A characterization of integral input-to-state stability for hybrid systems

This paper addresses characterizations of integral input-to-state stability (iISS) for hybrid systems. In particular, we give a Lyapunov characterization of iISS unifying and generalizing the existing theory for pure continuous-time and pure discrete-time systems. Moreover, iISS is related to dissipativity and detectability notions. Robustness of iISS to sufficiently small perturbations is also investigated. As an application of our results, we provide a maximum allowable sampling period guaranteeing iISS for sampled-data control systems with an emulated controller.

[1]  Luca Zaccarian,et al.  Stability properties of reset systems , 2008, Autom..

[2]  Eduardo D. Sontag,et al.  Mathematical Control Theory: Deterministic Finite Dimensional Systems , 1990 .

[3]  Dragan Nesic,et al.  Gronwall inequality for hybrid systems , 2014, Autom..

[4]  Chaohong Cai,et al.  Characterizations of input-to-state stability for hybrid systems , 2009, Syst. Control. Lett..

[5]  Rafal Goebel,et al.  Solutions to hybrid inclusions via set and graphical convergence with stability theory applications , 2006, Autom..

[6]  Dragan Nesic,et al.  Input-to-state stability for a class of hybrid dynamical systems via averaging , 2012, Math. Control. Signals Syst..

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

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

[9]  Chaohong Cai,et al.  Smooth Lyapunov Functions for Hybrid Systems—Part I: Existence Is Equivalent to Robustness , 2007, IEEE Transactions on Automatic Control.

[10]  Chaohong Cai,et al.  Smooth Lyapunov Functions for Hybrid Systems Part II: (Pre)Asymptotically Stable Compact Sets , 2008, IEEE Transactions on Automatic Control.

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

[12]  A. Teel,et al.  A smooth Lyapunov function from a class- ${\mathcal{KL}}$ estimate involving two positive semidefinite functions , 2000 .

[13]  Dragan Nesic,et al.  A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models , 2004, IEEE Transactions on Automatic Control.

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

[15]  Dragan Nesic,et al.  Explicit Computation of the Sampling Period in Emulation of Controllers for Nonlinear Sampled-Data Systems , 2009, IEEE Transactions on Automatic Control.

[16]  J. L. Mancilla-Aguilar,et al.  On converse Lyapunov theorems for ISS and iISS switched nonlinear systems , 2001 .

[17]  Dragan Nesic,et al.  A Trajectory-Based Approach for the Stability Robustness of Nonlinear Systems with Inputs , 2002, Math. Control. Signals Syst..

[18]  D. Nesic,et al.  A trajectory based approach for robustness of input-to-state stability , 2001, Proceedings of the 2001 American Control Conference. (Cat. No.01CH37148).

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

[20]  Ricardo G. Sanfelice,et al.  Invariance Principles for Hybrid Systems With Connections to Detectability and Asymptotic Stability , 2007, IEEE Transactions on Automatic Control.

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

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

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

[24]  Yuandan Lin,et al.  A Smooth Converse Lyapunov Theorem for Robust Stability , 1996 .

[25]  R. Sanfelice,et al.  Hybrid dynamical systems , 2009, IEEE Control Systems.

[26]  Ricardo G. Sanfelice,et al.  Hybrid Dynamical Systems: Modeling, Stability, and Robustness , 2012 .

[27]  Dragan Nesic,et al.  A Lyapunov Proof of an Improved Maximum Allowable Transfer Interval for Networked Control Systems , 2007, IEEE Transactions on Automatic Control.

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

[29]  A. Teel,et al.  A Smooth Lyapunov Function from a Class-kl Estimate Involving Two Positive Semideenite Functions , 1999 .

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

[31]  Eduardo D. Sontag,et al.  Mathematical Control Theory Second Edition , 1998 .

[32]  João Pedro Hespanha,et al.  Lyapunov conditions for input-to-state stability of impulsive systems , 2008, Autom..

[33]  Eduardo Sontag.  Further facts about input to state stabilization , 1990 .