# A Generalization of the Passivity Theorem and the Small Gain Theorem Based on $\rho $-Stability, with Application to a Parameter Adaptation Algorithm for Recursive Identification

## TL;DR

This paper extends classical passivity and small gain theorems using $
ho$-stability, allowing for broader stability conditions and applying these results to a parameter adaptation algorithm in recursive identification.

## Contribution

It introduces a generalized stability framework based on $
ho$-passivity, broadening the applicability of passivity and small gain theorems in control theory.

## Key findings

- The closed-loop remains stable under $
ho$-passivity conditions.
- The approach generalizes classical passivity and small gain theorems.
- Application demonstrated in a parameter adaptation algorithm for recursive identification.

## Abstract

The usual passivity theorem considers a closed-loop, the direct chain of which consists of a strictly passive stable operator $H_{1}$, and the feedback chain of which consists of a passive operator $H_{2}$. Then the closed-loop is stable. Let $\rho >1$. We show here that the closed-loop is still stable when the direct chain consists of a strictly $\rho ^{-1}$-passive $% \rho ^{-1}$-stable operator (a weaker condition than above) and the feedback chain consists of a $\rho $-passive operator (a stronger condition than above). Variations on the theme of the small gain theorem (incremental or not) can be made similarly. This approach explains the results obtained in a paper on identification which was recently published.