Positivity, monotonicity, and consensus on Lie groups

TL;DR

This paper extends the theory of differential positivity to systems on Lie groups, introducing invariant cone fields and a generalized Perron-Frobenius theory to analyze asymptotic behavior and consensus in nonlinear control.

Contribution

It develops a mathematical framework for differential positivity on Lie groups, including invariant cone fields and a generalized Perron-Frobenius theory with higher-rank cone fields.

Findings

Invariant cone fields can be generated on Lie groups from a single cone in the Lie algebra.

Differential positivity constrains the asymptotic behavior of systems on Lie groups.

Generalized Perron-Frobenius theory describes $k$-dimensional attractors in differentially positive systems.

Abstract

Dynamical systems whose linearizations along trajectories are positive in the sense that they infinitesimally contract a smooth cone field are called differentially positive. The property can be thought of as a generalization of monotonicity, which is differential positivity in a linear space with respect to a constant cone field. Differential positivity places significant constraints on the asymptotic behavior of trajectories under mild technical conditions. This paper studies differentially positive systems defined on Lie groups. The geometry of a Lie group allows for the generation of invariant cone fields over the tangent bundle given a single cone in the Lie algebra. We outline the mathematical framework for studying differential positivity of discrete and continuous-time dynamics on a Lie group with respect to an invariant cone field and motivate the use of this analysis framework in nonlinear control, and, in particular in nonlinear consensus theory. We also introduce a generalized notion of differential positivity of a dynamical system with respect to an extended notion of cone fields generated by cones of rank $k$. This new property provides the basis for a generalization of differential Perron-Frobenius theory, whereby the Perron-Frobenius vector field which shapes the one-dimensional attractors of a differentially positive system is replaced by a distribution of rank $k$ that results in $k$-dimensional integral submanifold attractors instead.