# A Geometric Method for Passivation and Cooperative Control of Equilibrium-Independent Passivity-Short Systems

## TL;DR

This paper introduces a geometric method to transform equilibrium-independent passivity-short systems into passive systems using a projective quadratic inequality approach, enabling better control and analysis of such systems.

## Contribution

The paper develops a novel geometric approach based on PQIs to find passivizing transformations for EIPS systems, including network applications.

## Key findings

- Explicit solution set for PQIs related to EIPS systems.
- Constructed input-output mappings that passivize EIPS systems.
- Demonstrated the method through numerous illustrative examples.

## Abstract

Equilibrium-independent passive-short (EIPS) systems are a class of systems that satisfy a passivity-like dissipation inequality with respect to any forced equilibria with non-positive passivity indices. This paper presents a geometric approach for finding a passivizing transformation for such systems, relying on their steady-state input-output relation and the notion of projective quadratic inequalities (PQIs). We show that PQIs arise naturally from passivity-shortage characteristics of an EIPS system, and the set of their solutions can be explicitly expressed. We leverage this connection to build an input-output mapping that transforms the steady-state input-output relation to a monotone relation, and show that the same mapping passivizes the EIPS system. We show that the proposed transformation can be implemented through a combination of feedback, feed-through, post- and pre-multiplication gains. Furthermore, we consider an application of the presented passivation scheme for the analysis of networks comprised of EIPS systems. Numerous examples are provided to illustrate the theoretical findings.