Serial interconnections of 1-contracting and 2-contracting systems

Ofir, Ron; Margaliot, Michael; Levron, Yoash; Slotine, Jean‐Jacques E.

doi:10.1109/cdc45484.2021.9682810

Cited by 5 publications

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“…Ref. [22] considered the series interconnection of k-contracting systems with k ∈ {1, 2}, and showed that such interconnections are in general not 2-contracting. Furthermore, the asymptotic analysis of such systems is more delicate than in the case of 1-contracting systems because the well-ordered behaviour of 2-contracting systems only holds in the time-invariant case, while connecting two sub-systems implies that at least one sub-system has an input from the other sub-system and is thus time-varying.…”

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confidence: 99%

A sufficient condition for $k$-contraction of the series connection of two systems

Ofir,

Margaliot,

Levron

et al. 2021

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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.

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confidence: 99%