Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)

[+] Author and Article Information
Rifat Sipahi1

Department of Mechanical and Industrial Engineering, Snell Engineering Center, Northeastern University, Boston, MA 02115rifat@coe.neu.edu

Nejat Olgac

Mechanical Engineering Department, University of Connecticut, Storrs, CT 06269-3139olgac@engr.uconn.edu


The author conducted this research at the Mechanical Engineering Department, University of Connecticut, Storrs, CT 06269-3139.

J. Dyn. Sys., Meas., Control 129(3), 245-251 (Aug 21, 2006) (7 pages) doi:10.1115/1.2718235 History: Received January 20, 2005; Revised August 21, 2006

A novel treatment for the stability of linear time invariant (LTI) systems with rationally independent multiple time delays is presented in this paper. The independence of delays makes the problem much more challenging compared to systems with commensurate time delays (where the delays have rational relations). We uncover some wonderful features for such systems. For instance, all the imaginary characteristic roots of these systems can be found exhaustively along a set of surfaces in the domain of the delays. They are called the “kernelsurfaces (curves for two-delay cases), and it is proven that the number of the kernel surfaces is manageably small and bounded. All possible time delay combinations, which yield an imaginary characteristic root, lie either on this kernel or its infinitely many “offspring” surfaces. Another hidden feature is that the root tendencies along these surfaces exhibit an invariance property. From these outstanding characteristics an efficient, exact, and exhaustive methodology results for the stability assessment. As an added uniqueness of this method, the systems under consideration do not have to be stable for zero delays. Several example case studies are presented, which are prohibitively difficult, if not impossible to solve using any other peer methodology known to the authors.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

R1=0 and the constraint of Eq. 30.

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Figure 2

The kernel formation of Case I and the only possible ωc’s

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Figure 3

Complete stability map of Case I, some RTs and NUs

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Figure 4

Color-coded ωc of the kernel for Case II

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Figure 5

Stability map of Case II

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Figure 6

Color-coded ωc of the kernel for Case III

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Figure 7

Stability map of Case III



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