Complementarity modeling of hybrid systems

A complementarity framework is described for the modeling of certain classes of mixed continuous/discrete dynamical systems. The use of such a framework is well known for mechanical systems with inequality constraints, but we give a more general formulation which also applies, for instance, to switching control systems. The main theoretical results in the paper are concerned with uniqueness of smooth continuations; the solution of this problem requires the construction of a map from the continuous state to the discrete state. A crucial technical tool is the so-called linear complementarity problem from mathematical programming, and we introduce various generalizations of this problem.

[1]  Per Lötstedt.  Mechanical Systems of Rigid Bodies Subject to Unilateral Constraints , 1982 .

[2]  B. C. Eaves,et al.  Equivalence of LCP and PLS , 1981, Math. Oper. Res..

[3]  M. Seetharama Gowda,et al.  The extended linear complementarity problem , 1995, Math. Program..

[4]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[5]  L. Vandenberghe,et al.  The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits , 1989, IEEE International Symposium on Circuits and Systems,.

[6]  P. S. Bauer.  Dissipative Dynamical Systems: I. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[7]  R. Narasimhan,et al.  Analysis on Real and Complex Manifolds , 1973 .

[8]  Bart De Schutter,et al.  The linear dynamic complementarity problem is a special case of the extended linear complementarity problem , 1998 .

[9]  Arjan van der Schaft,et al.  Non-linear dynamical control systems , 1990 .

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

[11]  B. Brogliato.  Nonsmooth Impact Mechanics: Models, Dynamics and Control , 1996 .

[12]  L. Vandenberghe,et al.  The generalized linear complementarity problem applied to the complete analysis of resistive piecewise-linear circuits , 1989 .

[13]  Thomas A. Henzinger,et al.  The Algorithmic Analysis of Hybrid Systems , 1995, Theor. Comput. Sci..

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

[15]  Patrick T. Harker,et al.  Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications , 1990, Math. Program..

[16]  H. Samelson,et al.  A partition theorem for Euclidean $n$-space , 1958 .

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

[18]  M. Marques.  Differential Inclusions in Nonsmooth Mechanical Problems , 1993 .

[19]  M. Marques,et al.  Differential Inclusions in Nonsmooth Mechanical Problems: Shocks and Dry Friction , 1993 .

[20]  Joos Vandewalle,et al.  The generalized linear complementarity problem and an algorithm to find all its solutions , 1992, Math. Program..