Cluster Treatment Of Characteristic Roots Research Articles

Extended Kronecker Summation for Cluster Treatment of LTI Systems with Multiple Delays

A new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant (LTI) dynamics with multiple delays. These hypersurfaces, as they have very recently been introduced in a concept paper [R. Sipahi and N. Olgac, Automatica, 41 (2005), pp. 1413–1422], form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots (CTCR). In fact, these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine the kernel and offspring we use the extraordinary features of the “extended Kronecker summation” operation in this paper. The end result is that the infinite‐dimensional problem reduces to a finite‐dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper, we are able to shorten the computational time considerably in determining these hypersurfaces. We demonstrate these concepts via some example case studies. One of the examples treats a 3‐delay system. For this case another interesting perspective, called the “building block,” is also utilized to display the kernel in three‐dimensional space in the domain of “spectral delays.”

Stability Robustness of Retarded LTI Systems with Single Delay and Exhaustive Determination of Their Imaginary Spectra

In this paper we consider the stability robustness of the general class of vector LTI (linear time invariant) equations with a single delay, $\dot{\bf x}(t) = {\bf A}{\bf x}(t) + {\bf B}{\bf x}(t-\tau)$, ${\bf x} \in {\bf R}^n$. The robustness is against the uncertain, but constant delay, $\tau \in {\bf R}^+$. We first present a set of novel propositions and state that the solution must start from the complete knowledge of imaginary spectra of the system, and the corresponding delays. The propositions claim that such spectra form a set of manageably small number of members, and this number is upper bounded by $n^2$ regardless of the composition of ${\bf A}$ and ${\bf B}$ matrices. They also claim that the infinite-dimensional system at hand has an outstanding discipline regarding these imaginary spectra. This discipline invites the recently developed concept called the cluster treatment of characteristic roots (CTCR). The CTCR procedure requires a complete and precise determination of the imaginary spectra of the system. There are many procedures in the literature to achieve this. They are, in fact, some variations of the five main methods of different levels of precision and complexity. There is, however, no study known to the authors for presenting a comparison among these methods. This paper addresses this need. We first offer an overview of each of the five methods and then compare their numerical performances over an example case study.

Complete Stability Analysis of Neutral‐Type First Order Two‐Time‐Delay Systems with Cross‐Talking Delays

The stability robustness of first order linear time invariant dynamics of neutral type with multiple time delays against delay uncertainties is taken into consideration. We depart from a simpler investigation of Hale and Huang [J. Math. Anal. Appl., 178 (1993), pp. 344–362], which studies the same problem for retarded‐type systems. On this basis we further introduce two challenging features by including (a) terms that add neutral dynamics and (b) an additional term that introduces cross‐talk between the multiple delays. To the best of the authors’ knowledge, the stability posture of this class of systems can be treated only by a unique procedure. It is known as cluster treatment of characteristic roots (CTCR), which was recently developed for retarded‐type dynamics. We first show the applicability of CTCR to the stability analysis of neutral‐type multiple‐delay dynamics. Next, we prove the well‐known “small‐delay” phenomenon for the dynamics at hand, interestingly, as a natural by‐product of the CTCR paradigm. Finally, we present several case studies to display the steps and the strengths of CTCR. This deployment is scalable to treat similar problems with higher order dynamics, which have direct ramifications to some practical control applications.

EXTENDED KRONECKER SUMMATION FOR DETERMINING THE KERNEL AND OFFSPRING OF LTI SYSTEMS WITH MULTIPLE DELAYS

A new concept is presented for determining the kernel and the offspring hypersurfaces for general LTI dynamics with multiple delays. These hypersurfaces, as they are very recently introduced in a concept paper (Sipahi and Olgac 2005), form the basis of the overriding paradigm which is called the “Cluster Treatment of Characteristic Roots (CTCR)”. In fact, these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots. To determine these kernel and offspring we use the extraordinary features of “Extended Kronecker Summation” operation in this paper. The end result is that the infinite dimensional problem reduces to a finite dimensional one (and preferably into an eigenvalue problem). Following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces. We demonstrate these concepts via some example case studies.

EXACT STABILITY ANALYSIS OF NEUTRAL SYSTEMS WITH CROSS-TALKING DELAYS

The stability of neutral systems with two cross-talking delays is investigated using the method of cluster treatment of characteristic roots (CTCR). There are two main outcomes of this study: (a) Generation of the well-known delay stabilizability condition as a by-product of the CTCR procedure. This is achieved by a small delay stability treatment over the system. We also demonstrate for the delay-stabilizable systems the exact bounds of the stability regions in the domain of the delays. (b) Validation of these stability regions using an alternative point-wise algorithm, which computes the right-most roots of the characteristic quasi-polynomial.

Complete Stability Map of First Order - Two Time Delay Systems with Delay Cross-Talk

First order linear time invariant dynamics is taken into account with two delays. A unique feature is included in this study, from the stability perspective: a term that represents the cross-talk between the two delays. This class of systems can be treated by a unique procedure, which is recently developed, known as Cluster Treatment of Characteristic Roots (CTCR). In the text we present several case studies to display the steps and the strengths of CTCR. We also include some comparison results with respect to a cornerstone study (Hale, and Huang, 1993), which gives a wonderful point of reference for this class of systems, however without the delay cross-talk effect.

“Delay Scheduling”: A New Concept for Stabilization in Multiple Delay Systems

A trajectory-tracking problem is considered for a linear time invariant (LTI) dynamics with a fixed control law. However, the feedback line is affected by multiple time delays. The stability of the dynamics becomes a complex problem. It is well known that time-delayed LTI systems may exhibit multiple stable operating zones (which we call pockets) in the space of the delays. Our aim in this paper is to locate and experimentally validate these pockets. For the analytical determination of the pockets we utilize a new methodology, the cluster treatment of characteristic roots (CTCR). The study results in several interesting conclusions. (i) The systems may exhibit better control performance (for instance, faster disturbance rejection) for larger time delays. (ii) Consequently, we propose a unique and interesting utilization of the time delays as agents to enhance the control performance, the delay scheduling technique.

Complete stability robustness of third-order LTI multiple time-delay systems

A unique procedure is presented in this paper, for a complete stability robustness of the third-order LTI multiple time-delay systems (LTI-MTDS). The uniqueness of the treatment is simply due to the fact that there is no comparable methodology, presently, in the literature. The end result of this procedure is an exhaustive and precise determination of the stable regions in the domain of time delays. The backbone of the method is a novel framework called “the cluster treatment of characteristic roots, (CTCR)”. CTCR is constructed over two fundamental propositions. The first proposition claims the existence of a bounded number of so-called “ kernel curves”, where the only imaginary characteristic roots occur. The second proposition is on an interesting directional invariance property of the crossing tendencies of these imaginary roots. For simplicity of conveyance and without loss of generality, the number of time delays is taken as two in this document. The new methodology is expandable to higher-order dynamics with more time delays than two, as the authors intend to demonstrate in future publications.

A Unique Methodology for Chatter Stability Mapping in Simultaneous Machining

A novel analytical tool is presented to assess the stability of simultaneous machining (SM) dynamics, which is also known as parallel machining. In SM, multiple cutting tools, which are driven by multiple spindles at different speeds, operate on the same workpiece. Its superior machining efficiency is the main reason for using SM compared with the traditional single tool machining (STM). When SM is optimized in the sense of maximizing the rate of metal removal constrained with the machined surface quality, typical “chatter instability” phenomenon appears. Chatter instability for single tool machining (STM) is broadly studied in the literature. When formulated for SM, however, the problem becomes notoriously more complex. There is practically no literature on the SM chatter, except a few ad hoc and inconclusive reports. This study presents a unique treatment, which declares the complete stability picture of SM chatter within the mathematical framework of multiple time-delay systems (MTDS). What resides at the core of this development is our own paradigm, which is called the cluster treatment of characteristic roots (CTCR). This procedure determines the regions of stability completely in the domain of the spindle speeds for varying chip thickness. The new methodology opens the research to some interesting directions. They, in essence, aim towards duplicating the well-known “stability lobes” concept of STM for simultaneous machining, which is clearly a nontrivial task.

The Cluster Treatment of Characteristic Roots and the Neutral Type Time-Delayed Systems

A new methodology is presented for assessing the stability posture of a general class of linear time-invariant—neutral time-delayed systems (LTI-NTDS). It is based on a “Cluster Treatment of Characteristic Roots CTCR” paradigm, which yields a procedure called the Direct Method (DM). The technique offers a number of unique features: It returns exact bounds of time delay for stability, as well as the number of unstable characteristic roots of the system in an explicit and nonsequentially evaluated function of time delay. As a direct consequence of the latter feature, the new methodology creates entirely, all existing stability intervals of delay, τ. It is shown that the Direct Method inherently enforces an intriguing necessary condition for τ-stabilizability, which is the main contribution of this paper. This, so-called, “small delay” effect, was recognized earlier for NTDS, only through some cumbersome mathematics. Furthermore, the Direct Method is also unique in handling systems with unstable starting posture for τ=0, which may be τ-stabilized for higher values of delay. Example cases are provided.