# Stability robustness of retarded LTI systems with single delay and exhaustive determination of their imaginary spectra

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.

#### 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.

#### Suggested Citation

Rifat Sipahi and Nejat Olgac. "Stability robustness of retarded LTI systems with single delay and exhaustive determination of their imaginary spectra" Mechanical and Industrial Engineering Faculty Publications (2006).
Available at: http://works.bepress.com/rsipahi/14

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