Qualitative equivalences of ISS and lp-gain stability properties for discrete-time nonlinear systems

Input-to-state stability (ISS) and inputoutput L2-gain are two important robust stability concepts in analyzing the interconnections of nonlinear dynamical systems. In this paper, we demonstrate several qualitative equivalences between different ISS related properties and lp-gain properties for discrete-time systems via nonlinear changes of coordinates. Prior to that, certain summation-to-summation estimates are shown to be equivalent to the standard definitions of ISS and integral ISS (iISS) which lead to sufficient Lyapunov function conditions to verify lp-gain properties. Combined with known results on the equivalence of 0-input global asymptotic stability and iISS, and a superposition principle, we subsequently outline interesting implicative relationships between various discrete-time robust stability properties.

[1]  M. Vidyasagar,et al.  Nonlinear systems analysis (2nd ed.) , 1993 .

[2]  Huan Zhang,et al.  Nonlinear L2-gain verification for nonlinear systems , 2012, Syst. Control. Lett..

[3]  Miroslav Krstic,et al.  Nonlinear and adaptive control de-sign , 1995 .

[4]  Peter M. Dower,et al.  Nonlinear systems with nonlinear ℒ2-gain , 2014, 53rd IEEE Conference on Decision and Control.

[5]  Mrdjan J. Jankovic,et al.  Constructive Nonlinear Control , 2011 .

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

[7]  M. James,et al.  Extending H-infinity Control to Nonlinear Systems: Control of Nonlinear Systems to Achieve Performance Objectives , 1987 .

[8]  Eduardo D. Sontag,et al.  Input to state stability and allied system properties , 2011 .

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

[10]  G. Zames.  On the input-output stability of time-varying nonlinear feedback systems Part one: Conditions derived using concepts of loop gain, conicity, and positivity , 1966 .

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

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

[13]  Andrew R. Teel,et al.  On the Robustness of KL-stability for Difference Inclusions: Smooth Discrete-Time Lyapunov Functions , 2005, SIAM J. Control. Optim..

[14]  Peter M. Dower,et al.  Equivalences of Stability Properties for Discrete-Time Nonlinear Systems , 2015 .

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

[16]  D. Angeli.  Intrinsic robustness of global asymptotic stability , 1999 .

[17]  Dragan Nesic,et al.  Summability characterizations of uniform exponential and asymptotic stability of sets for difference inclusions , 2010 .

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

[19]  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.

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

[21]  C. Desoer,et al.  Feedback Systems: Input-Output Properties , 1975 .

[22]  Lars Grüne,et al.  ISS-Lyapunov Functions for Discontinuous Discrete-Time Systems , 2014, IEEE Transactions on Automatic Control.

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

[24]  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.

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

[26]  Zhong-Ping Jiang,et al.  Input-to-state stability for discrete-time nonlinear systems , 1999 .

[27]  Antonio Loría,et al.  Integral Characterizations of Uniform Asymptotic and Exponential Stability with Applications , 2002, Math. Control. Signals Syst..

[28]  A. Schaft.  L2-Gain and Passivity Techniques in Nonlinear Control. Lecture Notes in Control and Information Sciences 218 , 1996 .

[29]  M. Vidyasagar,et al.  New relationships between input-output and Lyapunov stability , 1982 .