Abstract
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.
Keywords
linear time‐delayed dynamics, quasi polynomial, imaginary spectra detection, robust stability
Disciplines
Industrial Engineering | Mechanical Engineering
Publisher
SIAM
Publication Date
11-16-2006
Rights Information
© 2006 Society for Industrial and Applied Mathematics
Rights Holder
SIAM
Recommended Citation
Sipahi, Rifat and Olgac, Nejat, "Stability robustness of retarded LTI systems with single delay and exhaustive determination of their imaginary spectra" (2006). Mechanical and Industrial Engineering Faculty Publications. Paper 41.
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Notes
Originally published in SIAM J. Control Optim. 45 (2006), pp. 1680-1696. DOI:10.1137/050633238
Dr. Sipahi is affiliated with Northeastern University as of the time of deposit.