Passivity and complementarity

This paper studies the interaction between the notions of passivity of systems theory and complementarity of mathematical programming in the context of complementarity systems. These systems consist of a dynamical system (given in the form of state space representation) and complementarity relations. We study existence, uniqueness, and nature of solutions for this system class under a passivity assumption on the dynamical part. A complete characterization of the initial states and the inputs for which a solution exists is given. These initial states are called consistent states. For the inconsistent states, we introduce a solution concept in the framework of distributions.

[1]  Jong-Shi Pang,et al.  Differential variational inequalities , 2008, Math. Program..

[2]  Arjan van der Schaft,et al.  The complementary-slackness class of hybrid systems , 1996, Math. Control. Signals Syst..

[3]  M. Çamlibel.  Complementarity Methods in the Analysis of Piecewise Linear Dynamical Systems , 2001 .

[4]  .. P. M. H. Heemels,et al.  The Rational Complementarity ProblemW , 1998 .

[5]  J. Moreau,et al.  Nonsmooth Mechanics and Applications , 1989 .

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

[7]  M. Kanat Camlibel,et al.  On Linear Passive Complementarity Systems , 2002, Eur. J. Control.

[8]  S. Sastry,et al.  Zeno hybrid systems , 2001 .

[9]  Richard W. Cottle,et al.  Linear Complementarity Problem. , 1992 .

[10]  W. Heemels.  Linear complementarity systems : a study in hybrid dynamics , 1999 .

[11]  A. Schaft,et al.  Switched networks and complementarity , 2003 .

[12]  J. M. Schumacher,et al.  Complementarity systems in optimization , 2004, Math. Program..

[13]  Vincent Acary,et al.  On the equivalence between complementarity systems, projected systems and differential inclusions , 2006, Syst. Control. Lett..

[14]  M. Kanat Camlibel,et al.  Algebraic Necessary and Sufficient Conditions for the Controllability of Conewise Linear Systems , 2008, IEEE Transactions on Automatic Control.

[15]  J. Moreau.  Liaisons unilatérales sans frottement et chocs inélastiques , 1983 .

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

[17]  Harry L. Trentelman,et al.  Control theory for linear systems , 2002 .

[18]  M. Kanat Camlibel,et al.  Lyapunov Stability of Complementarity and Extended Systems , 2006, SIAM J. Optim..

[19]  L. Schwartz.  Théorie des distributions , 1966 .

[20]  Jong-Shi Pang,et al.  Linear Complementarity Systems: Zeno States , 2005, SIAM J. Control. Optim..

[21]  M. Kanat Camlibel,et al.  Conewise Linear Systems: Non-Zenoness and Observability , 2006, SIAM J. Control. Optim..

[22]  B. Brogliato,et al.  Well-posedness, stability and invariance results for a class of multivalued Lur'e dynamical systems , 2011 .

[23]  W. Heemels,et al.  On the dynamic analysis of piecewise-linear networks , 2002 .

[24]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[25]  J. Moreau,et al.  Unilateral Contact and Dry Friction in Finite Freedom Dynamics , 1988 .

[26]  Bernard Brogliato,et al.  Absolute stability and the Lagrange-Dirichlet theorem with monotone multivalued mappings , 2001, Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228).

[27]  Bernard Brogliato,et al.  Some results on the controllability of planar variational inequalities , 2005, Syst. Control. Lett..

[28]  Arjan van der Schaft,et al.  An Introduction to Hybrid Dynamical Systems, Springer Lecture Notes in Control and Information Sciences 251 , 1999 .

[29]  Jong-Shi Pang,et al.  Strongly Regular Differential Variational Systems , 2007, IEEE Transactions on Automatic Control.

[30]  Jan C. Willems,et al.  Dissipative Dynamical Systems , 2007, Eur. J. Control.

[31]  Arjan van der Schaft,et al.  Complementarity modelling of hybrid systems , 1997 .

[32]  W. P. M. H. Heemels,et al.  The Complementarity Class of Hybrid Dynamical Systems , 2003, Eur. J. Control.

[33]  Jong-Shi Pang,et al.  Solution dependence on initial conditions in differential variational inequalities , 2008, Math. Program..

[34]  M. Kanat Camlibel,et al.  Popov-Belevitch-Hautus type controllability tests for linear complementarity systems , 2007, Syst. Control. Lett..

[35]  B. Brogliato,et al.  Existence and Uniqueness of Solutions for Non-Autonomous Complementarity Dynamical Systems , 2010 .

[36]  W. P. M. H. Heemels,et al.  Linear Complementarity Systems , 2000, SIAM J. Appl. Math..

[37]  A. J. van der Schaft,et al.  Complementarity modeling of hybrid systems , 1998, IEEE Trans. Autom. Control..

[38]  Paul A. McCollum,et al.  Laplace transform tables and theorems , 1965 .

[39]  Bernard Brogliato,et al.  Some perspectives on the analysis and control of complementarity systems , 2003, IEEE Trans. Autom. Control..

[40]  J. Moreau.  Quadratic Programming in Mechanics: Dynamics of One-Sided Constraints , 1966 .

[41]  J. Stoer,et al.  Convexity and Optimization in Finite Dimensions I , 1970 .

[42]  M. Kanat Camlibel,et al.  A full characterization of stabilizability of bimodal piecewise linear systems with scalar inputs , 2008, Autom..

[43]  M. Çamlibel,et al.  Popov–Belevitch–Hautus type tests for the controllability of linear complementarity systems , 2007 .

[44]  W. Rudin.  Principles of mathematical analysis , 1964 .

[45]  Thomas A. Henzinger,et al.  Hybrid Systems: Computation and Control , 1998, Lecture Notes in Computer Science.

[46]  M. Kanat Camlibel,et al.  On the Controllability of Bimodal Piecewise Linear Systems , 2004, HSCC.

[47]  Arjan van der Schaft,et al.  Modelling, Well-Posedness, Stability of Switched Electrical Networks , 2003, HSCC.

[48]  Daniel Goeleven,et al.  Stability and instability matrices for linear evolution variational inequalities , 2004, IEEE Transactions on Automatic Control.