Passivity and passivity indices of nonlinear systems under operational limitations using approximations

Abstract

In this paper, we will discuss how operational limitations affect input–output behaviours of the system. In particular, we will provide a formulation 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.

Keywords:

• Operational limitations
• passivity indices
• approximation
• nonlinear systems
• dissipativity

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Hasan Zakeri  http://orcid.org/0000-0001-6136-8333

Panos J. Antsaklis  http://orcid.org/0000-0002-6836-5028

Additional information

Funding

This work was supported by the Army Research Office (ARO) under grant no. ARL W911NF-17-1-0072.