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

TL;DR

This paper establishes a sufficient condition for the $k$-contraction property in the series connection of two systems, extending the understanding of how interconnections preserve contraction properties and their implications for system behavior.

Contribution

It introduces a new formula for the $k$th compounds of block-diagonal matrices and applies it to derive conditions for $k$-contraction in interconnected systems.

Findings

Series interconnection preserves $k$-contraction under certain conditions.

New formula for $k$th compounds of block-diagonal matrices.

Conditions for $2$-contracting systems with decaying inputs to maintain ordered behavior.

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 $k\in\{1,\dots,n\}$. 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 $k$th 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.