Trapping Regions for Quadratic Systems with Generalized Lossless Nonlinearities

TL;DR

This paper introduces a generalized framework for computing trapping regions in quadratic systems with lossless nonlinearities, expanding applicability and improving bounds over prior methods.

Contribution

It presents an efficient parameterization for broader classes of systems and formulates conditions for ellipsoidal trapping regions, surpassing existing spherical-region approaches.

Findings

Successfully identified trapping regions in a 4D system where previous methods failed.

Achieved smaller trapping regions in an aerodynamics model compared to existing techniques.

Correctly identified a globally stable equilibrium in a simple academic example.

Abstract

A trapping region is a compact set that is forward invariant with respect to the dynamics. Existence of a trapping region certifies boundedness of trajectories, and the size of the set provides an estimate of the ultimate bound. Prior work on trapping region analysis has focused on quadratic systems with energy-preserving (lossless) nonlinearities. In this work, we focus on a generalization of the lossless property and present an efficient parameterization that enables optimal trapping region computation for a broader class of quadratic systems than afforded by existing methods. We also formulate conditions for ellipsoidal trapping regions, whereas spherical regions have been the focus of prior works. Three numerical examples are used to demonstrate the proposed framework: (1) a four dimensional system for which the prior state-of-the art is incapable of identifying a trapping region; (2) a low-order unsteady aerodynamics model for which the proposed approach yields trapping regions approximately an order of magnitude smaller than prevailing methods; and (3) a two-state academic example in which the proposed approach correctly identifies a globally asymptotically stable equilibrium point.