An exact method for the stability analysis of time-delayed linear time-invariant (LTI) systems

Abstract

A general class of linear time invariant systems with time delay is studied. Recently, they attracted considerable interest in the sys-tems and control community. The complexity arises due to the exponen-tial type transcendental terms in their characteristic equation. The tran-scendentality brings infinitely many characteristic roots, which are cum-bersome to elaborate as evident from the literature. A number of method-ologies have been suggested with limited ability to assess the stability in the parametric domain of time delay. This study offers an exact, structured and robust methodology to bring a closure to the question at hand. Ultimately we present a unique explicit analytical expression in terms of the system parameters which not only reveals the stability regions (pockets) in the do-main of time delay, but it also declares the number of unstable character-istic roots at any given pocket. The method starts with the determination of all possible purely imaginary (resonant) characteristic roots for any pos-itive time delay. To achieve this a simplifying substitution is used for the transcendental terms in the characteristic equation. It is proven that the number of such resonant roots for a given dynamics is finite. Each one of these roots is created by infinitely many time delays, which are periodi-cally distributed. An interesting property is also claimed next, that the root crossing directions at these locations are invariant with respect to the delay and dependent only on the crossing frequency. These two unique findings facilitate a simple and practical stability method, which is the highlight of the work.

An adaptive algorithm for efficient computation of level curves of surfaces

“…Delay systems such as (3) are particularly important in control theory, where the stability effects of delays are a crucial problem [10,12]. Important applications can be found also in machining tool such as milling, turning and drilling where the role of parameters such as spindle speed and feed are stability determining [8,11]: these are second order systems with time dependent coefficients and the interest is in the stability of periodic solutions. The asymptotic stability of the zero solution of (3) is determined by the position on C of the rightmost characteristic root, i.e.…”

Full Order Unknown Inputs Observer for Multiple Time-Delay Systems

“…Their analysis and synthesis methods are not belonging, in that case, to those of standard ones. 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].…”

Inverting Control, a New Strategy on Time Delayed Systems

“…There is however, very limited control synthesis studies in the literature on time delayed systems, again mainly due to the notoriety of the problem (Niculescu 2001;Filipovic and Olgac 2002;Insperger and Stépán 2005). Author's group has contributed in this field, both from the stability analysis and the control synthesis aspects of the research (Olgac and Sipahi 2002;Olgac, Ergenc and Sipahi 2005;Sipahi and Olgac 2005;Olgac and Sipahi 2006; * Author was affiliated with University of Connecticut when the work was done. Ergenc, Olgac and Fazelinia, 2007).…”

“…(Olgac and Sipahi 2002;Olgac and Sipahi 2004;Sipahi and Olgac 2005;Olgac and Sipahi 2006; as well as the strategies for stabilizing (or disturbance rejecting) control (Olgac, Ergenc and Sipahi 2005). …”