Dissipativity theory for discontinuous dynamical systems: Basic input, state, and output properties, and finite-time stability of feedback interconnections

In this paper we develop dissipativity theory for discontinuous dynamical systems. Specifically, using set-valued supply rate maps and set-valued connective supply rate maps consisting of locally Lebesgue integrable supply rates and connective supply rates, respectively, and set-valued storage maps consisting of piecewise continuous storage functions, dissipativity properties for discontinuous dynamical systems are presented. Furthermore, extended Kalman-Yakubovich-Popov set-valued conditions, in terms of the discontinuous system dynamics, characterizing dissipativity via generalized Clarke gradients and locally Lipschitz continuous storage functions are derived. Finally, these results are used to develop feedback interconnection stability results for discontinuous dynamical systems by appropriately combining the set-valued storage maps for the forward and feedback systems.

[1]  Dennis S. Bernstein,et al.  Finite-Time Stability of Continuous Autonomous Systems , 2000, SIAM J. Control. Optim..

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

[3]  David J. Hill,et al.  Stability results for nonlinear feedback systems , 1977, Autom..

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

[5]  Christopher Edwards,et al.  Sliding mode control : theory and applications , 1998 .

[6]  Vadim I. Utkin,et al.  Sliding Modes in Control and Optimization , 1992, Communications and Control Engineering Series.

[7]  M. Branicky.  Multiple Lyapunov functions and other analysis tools for switched and hybrid systems , 1998, IEEE Trans. Autom. Control..

[8]  Carla A. Schwartz,et al.  A minimum energy approach to switching control for mechanical systems , 1997 .

[9]  W. Haddad,et al.  Nonlinear Dynamical Systems and Control: A Lyapunov-Based Approach , 2008 .

[10]  P. Moylan,et al.  Dissipative Dynamical Systems: Basic Input-Output and State Properties , 1980 .

[11]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[12]  J. Willems.  Dissipative dynamical systems part I: General theory , 1972 .

[13]  Bernhard Maschke,et al.  Dissipative Systems Analysis and Control , 2000 .

[14]  P. Moylan,et al.  The stability of nonlinear dissipative systems , 1976 .

[15]  Shouchuan Hu.  Differential equations with discontinuous right-hand sides☆ , 1991 .

[16]  Vijay Kumar,et al.  Decentralized Algorithms for Multi-Robot Manipulation via Caging , 2004, Int. J. Robotics Res..

[17]  João Pedro Hespanha,et al.  Uniform stability of switched linear systems: extensions of LaSalle's Invariance Principle , 2004, IEEE Transactions on Automatic Control.

[18]  Francesco Bullo,et al.  Coordination and Geometric Optimization via Distributed Dynamical Systems , 2003, SIAM J. Control. Optim..

[19]  Wassim M. Haddad,et al.  Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control , 2006 .

[20]  Jun Zhao,et al.  Dissipativity Theory for Switched Systems , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[21]  J. Willems.  Dissipative dynamical systems Part II: Linear systems with quadratic supply rates , 1972 .

[22]  W.M. Haddad,et al.  Semistability theory for differential inclusions with applications to consensus problems in dynamical networks with switching topology , 2008, 2008 American Control Conference.

[23]  D. Hill,et al.  A notion of passivity for switched systems with state-dependent switching , 2006 .

[24]  Mario di Bernardo,et al.  Piecewise smooth dynamical systems , 2008, Scholarpedia.

[25]  M. Zefran,et al.  A notion of passivity for hybrid systems , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

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

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

[28]  Friedrich Pfeiffer,et al.  Multibody Dynamics with Unilateral Contacts , 1996 .