A Sufficient Condition for k-Contraction of the Series Connection of Two Systems
Name
2105.07678v1.pdf
Description
Accepted version
Size
1003.58 KB
Format
Adobe PDF
Checksum (MD5)
ba24dbd375462e39ec69818bdcb1f810
Author(s)
Ofir, Ron
Margaliot, Michael
Levron, Yoash
Slotine, Jean-Jacques
Date Issued
September 2022
Journal
IEEE Transactions on Automatic Control
Publisher
Institute of Electrical and Electronics Engineers
Citation
R. Ofir, M. Margaliot, Y. Levron and J. -J. Slotine, "A Sufficient Condition for k -Contraction of the Series Connection of Two Systems," in IEEE Transactions on Automatic Control, vol. 67, no. 9, pp. 4994-5001, Sept. 2022.
Version
Author's final manuscript
Abstract
The flow of contracting systems contracts 1-dimensional parallelotopes, i.e., line segments, at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system.
A generalization of contracting systems is k-contracting systems, where k∈{1,…,n}. The flow of such systems contracts the volume of k-dimensional parallelotopes at an exponential rate, and in particular they reduce to contracting systems when k=1. It was shown by Muldowney and Li that time-invariant 2-contracting systems have a well-ordered asymptotic behaviour: all bounded trajectories converge to the set of equilibria.
Here, we derive a sufficient condition guaranteeing that the system obtained from the series interconnection of two sub-systems is k-contracting. This is based on a new formula for the kth multiplicative and additive compounds of a block-diagonal matrix, which may be of independent interest. As an application, we find conditions guaranteeing that 2-contracting systems with an exponentially decaying input retain the well-ordered behaviour of time-invariant 2-contracting systems.
MIT Department
Massachusetts Institute of Technology. Department of Mechanical Engineering
Massachusetts Institute of Technology. Department of Brain and Cognitive Sciences
Terms of Use
Creative Commons Attribution-Noncommercial-ShareAlike
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1109/tac.2022.3177715
