REVISITING THE IISS SMALL-GAIN THEOREM THROUGH TRANSIENT PLUS ISS SMALL-GAIN REGULATION

Recently, the small-gain theorem for input-to-state stable (ISS) systems has been extended to the class of integral input-to-state stable (iISS) systems. Feedback connections of two iISS 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 ISS for guaranteeing globally asymptotic stability and iISS of the overall system. Making use of this necessary condition for the stability, this paper gives a new interpretation to the iISS small gain theorem as transient plus ISS small-gain regulation. The observation provides useful information for designing and analyzing nonlinear control systems based on the iISS small-gain theorem.

[1]  Hiroshi Ito,et al.  A Lyapunov Approach to Cascade Interconnection of Integral Input-to-State Stable Systems , 2010, IEEE Transactions on Automatic Control.

[2]  David Angeli,et al.  A Unifying Integral ISS Framework for Stability of Nonlinear Cascades , 2001, SIAM J. Control. Optim..

[3]  Zhong-Ping Jiang,et al.  Small-gain theorem for ISS systems and applications , 1994, Math. Control. Signals Syst..

[4]  Hiroshi Ito,et al.  State-Dependent Scaling Problems and Stability of Interconnected iISS and ISS Systems , 2006, IEEE Transactions on Automatic Control.

[5]  Alessandro Astolfi,et al.  A tight small gain theorem for not necessarily ISS systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[6]  Zhong-Ping Jiang,et al.  A new small‐gain theorem with an application to the stabilization of the chemostat , 2012 .

[7]  Eduardo Sontag.  Smooth stabilization implies coprime factorization , 1989, IEEE Transactions on Automatic Control.

[8]  Zhong-Ping Jiang,et al.  A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems , 1996, Autom..

[9]  Eduardo Sontag.  Comments on integral variants of ISS , 1998 .

[10]  A. Teel.  A nonlinear small gain theorem for the analysis of control systems with saturation , 1996, IEEE Trans. Autom. Control..

[11]  David Angeli,et al.  A characterization of integral input-to-state stability , 2000, IEEE Trans. Autom. Control..

[12]  Randy A. Freeman,et al.  Interpreting the iISS small-gain theorem as transient plus ISS small-gain regulation , 2010, 49th IEEE Conference on Decision and Control (CDC).

[13]  Zhong-Ping Jiang,et al.  Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective , 2009, IEEE Transactions on Automatic Control.

[14]  Hiroshi Ito,et al.  Necessary conditions for stability of networks of iISS systems , 2010, 49th IEEE Conference on Decision and Control (CDC).

[15]  Eduardo Sontag,et al.  On characterizations of the input-to-state stability property , 1995 .

[16]  David Angeli,et al.  Separation Principles for Input-Output and Integral-Input-to-State Stability , 2004, SIAM J. Control. Optim..

[17]  David Angeli,et al.  Integral Input to State Stable systems in cascade , 2008, Syst. Control. Lett..