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

Henri Bourlès

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arXiv · 1902.03015

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

Henri Bourlès

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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.

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BibTeX RIS

Henri Bourlès. 2019-02-08. 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. https://arxiv.org/abs/1902.03015

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