters for which such critical roots exist will divide the parameter space into several regions, each region being characterized by the same number of unstable characteristic roots (see Michiels & Niculescu (2007) and Stépán (1989) on the so-called D-or t-decomposition methods). In the case of a two-dimensional parameter space, the boundaries of such regions are called stability crossing curves, as suggested by Stépán (1989) and Gu et al (2005). The points on these curves are the parameters (either delay or controller gain parameters) that engender s = ju solutions in the characteristic equation (3.3), where u ≥ 0 without loss of generality.…”

“…Capturing these stability phenomena is far from trivial, and one of the main research interests in such cases is to find appropriate algorithms for characterizing them globally. In this context, some interesting ideas proposed in the literature include appropriate frequency-sweeping tests (Chen & Latchman 1995;Gu et al 2003Gu et al , 2005), which are also exploited here to assess the exponential stability with respect to delays, drivers' measures of aggressiveness and gains of the automated controllers.…”