Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators

Associated with a skew-symmetric linear operator on the spatial domain $[a,b]$ we define a Dirac structure which includes the port variables on the boundary of this spatial domain. This Dirac structure is a subspace of a Hilbert space. Naturally, associated with this Dirac structure is an infinite-dimensional system. We parameterize the boundary port variables for which the \( C_{0} \)-semigroup associated with this system is contractive or unitary. Furthermore, this parameterization is used to split the boundary port variables into inputs and outputs. Similarly, we define a linear port controlled Hamiltonian system associated with the previously defined Dirac structure and a symmetric positive operator defining the energy of the system. We illustrate this theory on the example of the Timoshenko beam.

[1]  Irene Dorfman,et al.  Dirac Structures and Integrability of Nonlinear Evolution Equations , 1993 .

[2]  O. Staffans Well-Posed Linear Systems , 2005 .

[3]  A. Jamiołkowski Book reviewApplications of Lie groups to differential equations : Peter J. Olver (School of Mathematics, University of Minnesota, Minneapolis, U.S.A): Graduate Texts in Mathematics, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1986, XXVI+497pp. , 1989 .

[4]  A. Schaft,et al.  The Hamiltonian formulation of energy conserving physical systems with external ports , 1995 .

[5]  A. Schaft,et al.  On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems , 1999 .

[6]  Marius Tucsnak,et al.  Well-posed linear systems a survey with emphasis on conservative systems , 2001 .

[7]  Goran Golo,et al.  Interconnection structures in port-based modelling: tools for analysis and simulation , 2002 .

[8]  R. Phillips,et al.  DISSIPATIVE OPERATORS AND HYPERBOLIC SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS , 1959 .

[9]  V. Gorbachuk,et al.  Boundary Value Problems for Operator Differential Equations , 1990 .

[10]  Bernhard Maschke,et al.  Interconnection and Structure in Physical Systems' Dynamics , 1998 .

[11]  Arjan van der Schaft,et al.  Compositional modelling of distributed-parameter systems , 2005 .

[12]  Alan S. Perelson,et al.  System Dynamics: A Unified Approach , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[13]  Harish K. Pillai,et al.  Lossless and Dissipative Distributed Systems , 2001, SIAM J. Control. Optim..

[14]  Arjan van der Schaft,et al.  Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems , 2002, Autom..

[15]  A. V. der,et al.  An Intrinsic Hamiltonian Formulation of Network Dynamics : Non-standard Poisson Structures and Gyrators , 2001 .

[16]  Bernhard Maschke,et al.  Modelling and Control of Mechanical Systems , 1997 .

[17]  A. J. V. D. Schafta,et al.  Hamiltonian formulation of distributed-parameter systems with boundary energy flow , 2002 .

[18]  A. Schaft,et al.  Port controlled Hamiltonian representation of distributed parameter systems , 2000 .

[19]  Hans Zwart,et al.  Boundary Control Systems and the System Node , 2005 .

[20]  A. Schaft,et al.  Port-controlled Hamiltonian systems : modelling origins and systemtheoretic properties , 1992 .

[21]  A. Parsian,et al.  Dirac structures on Hilbert spaces , 1999 .

[22]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[23]  Arjan van der Schaft,et al.  Interconnected mechanical systems , 1997 .

[24]  Romeo Ortega,et al.  Putting energy back in control , 2001 .

[25]  P. V. Remoortere Physical systems theory in terms of bond graphs : P.C. Breedveld: Vakgroep Besturingsystemen en Computertechniek, THT, Afdeling Electrotechniek, Postbus 217, 7500 AE Enschede, The Netherlands. 1984, 200 pages, ISBN 90-9000599-4 , 1984 .

[26]  Arjan van der Schaft,et al.  Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation , 1998, Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171).

[27]  A. Schaft,et al.  A Hamiltonian formulation of the Timoshenko beam , 2002 .

[28]  A. J. van der Schaft,et al.  Port-controlled Hamiltonian Systems:Modelling Origins and System-Theoretic Properties , 1992 .

[29]  Arjan van der Schaft,et al.  Interconnected mechanical systems, part I: geometry of interconnection and implicit Hamiltonian systems , 1997 .

[30]  G. Golo,et al.  Tools for analysis of Dirac structures on Hilbert spaces , 2004 .