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

Hasan Zakeri, Panos J. Antsaklis

TL;DR

This paper addresses the challenge of characterizing passivity, dissipativity, and passivity indices for nonlinear systems under input and state constraints by adopting local, region-bounded analyses. It introduces two polynomial approximation schemes—Taylor expansions and multivariate Bernstein polynomials—facilitating tractable, SOS-compatible feasibility conditions that certify local $QSR$-dissipativity, passivity, and the corresponding indices. The approaches offer complementary trade-offs: Taylor-based methods provide intuitive modeling and broad applicability at the cost of large optimization problems, while Bernstein-based methods yield smaller, more scalable problems with explicit error bounds on a fixed region. Together, they enable systematic Lyapunov-function synthesis and robust interconnection analysis for cyber-physical systems operating within predefined limits, with demonstrations on nonlinear and pendulum-like dynamics. The results advance local-energy-based design under operational constraints and open paths for more efficient index computation and extensions to broader constraint sets.

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.

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

TL;DR

This paper addresses the challenge of characterizing passivity, dissipativity, and passivity indices for nonlinear systems under input and state constraints by adopting local, region-bounded analyses. It introduces two polynomial approximation schemes—Taylor expansions and multivariate Bernstein polynomials—facilitating tractable, SOS-compatible feasibility conditions that certify local -dissipativity, passivity, and the corresponding indices. The approaches offer complementary trade-offs: Taylor-based methods provide intuitive modeling and broad applicability at the cost of large optimization problems, while Bernstein-based methods yield smaller, more scalable problems with explicit error bounds on a fixed region. Together, they enable systematic Lyapunov-function synthesis and robust interconnection analysis for cyber-physical systems operating within predefined limits, with demonstrations on nonlinear and pendulum-like dynamics. The results advance local-energy-based design under operational constraints and open paths for more efficient index computation and extensions to broader constraint sets.

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.

Paper Structure

This paper contains 11 sections, 15 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Passivity and Dissipativity
  4. Passivity Indices
  5. Passivity Under Operational Limitations
  6. Polynomial Approximations
  7. Approach Based on Taylor's Theorem
  8. Approach Based on Bernstein Polynomials
  9. Passivity and Passivity Indices
  10. Examples
  11. Conclusions

Key Result

Theorem 1

Suppose $f(\cdot)$ is a real-valued and continuous function defined on the compact real interval $[a, b].$ Then for every $\varepsilon> 0,$ there exists a polynomial $p(x)$ (which might depend on $\varepsilon$) such that for all $x \in [a, b],$ we have $\vert f(x) - p(x)\vert < \varepsilon,$ or equi

Figures (1)

  • Figure 1: OFP index $\rho$ versus upper bound $r$ on state norm zakeri.acc2016

Theorems & Definitions (35)

  • Definition 1
  • Definition 2: Passivity hill.moylan.76willem.2
  • Definition 3: Input Feed-forward Passivity Index
  • Definition 4: Output Feedback Passivity
  • Definition 5: Local Dissipativity
  • Definition 6
  • Definition 7
  • Definition 8
  • Theorem 1: Weierstrass Approximation Theorem Apostol
  • Theorem 2
  • ...and 25 more