Revisiting the <scp>IISS</scp> Small‐Gain Theorem through Transient Plus <scp>ISS</scp> Small‐Gain Regulation

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  • Hiroshi Ito
    Department of Systems Design and Informatics Kyushu Institute of Technology 680‐4 Kawazu Iizuka 820‐8502 Japan
  • Randy A. Freeman
    Department of Electrical Engineering and Computer Science Northwestern University Evanston IL 60208‐3118 USA
  • Antoine Chaillet
    L2S, EECI Univ. Paris Sud 11 — Supélec 3 rue Joliot Curie 91192 Gif‐sur‐Yvette France

説明

<jats:title>Abstract</jats:title><jats:p>Recently, the small‐gain theorem for input‐to‐state stable (<jats:styled-content style="fixed-case">ISS</jats:styled-content>) systems has been extended to the class of integral input‐to‐state stable (<jats:styled-content style="fixed-case">iISS</jats:styled-content>) systems. Feedback connections of two <jats:styled-content style="fixed-case">iISS</jats:styled-content> systems are robustly stable with respect to disturbance if an extended small‐gain condition is satisfied. It has been proved that at least one of the two iISS subsystems needs to be <jats:styled-content style="fixed-case">ISS</jats:styled-content> for guaranteeing globally asymptotic stability and <jats:styled-content style="fixed-case">iISS</jats:styled-content> of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the <jats:styled-content style="fixed-case">iISS</jats:styled-content> small gain theorem as transient plus <jats:styled-content style="fixed-case">ISS</jats:styled-content> small‐gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the <jats:styled-content style="fixed-case">iISS</jats:styled-content> small‐gain theorem.</jats:p>

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