Positive Damping Region: A Graphic Tool for Passivization Analysis with Passivity Index

TL;DR

This paper introduces a geometric graphical tool called the positive damping region for analyzing and designing passivization of linear systems using passivity indices, applicable to both SISO and MIMO systems.

Contribution

It develops a visual framework linking passivization criteria to the Nyquist plot and Rayleigh quotient, enabling easier analysis and control design trade-offs.

Findings

The positive damping region characterizes passivizability with a given passivity index.

The tool facilitates analysis of frequency bands and maximum passivity index.

Application demonstrated in power system stability and device controller design.

Abstract

This paper presents a geometric framework for analyzing output-feedback and input-feedforward passivization of linear time-invariant systems. We reveal that a system is passivizable with a given passivity index when the Nyquist plot for SISO systems or the Rayleigh quotient of the transfer function for MIMO systems lies within a specific, index-dependent region in the complex plane, termed the positive damping region. The criteria enable a convenient graphic tool for analyzing the passivization, the associated frequency bands, the maximum achievable passivity index, and the waterbed effect between them. Additionally, the tool can be encoded into classical tools such as the Nyquist plot, the Nichols plot, and the generalized KYP lemma to aid control design. Finally, we demonstrate its application in passivity-based power system stability analysis and discuss its implications for electrical engineers regarding device controller design trade-offs.