Generalization of cluster treatment of characteristic roots for robust stability of multiple time‐delayed systems

Abstract

SUMMARYA new perspective is presented for studying the stability robustness of nth order systems with p rationally independent delays. It deploys a holographic mapping procedure over the delay space into a new coordinate system in order to achieve the objective. This mapping collapses the entire set of potential stability switching points on a manageably small number of hypersurfaces, which are explicitly defined in the new domain. This property considerably alleviates the problem, which is otherwise infinite dimensional, and therefore notoriously complex to handle. We further declare some unrecognized features of these switching hypersurfaces, that they are (a) encapsulated within a higher-dimensional cube with edges of length 2 , which we name the 'building block', and (b) the 'offspring' of this building block, which represent the secondary stability switchings, appear within the adjacent and identical building blocks (cubes) stacked up next to each other. The final outlook is an exclusive representation of stability for this general class of systems at any arbitrary point in the delay space. Two example case studies are also provided, which are not possible to analyze using any other methodology known to the authors.

A Panoramic Sketch about the Robust Stability of Time-Delay Systems and Its Applications

“…e Smith predictors allow to use a controller structure which takes the delay out of the control loop, which reduce the stability analysis to the one of a free-delay system. e employment of the Rekasius transformation implies an infinity-to-one holographic mapping (the mapping is asymmetric), and it is also impossible to track all of the infinitely many roots, especially, since the dominant root cannot be declared, as mentioned in [206]. e Padé approximation has been used to approximate the exponential function e − sτ , s ∈ C, through rational approximation of the form (P mn (sτ)/P nm (sτ)), where…”

Exact Upper and Lower Bounds of Crossing Frequency Set and Delay Independent Stability Test for Multiple Time Delayed Systems

“…Illustrative example in this section is taken from Sipahi et al (2008) where the system has three delays. Firstly, lower and upper bounds of CFS are computed on 3D delay domain, Ω 3D ∈ [ω 3D ,ω 3D ], next τ 3 = 1 is kept fixed, without loss of generality, and lower and upper bounds of CFS are computed on 2D delay domain, Ω 2D ∈ [ω 2D ,ω 2D ].…”

LMI-Based Analysis and Stabilization of Nonlinear Descriptors with Multiple Delays via Delayed Nonlinear Controller Schemes

“…For example, in [40][41][42], a proportional control with an appropriate delay replaces a traditional proportional-derivative one; thus, the system response is fast and insensitive to high-frequency noise. In [43], a scheme called time-delayed feedback control (TDFC) is proposed, originating different investigations [44][45][46][47][48][49][50][51][52][53][54][55].…”