Stability analysis of multiple time delayed systems using 'building block' concept

Abstract

An intriguing perspective is presented in studying the stability robustness of systems with multiple independent and uncertain delays. It is based on a very practical mapping, that creates a dramatic reduction in the dimension of the problem from infinity to manageably small. In essence, this mapping collapses the entire set of potential stability switching points onto a small (upperbounded) number of building hypersurfaces within, what we call a building block. We demonstrate that these building hypersurfaces can be implicitly defined and it is also shown that the exhaustive detection of these building hypersurfaces is necessary and sufficient in order to arrive at the complete stability robustness picture in the domain of the delays. This novel perspective serves very well for the preparatory steps of the authors' earlier contribution in the area, Cluster Treatment of Characteristic Roots (CTCR). A complete example case study is also provided.

Stability analysis of multiple time-delayed system

“…To assist the readers, an overview of the CTCR paradigm is presented here, borrowing from [19][20][21][22][23][24][25]. Since [19][20][21][22][23][24][25] focus mainly on the stability analysis of multiple time-delayed system with all delays being uncertain, here we suppose that K in (2) is a constant vector.…”

“…Based on the Hermit-Biehler Theorem and D-partition method, some pieces of paper discussed the stabilization problem of linear systems with time delay using low-order controllers and the complete sets of the loworder controllers that can guarantee the system stability are exactly presented [15][16][17][18]. The combination of a very recent paradigm Cluster Treatment of Characteristic Roots (CTCR) and a recently proposed concept "Building Block" (BB) [19][20][21][22][23][24][25] serve as a powerful tool to uncover completely the stability region of uncertain parameters in time-delay space. However, with nondeterministic polynomial, time hard numerical complexity is still remaining.…”

“…However, with nondeterministic polynomial, time hard numerical complexity is still remaining. Most of the literatures focus on the determination of stability regions of uncertain parameters either in coefficient space [15][16][17][18] or time-delay space [19][20][21][22][23][24][25].…”

Stability crossing set for systems with three scalar delay channels

“…In the last decade or so, many analyses have been conducted on systems with two or three delays with various generality. Examples include systems with two delays without cross term [17], multiple delay systems with cross terms [18], polynomial eigenvalue approach [19], and three variable delays with additional fixed delays [20].…”

Full Order Unknown Inputs Observer for Multiple Time-Delay Systems

“…Massive research activities have been developed to solve stability and stabilization problems of delay systems (see for example [6,7,8,9,10,11,12,13,14] and references therein). In that context, many interests have been also given to systems with multiple time-delays [15,16,17,18,19] for which the well-known limitations are the presence of delays either in the states of the plant, in the inputs as well in the outputs [20,21]. The problem of estimating state variables is of much significance in many applications [22,23].…”