On Linear Passive Complementarity Systems

We study the notion of passivity in the context of complementarity systems, which form a class of nonsmooth dynamical systems that is obtained from the coupling of a standard input/output system to complementarity conditions as used in mathematical programming. In terms of electrical circuits, the systems that we study may be viewed as passive networks with ideal diodes. Extending results from earlier work, we consider here complementarity systems with external inputs. It is shown that the assumption of passivity of the underlying input/output dynamical system plays an important role in establishing existence and uniqueness of solutions. We prove that solutions may contain delta functions but no higher-order impulses. Several characterizations are provided for the state jumps that may occur due to inconsistent initialization or to input discontinuities. Many of the results still hold when the assumption of passivity is replaced by the assumption of “passifiability by pole shifting”. The paper ends with some remarks on stability.

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

[2]  Vasile Mihai Popov,et al.  Hyperstability of Control Systems , 1973 .

[3]  L. Silverman,et al.  System structure and singular control , 1983 .

[4]  Arjan van der Schaft,et al.  Uniqueness of solutions of linear relay systems , 1999, Autom..

[5]  Johannes Schumacher,et al.  Rational complementarity problem , 1998 .

[6]  Johannes Schumacher,et al.  The nature of solutions to linear passive complementarity systems , 1999, Proceedings of the 38th IEEE Conference on Decision and Control (Cat. No.99CH36304).

[7]  C. A. Desoer,et al.  Nonlinear Systems Analysis , 1978 .

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

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

[10]  Johannes Schumacher,et al.  Well-posedness of a class of linear networks with ideal diodes , 2000 .

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

[12]  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).

[13]  Allen R. Stubberud,et al.  Schaum's outline of theory and problems of feedback and control systems , 1967 .

[14]  David G. Luenberger,et al.  Linear and nonlinear programming , 1984 .

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

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

[17]  Thomas A. Henzinger,et al.  Hybrid Systems III , 1995, Lecture Notes in Computer Science.

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

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

[20]  Van Bokhoven.  Piecewise-linear modelling and analysis , 1981 .

[21]  V. Borkar,et al.  A unified framework for hybrid control: model and optimal control theory , 1998, IEEE Trans. Autom. Control..

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

[23]  J. Ian Richards,et al.  Theory of Distributions: A general definition of multiplication and convolution for distributions , 1990 .

[24]  W. D. Cairns.  THE MATHEMATICAL ASSOCIATION OF AMERICA. , 1917, Science.

[25]  Johannes Schumacher,et al.  On event driven simulation of electrical circuits with ideal diodes , 2001 .

[26]  P. Maher,et al.  Handbook of Matrices , 1999, The Mathematical Gazette.

[27]  G. Dantzig,et al.  COMPLEMENTARY PIVOT THEORY OF MATHEMATICAL PROGRAMMING , 1968 .

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