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 “kernel” surfaces (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.
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e-mail: rifat@coe.neu.edu
e-mail: olgac@engr.uconn.edu
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May 2007
Technical Papers
Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)
Rifat Sipahi,
Rifat Sipahi
Department of Mechanical and Industrial Engineering, Snell Engineering Center,
e-mail: rifat@coe.neu.edu
Northeastern University
, Boston, MA 02115
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Nejat Olgac
Nejat Olgac
Mechanical Engineering Department,
e-mail: olgac@engr.uconn.edu
University of Connecticut
, Storrs, CT 06269-3139
Search for other works by this author on:
Rifat Sipahi
Department of Mechanical and Industrial Engineering, Snell Engineering Center,
Northeastern University
, Boston, MA 02115e-mail: rifat@coe.neu.edu
Nejat Olgac
Mechanical Engineering Department,
University of Connecticut
, Storrs, CT 06269-3139e-mail: olgac@engr.uconn.edu
J. Dyn. Sys., Meas., Control. May 2007, 129(3): 245-251 (7 pages)
Published Online: August 21, 2006
Article history
Received:
January 20, 2005
Revised:
August 21, 2006
Citation
Sipahi, R., and Olgac, N. (August 21, 2006). "Kernel and Offspring Concepts for the Stability Robustness of Multiple Time Delayed Systems (MTDS)." ASME. J. Dyn. Sys., Meas., Control. May 2007; 129(3): 245–251. https://doi.org/10.1115/1.2718235
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