A Characterization of All Passivizing Input-Output Transformations of a Passive-Short System

TL;DR

This paper classifies all input-output transformations that can passivize a system with a passivity shortage, providing a comprehensive framework for system passivation and applications like synchronization and gain minimization.

Contribution

It offers a complete characterization of passivizing input-output transformations for systems with passivity deficits, extending passivity theory tools.

Findings

Classified all passivizing transformations for SISO and MIMO systems.

Applied results to synchronize non-passive agents in a network.

Enabled optimization of system performance via passivation techniques.

Abstract

Passivity theory is one of the cornerstones of control theory, as it allows one to prove stability of a large-scale system while treating each component separately. In practice, many systems are not passive, and must be passivized in order to be included in the framework of passivity theory. Input-output transformations are the most general tool for passivizing systems, generalizing output-feedback and input-feedthrough. In this paper, we classify all possible input-output transformations that map a system with given shortage of passivity to a system with prescribed excess of passivity. We do so by using the connection between passivity theory and cones for SISO systems, and using the S-lemma for MIMO systems. We also present several possible applications of our results, including simultaneous passivation of multiple systems or with respect to multiple equilibria, as well as optimization problems such as $\mathcal{L}_2$-gain minimization. We also exhibit our results in a case study about synchronization in a network of non-passive faulty agents.