On Contraction Analysis for Non-linear Systems

Abstract

This paper derives new results in nonlinear system analysis using methods inspired from fluid mechanics and differential geometry. Based on a differential analysis of convergence, these results may be viewed as generalizing the classical Krasovskii theorem, and, more loosely, linear eigenvalue analysis. A central feature is that convergence and limit behavior are in a sense treated separately, leading to significant conceptual simplifications. The approach is illustrated by controller and observer designs for simple physical examples.

A mathematical model for the dynamics of clustering

“…conditions is strongly related to the notions of 'convergent systems' [22] and 'contracting systems' [16]. However, the system (1) exhibits a weaker property than uniform contraction (which is imposed in [22,16] and related literature), and consequently the function V does not necessarily approach zero.…”

Convergent dynamics, a tribute to Boris Pavlovich Demidovich

“…Incremental stability, contraction analysis are some of the terms related to stability properties of solutions with respect to each other. In the mid 1990s, Lohmiller and Slotine (see [11] and references therein) independently obtained and extended the result of Demidovich. In particular, they pointed out that systems satisfying the (extended) Demidovich condition, enjoy certain properties of asymptotically stable linear systems that are not encountered in general asymptotically stable nonlinear systems.…”

“…Several papers studying such stability properties in their own respect appeared [6,11,1,8]. In this paper, we would like to look at such properties from a historical perspective and to pay tribute to B.P.…”

Phase Synchronization Control of Robotic Networks on Periodic Ellipses with Adaptive Network Topologies

“…16 Consider a nonlinear non-autonomous systemẋ = f (x, t) which is contracting with a contraction rate λ. Let P 1 (t) be a trajectory of the system.…”

“…This justifies the combined synchronization and tracking control framework for robotic networks, first introduced in. 3 For the nonlinear stability proofs, we use contraction analysis, 16 which has recently been successfully applied to network systems. 3,21,31 Third, the proposed coordinate transformation method and the phase angle shift method facilitate a phase angle shift in any ellipse in 3D space so that the elliptical motions of the networked EL system can be described by the combination of circular and sinusoidal motions in a new coordinate system.…”