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

Authors

Y. Le Gorrec, H. Zwart, B. Maschke

Abstract

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.

Citation

• Journal: SIAM Journal on Control and Optimization
• Year: 2005
• Volume: 44
• Issue: 5
• Pages: 1864–1892
• Publisher: Society for Industrial & Applied Mathematics (SIAM)
• DOI: 10.1137/040611677

BibTeX

@article{Le_Gorrec_2005,
  title={{Dirac structures and Boundary Control Systems associated with Skew-Symmetric Differential Operators}},
  volume={44},
  ISSN={1095-7138},
  DOI={10.1137/040611677},
  number={5},
  journal={SIAM Journal on Control and Optimization},
  publisher={Society for Industrial & Applied Mathematics (SIAM)},
  author={Le Gorrec, Y. and Zwart, H. and Maschke, B.},
  year={2005},
  pages={1864--1892}
}

Download the bib file

References

• P. C. Breedveld, Physical Systems Theory in Terms of Bond Graphs , Ph.D. thesis, Technische Hogeschool Twente, Enschede, The Netherlands, 1984.
• Courant, T. J. Dirac manifolds. Transactions of the American Mathematical Society vol. 319 631–661 (1990) – 10.1090/s0002-9947-1990-0998124-1
• Curtain, R. F. & Zwart, H. An Introduction to Infinite-Dimensional Linear Systems Theory. Texts in Applied Mathematics (Springer New York, 1995). doi:10.1007/978-1-4612-4224-6 – 10.1007/978-1-4612-4224-6
• Dalsmo, M. & van der Schaft, A. On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems. SIAM Journal on Control and Optimization vol. 37 54–91 (1998) – 10.1137/s0363012996312039
• Dorfman Irene, Dirac structures and integrability of nonlinear evolution equations (1993)
• G. Golo, V. Talasila, and A. J. van der Schaft, A Hamiltonian formulation of the Timoshenko beam , in Proceedings of the Mechatronics, Drebbel Institute for Mechatronics, University of Twente, Enschede, The Netherlands, 2002, pp. 544–553. Full paper on accompanying CD‐ROM.
• G. Golo, Interconnection Structures in Port‐Based Modelling: Tools for Analysis and Simulation , Ph.D. thesis, University of Twente, Enschede, The Netherlands, 2002.
• G. Golo, O. V. Iftime, H. Zwart, and A. J. van der Schaft, Tools for Analysis of Dirac Structures on Hilbert Spaces , Memorandum Faculteit TW 1729, Universiteit Twente, Enschede, 2004, http://www.math.utwente.nl/ssb/annrep04/publs04.htm.
• V. I. Gorbachuk and M. L. Gorbachuk, Boundary Value Problems for Operator Differential Equations , Math. Appl. (Soviet Ser.) 48, Kluwer Academic Publishers, Norwell, MA, 1991.
• (Expanded and revised translation of Granichnye zadachi dia differentsial’no‐operatornykh uravneii.)
• Y. Le Gorrec, H. Zwart, and B. M. Maschke, Dirac Structures and Boundary Control Systems Associated with Skew‐Symmetric Differential Operators , Memorandum Faculteit TW 1730, Universiteit Twente, Enschede, 2004, http://www.math.utwente.nl/ssb/ annrep04/publs04.htm.
• D. C. Karnopp, D. L. Margolis, and R. C. Rosenberg, System Dynamics: A Unified Approach , Wiley, New York, 1990.
• B. M. Maschke, Interconnection and structure in physical systems’ dynamics , in Proceedings of the 5th NOLCOS, NOLCOS’98, Enschede, 1998, pp. 291–296.
• Maschke, B., Ortega, R. & Van Der Schaft, A. J. Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Transactions on Automatic Control vol. 45 1498–1502 (2000) – 10.1109/9.871758
• B. M. Maschke and A. J. van der Schaft, Port controlled Hamiltonian systems: Modeling origins and system theoretic properties , in Proceedings of the 3rd NOLCOS, NOLCOS’92, Bordeaux, 1992, pp. 282–288.
• B. M. Maschke and A. J. van der Schaft, Interconnected mechanical systems, Part 2: The dynamics of spatial mechanical networks , in Modelling and Control of Mechanical Systems, Imperial College Press, London, 1997, pp. 17–30.
• B. M. Maschke and A. J. van der Schaft, Port controlled Hamiltonian representation of distributed parameter systems , in Workshop on Modeling and Control of Lagrangian and Hamiltonian Systems, Princeton, NJ, 2000.
• BernhardMaschke, Arjanvan der Schaft, Compositional modelling of distributed‐parameter systems, Lecture Notes in Control and Inform. Sci., Vol. 311, Springer, London, 2005, 115–154MR2130105
• Maschke, B. M., Van Der Schaft, A. J. & Breedveld, P. C. An intrinsic hamiltonian formulation of network dynamics: non-standard poisson structures and gyrators. Journal of the Franklin Institute vol. 329 923–966 (1992) – 10.1016/s0016-0032(92)90049-m
• B. M. Maschke and A. J. van der Schaft, Canonical interdomain coupling in distributed parameter systems: An extension of the symplectic gyrator , in Proceedings of the International Mechanical Engineering Congress and Exposition, New York, 2001.
• Olver, P. J. Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics (Springer New York, 1993). doi:10.1007/978-1-4612-4350-2 – 10.1007/978-1-4612-4350-2
• Putting energy back in control. IEEE Control Systems vol. 21 18–33 (2001) – 10.1109/37.915398
• Ortega, R., van der Schaft, A., Maschke, B. & Escobar, G. Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica vol. 38 585–596 (2002) – 10.1016/s0005-1098(01)00278-3
• Parsian, A. & Deh Abad, A. S. Dirac structures on Hilbert spaces. International Journal of Mathematics and Mathematical Sciences vol. 22 97–108 (1999) – 10.1155/s0161171299220972
• Phillips, R. S. Dissipative operators and hyperbolic systems of partial differential equations. Transactions of the American Mathematical Society vol. 90 193–254 (1959) – 10.1090/s0002-9947-1959-0104919-1
• Pillai, H. K. & Willems, J. C. Lossless and Dissipative Distributed Systems. SIAM Journal on Control and Optimization vol. 40 1406–1430 (2002) – 10.1137/s0363012900368028
• Staffans, O. Well-Posed Linear Systems. (2005) doi:10.1017/cbo9780511543197 – 10.1017/cbo9780511543197
• A. J. van der Schaft, L2‐Gain and Passivity Techniques in Nonlinear Control , 2nd revised and enlarged ed., Comm. Control Engrg. Ser., Springer‐Verlag, London, 1999;
• 1st ed., Lecture Notes in Control and Inform. Sci. 218, Springer‐Verlag, Berlin, 1996.
• van der Schaft A. J., Archiv. Elektronik Übertragungstechnik (1995)
• A. J. van der Schaft and B. M. Maschke, Interconnected mechanical systems, Part 1: Geometry of interconnection and implicit Hamiltonian systems , in Modelling and Control of Mechanical Systems, Imperial College Press, London, 1997, pp. 1–16.
• van der Schaft, A. J. & Maschke, B. M. Hamiltonian formulation of distributed-parameter systems with boundary energy flow. Journal of Geometry and Physics vol. 42 166–194 (2002) – 10.1016/s0393-0440(01)00083-3
• J. A. Villegas, Y. Le Gorrec, H. Zwart, and A. J. van der Schaft, Boundary control systems and the system nodes , in 16th IFAC World Congress, Praha, 2005.
• H. Zwart, Y. Le Gorrec, B. M. Maschke, and J. A. Villegas, Well‐posedness of a class of boundary control systems , in 44th IEEE CDC‐ECC, Seville, Spain, submitted.