Passivity and Passivity Indices of Nonlinear Systems Under Operational Limitations using Approximations

TL;DR

This paper develops methods to analyze the passivity and passivity indices of nonlinear systems under operational constraints using polynomial approximations, enabling stability and dissipativity assessments.

Contribution

It introduces local passivity formulations for nonlinear systems with operational limitations using Taylor and Bernstein polynomial approximations, including optimization techniques for passivity analysis.

Findings

Conditions for stability, dissipativity, and passivity are derived.

Two polynomial approximation methods are compared for passivity analysis.

Optimization approaches are proposed to efficiently compute passivity indices.

Abstract

In this paper, we will discuss how operational limitations affect input-output behaviours of the system. In particular, we will provide formulations for passivity and passivity indices of a nonlinear system given operational limitations on the input and state variables. This formulation is presented in the form of local passivity and indices. We will provide optimisation based formulation to derive passivity properties of the system through polynomial approximations. Two different approaches are taken to approximate the nonlinear dynamics of a system through polynomial functions; namely, Taylor's theorem and a multivariate generalisation of Bernstein polynomials. For each approach, conditions for stability, dissipativity, and passivity of a system, as well as methods to find its passivity indices, are given. Two different methods are also presented to reduce the size of the optimisation problem in Taylor's theorem approach. Examples are provided to show the applicability of the results.