Relationships Between Subclasses of Integral Input-to-State Stability

A certain “qualitative equivalence” has been recently demonstrated between integral input-to-state stability (iISS) and a nonlinear <inline-formula><tex-math notation="LaTeX">$\mathcal{L}_2$</tex-math></inline-formula> -gain property. Furthermore, it has been observed that the properties of strong iISS and nonlinear <inline-formula> <tex-math notation="LaTeX">$\mathcal{L}_2$</tex-math></inline-formula>-gain are preserved when multiple systems satisfying the respective property are connected in cascade. This technical note clarifies the relationships between various input-to-state stability and (non)linear <inline-formula><tex-math notation="LaTeX">$\mathcal{L}_p$</tex-math> </inline-formula>-gain properties.

[1]  Peter M. Dower,et al.  Nonlinear L2-gain verification for bilinear systems , 2014, 2014 4th Australian Control Conference (AUCC).

[2]  Fabian R. Wirth,et al.  Asymptotic stability equals exponential stability, and ISS equals finite energy gain---if you twist your eyes , 1998, math/9812137.

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

[4]  Peter M. Dower,et al.  A dynamic programming approach to the approximation of nonlinear L2-gain , 2008, 2008 47th IEEE Conference on Decision and Control.

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

[6]  Peter M. Dower,et al.  Input-to-State Stability, Integral Input-to-State Stability, and ${\cal L}_{2} $-Gain Properties: Qualitative Equivalences and Interconnected Systems , 2016, IEEE Transactions on Automatic Control.

[7]  Eduardo Sontag,et al.  Forward Completeness, Unboundedness Observability, and their Lyapunov Characterizations , 1999 .

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

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

[10]  Hiroshi Ito,et al.  Combining iISS and ISS With Respect to Small Inputs: The Strong iISS Property , 2014, IEEE Transactions on Automatic Control.

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

[12]  Eduardo Sontag.  Input to State Stability: Basic Concepts and Results , 2008 .

[13]  Björn Rüffer,et al.  Connection between cooperative positive systems and integral input-to-state stability of large-scale systems , 2010, Autom..

[14]  Eduardo Sontag,et al.  Further Equivalences and Semiglobal Versions of Integral Input to State Stability , 1999, math/9908066.

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

[16]  David J. Hill.  A generalization of the small-gain theorem for nonlinear feedback systems , 1991, Autom..

[17]  Huan Zhang,et al.  A weak L2-gain property for nonlinear systems , 2012, 2012 IEEE 51st IEEE Conference on Decision and Control (CDC).

[18]  Christopher M. Kellett,et al.  A compendium of comparison function results , 2014, Math. Control. Signals Syst..

[19]  Fabian R. Wirth,et al.  Nonlinear Scaling of (i)ISS-Lyapunov Functions , 2016, IEEE Transactions on Automatic Control.

[20]  Hiroshi Ito,et al.  Strong iISS is preserved under cascade interconnection , 2014, Autom..

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