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A sufficient condition for 2-contraction of a feedback interconnection

Ron Ofir, Francesco Bullo, Michael Margaliot

Abstract

Multistationarity - the existence of multiple equilibrium points - is a common phenomenon in dynamical systems from a variety of fields, including neuroscience, opinion dynamics, systems biology, and power systems. A recently proposed generalization of contraction theory, called $k$-contraction, is a promising approach for analyzing the asymptotic behaviour of multistationary systems. In particular, all bounded trajectories of a time-invariant 2-contracting system converge to an equilibrium point, but the system may have multiple equilibrium points where more than one is locally stable. An important challenge is to study $k$-contraction in large-scale interconnected systems. Inspired by a recent small-gain theorem for 2-contraction by Angeli et al., we derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on (i) deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], (ii) a hierarchical approach for proving standard contraction, and (iii) a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh-Nagumo neurons.

A sufficient condition for 2-contraction of a feedback interconnection

Abstract

Multistationarity - the existence of multiple equilibrium points - is a common phenomenon in dynamical systems from a variety of fields, including neuroscience, opinion dynamics, systems biology, and power systems. A recently proposed generalization of contraction theory, called -contraction, is a promising approach for analyzing the asymptotic behaviour of multistationary systems. In particular, all bounded trajectories of a time-invariant 2-contracting system converge to an equilibrium point, but the system may have multiple equilibrium points where more than one is locally stable. An important challenge is to study -contraction in large-scale interconnected systems. Inspired by a recent small-gain theorem for 2-contraction by Angeli et al., we derive a new sufficient condition for 2-contraction of a feedback interconnection of two nonlinear dynamical systems. Our condition is based on (i) deriving new formulas for the 2-multiplicative [2-additive] compound of block matrices using block Kronecker products [sums], (ii) a hierarchical approach for proving standard contraction, and (iii) a network small-gain theorem for Metzler matrices. We demonstrate our results by deriving a simple sufficient condition for 2-contraction in a network of FitzHugh-Nagumo neurons.
Paper Structure (17 sections, 16 theorems, 113 equations, 6 figures)

This paper contains 17 sections, 16 theorems, 113 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compound matrices
  4. $k$-contraction
  5. Kronecker products and sums
  6. Relation between compound matrices and Kronecker products
  7. Block Kronecker products and sums
  8. Contraction with respect to hierarchic norms
  9. Network small-gain theorem for Metzler matrices
  10. Main result
  11. Small-gain theorem for 2-contraction
  12. Proof of main result
  13. Explicit expression for the 2-compounds of a block matrix
  14. Bounds on the norms of the blocks
  15. Proof of Theorem \ref{['thm:2cont_feedback']}
  16. ...and 2 more sections

Key Result

Proposition 7

Let $A\in\mathbb R^{n\times n}$, and fix $k\in\{1,\dots,n\}$. The entry of $A^{[k]}$ corresponding to $(\alpha,\beta)$ is equal to:

Figures (6)

  • Figure 1: Digraph $\mathcal{G}$ associated to the Metzler matrix $M$ in equation \ref{['eq:simpleM']}.
  • Figure 2: Definition of the matrix $S(x,z)$ in Thm. \ref{['thm:2cont_feedback']}.
  • Figure 3: The $2$-multiplicative compound of a block matrix.
  • Figure 4: The 2-additive compound of a block matrix.
  • Figure 5: Bounding the scaled hierarchic norm of $\tilde{J}^{[2]}$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 1
  • Example 2
  • Example 3
  • Definition 4
  • Example 5
  • Example 6
  • Proposition 7
  • Lemma 8: Lancaster1972normstensors,Wu2022AlphaContraction
  • Example 9
  • Proposition 10
  • ...and 20 more