Nonpathological ISS-Lyapunov Functions for Interconnected Differential Inclusions

This article concerns robustness analysis for interconnections of two dynamical systems (described by upper semicontinuous differential inclusions) using a generalized notion of derivatives associated with locally Lipschitz Lyapunov functions obtained from a finite family of differentiable functions. We first provide sufficient conditions for input-to-state stability (ISS) for differential inclusions, using a class of non-smooth (but locally Lipschitz) candidate Lyapunov functions and the concept of Lie generalized derivative. In general our conditions are less conservative than the more common Clarke derivative based conditions. We apply our result to state-dependent switched systems, and to the interconnection of two differential inclusions. As an example, we propose an observer-based controller for certain nonlinear two-mode state-dependent switched systems.

[1]  George Voutsadakis,et al.  Introduction to Set Theory , 2021, A Problem Based Journey from Elementary Number Theory to an Introduction to Matrix Theory.

[2]  C. Kellett.  Classical Converse Theorems in Lyapunov's Second Method , 2015, 1502.04809.

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

[4]  Matteo Della Rossa,et al.  Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems. (Fonctions de Lyapunov non-lisses pour l'analyse de stabilité de systèmes hybrides) , 2020 .

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

[6]  W. P. M. H. Heemels,et al.  Input-to-State Stability of Discontinuous Dynamical Systems with an Observer-Based Control Application , 2007, HSCC.

[7]  L. Zaccarian,et al.  Smooth Approximation of Patchy Lyapunov Functions for Switched Systems , 2019, IFAC-PapersOnLine.

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

[9]  Anup Parikh,et al.  Invariance-Like Results for Nonautonomous Switched Systems , 2016, IEEE Transactions on Automatic Control.

[10]  A. Bacciotti,et al.  Nonsmooth optimal regulation and discontinuous stabilization , 2003 .

[11]  Dragan Nesic,et al.  Lyapunov-Based Small-Gain Theorems for Hybrid Systems , 2014, IEEE Transactions on Automatic Control.

[12]  Luca Zaccarian,et al.  Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems , 2020, Mathematics of Control, Signals, and Systems.

[13]  Debasish Chatterjee,et al.  Input-to-state stability of switched systems and switching adaptive control , 2007, Autom..

[14]  A.R. Teel,et al.  Homogeneous hybrid systems and a converse Lyapunov theorem , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[15]  Eduardo Sontag,et al.  Changing supply functions in input/state stable systems , 1995, IEEE Trans. Autom. Control..

[16]  Warren E. Dixon,et al.  On reduction of differential inclusions and Lyapunov stability , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

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

[18]  Yu. S. Ledyaev,et al.  Asymptotic Stability and Smooth Lyapunov Functions , 1998 .

[19]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[20]  Lars Grüne,et al.  Linear programming based Lyapunov function computation for differential inclusions , 2011 .

[21]  Hiroshi Ito,et al.  An intuitive modification of max-separable Lyapunov functions to cover non-ISS systems , 2019, Autom..

[22]  Chaohong Cai,et al.  Robust Input-to-State Stability for Hybrid Systems , 2013, SIAM J. Control. Optim..

[23]  Bayu Jayawardhana,et al.  Input-to-State Stability of Differential Inclusions with Applications to Hysteretic and Quantized Feedback Systems , 2009, SIAM J. Control. Optim..

[24]  L. Zaccarian,et al.  Max-min Lyapunov functions for switched systems and related differential inclusions , 2020, Autom..

[25]  Aneel Tanwani,et al.  ISS Lyapunov Functions for Cascade Switched Systems and Sampled-Data Control , 2019, Autom..

[26]  Zhong-Ping Jiang,et al.  Robust Stability of Networks of iISS Systems: Construction of Sum-Type Lyapunov Functions , 2013, IEEE Transactions on Automatic Control.

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

[28]  Guosong Yang,et al.  A Lyapunov-based small-gain theorem for interconnected switched systems , 2015, Syst. Control. Lett..

[29]  Anders Rantzer,et al.  Computation of piecewise quadratic Lyapunov functions for hybrid systems , 1997, 1997 European Control Conference (ECC).

[30]  Andrew R. Teel,et al.  On using norm estimators for event-triggered control with dynamic output feedback , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[31]  W. P. M. H. Heemels,et al.  Input-to-state stability and interconnections of discontinuous dynamical systems , 2008, Autom..

[32]  Ricardo G. Sanfelice,et al.  Input-Output-to-State Stability Tools for Hybrid Systems and Their Interconnections , 2014, IEEE Transactions on Automatic Control.

[33]  J. Cortés.  Discontinuous dynamical systems , 2008, IEEE Control Systems.

[34]  Francesco Vasca,et al.  Asymptotic Stability of Piecewise Affine Systems With Filippov Solutions via Discontinuous Piecewise Lyapunov Functions , 2021, IEEE Transactions on Automatic Control.

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

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

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

[38]  A. Bacciotti,et al.  Stability and Stabilization of Discontinuous Systems and Nonsmooth Lyapunov Functions , 1999 .

[39]  Clyde F. Martin,et al.  A Converse Lyapunov Theorem for a Class of Dynamical Systems which Undergo Switching , 1999, IEEE Transactions on Automatic Control.

[40]  Aleksej F. Filippov,et al.  Differential Equations with Discontinuous Righthand Sides , 1988, Mathematics and Its Applications.

[41]  A. Teel,et al.  Global stabilizability and observability imply semi-global stabilizability by output feedback , 1994 .

[42]  Daniel Liberzon,et al.  Switching in Systems and Control , 2003, Systems & Control: Foundations & Applications.

[43]  K. Deimling.  Multivalued Differential Equations , 1992 .

[44]  S. Scholtes.  Introduction to Piecewise Differentiable Equations , 2012 .

[45]  L. Rosier.  Homogeneous Lyapunov function for homogeneous continuous vector field , 1992 .

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

[47]  Francesca Maria Ceragioli,et al.  Discontinuous ordinary differential equations and stabilization , 2000 .

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